Science, Irving K. Barber Faculty of (Okanagan)
Chemistry, Department of (Okanagan)
DSpace
UBCO
Sadeghifar, Alireza
Doctor of Philosophy - PhD
University of British Columbia
The process of H₂ adsorption on ionic surfaces has long been of importance technologically as well as of interest to “pure” science because they constitute simple ideal systems that may be subject to quantum effects. Experiments have shown that these systems can form a sequence of structures that undergo phase transitions. In particular, there is evidence that H₂/MgO(100) exhibits the following sequence of structural phase transitions: c(2×2) → c(4×2) → c(6×2). The details of these structures and their stabilities are not yet understood. This thesis presents an investigation of the H₂/NaCl(100) and H₂/MgO(100) systems where the detailed structures of these films and their stabilities are determined through the use of Monte Carlo simulations in the canonical ensemble. Key to this study is a hierarchy of newly developed H₂-H₂ and H₂–surface interaction potentials that depend on the quantum mechanical rotational state of the molecule(s) and are presented here for the first time. Using these potentials, it was found that the structure of the H₂ films depend on whether the molecules were in the para (J=0) or ortho (J=1) rotational state, or alternatively consisted of a mixture of molecules in these different rotational states. In addition, the external vibrational frequencies of the H₂ molecules with respect to the surface were calculated. Overall, the structure, stability, and external vibrational frequencies of the H₂/NaCl(100) and H₂/MgO(100) systems were found to match or be consistent with experimental results. They also provided details, e.g. molecular locations, that are not accessible experimentally. Consequently, a more complete and coherent picture of the H₂/NaCl(100) and H₂/MgO(100) systems has emerged.
Structure and Stability of Quantum H2 Films on Ionic Surfaces: A Theoretical Study by Alireza Sadeghifar B.Sc., University of Mazandaran, Iran, 2003 M.Sc., Amirkabir University of Technology (Tehran Polytechnic), Iran, 2010 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE COLLEGE OF GRADUATE STUDIES (Chemistry) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) August 2020 © Alireza Sadeghifar, 2020 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Structure and Stability of Quantum H2 Films on Ionic Surfaces: A Theoretical Study submitted by Alireza Sadeghifar in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemistry Examining Committee: Dr. David B. Jack, Department of Chemistry, Irving K. Barber Faculty of Science, UBC Supervisor Dr. Gino A. DiLabio, Department of Chemistry, Irving K. Barber Faculty of Science, UBC Supervisory Committee Member Dr. Isaac T. S. Li, Department of Chemistry, Irving K. Barber Faculty of Science, UBC Supervisory Committee Member Dr. Lukas Bichler, School of Engineering, UBC University Examiner Dr. Jan Peter Toennies, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany External Examiner iii Abstract The process of H2 adsorption on ionic surfaces has long been of importance technologically as well as of interest to “pure” science because they constitute simple ideal systems that may be subject to quantum effects. Experiments have shown that these systems can form a sequence of structures that undergo phase transitions. In particular, there is evidence that H2/MgO(100) exhibits the following sequence of structural phase transitions: c(2×2) → c(4×2) → c(6×2). The details of these structures and their stabilities are not yet understood. This thesis presents an investigation of the H2/NaCl(100) and H2/MgO(100) systems where the detailed structures of these films and their stabilities are determined through the use of Monte Carlo simulations in the canonical ensemble. Key to this study is a hierarchy of newly developed H2-H2 and H2–surface interaction potentials that depend on the quantum mechanical rotational state of the molecule(s) and are presented here for the first time. Using these potentials, it was found that the structure of the H2 films depend on whether the molecules were in the para (𝐽 = 0) or ortho (𝐽 = 1) rotational state, or alternatively consisted of a mixture of molecules in these different rotational states. In addition, the external vibrational frequencies of the H2 molecules with respect to the surface were calculated. Overall, the structure, stability, and external vibrational frequencies of the H2/NaCl(100) and H2/MgO(100) systems were found to match or be consistent with experimental results. They also provided details, e.g. molecular locations, that are not accessible experimentally. Consequently, a more complete and coherent picture of the H2/NaCl(100) and H2/MgO(100) systems has emerged. iv Lay Summary The formation of H2 films on ionic surfaces at low temperatures and pressures attracted considerable interest when it was found that the properties of H2 films depend on the rotational state (para-H2 or ortho-H2) of the H2 molecules. These unusual properties are mostly caused by quantum effects. Experimental studies have helped us a lot in understanding these systems, but there remained ambiguities and questions in these studies that are mainly out of the reach of the experimental equipment and techniques. Nano-scale computer simulations can act as a powerful tool in resolving this lack of knowledge. They can also help further explore the properties of these systems under these extreme conditions. One key point in the simulations though, is the importance of including the rotational state of molecules when calculating their interactions amongst themselves as well as with the underlying ionic surface. This study proposes the first model of this type with the purpose of being used in large-scale molecular simulations. v Preface The idea behind this project was initially identified by Dr. David Jack after his PhD student’s project, Dr. Jamal N. Dawoud (with the help from Dr. Abdulwahab K. Sallabi), on the adsorption of H2 molecules on ionic surfaces. The original work related to this project was developed by Dr. David Jack and several of his work study students, especially his B.Sc. honors student Brandon A. Kootnekoff. Their model is, in fact, another version of the first model (Model I) used in this study. It was later used in the Monte Carlo simulations of helicoptering H2 layers over MgO by Dr. Jack’s B.Sc. honors student Christina F. Livingston. During the first months of my PhD, I transferred their Monte Carlo code from FORTRAN 77 to FORTRAN 2003, collected some more data and analyzed them under the supervision of Dr. David Jack, and the results of this study was initially presented in a conference in Turkey; David B. Jack, Christina F. Livingston, Alireza Sadeghifar, et al. Rotational ordering in H2 films on MgO(001), European Conference on Surface Science (ECOSS 30), August 2014, Antalya, Turkey. We then completed this work even further and the results were published as an article in a journal; David B. Jack, Alireza Sadeghifar, Christina F. Livingston, Brandon A. Kootnekoff et al. Structure of helicoptering H2 molecules on MgO (001). Surface Science 641, 314-319 (2015). Although, parts of Chapters 1-5 of this thesis (especially parts that are related to the development of Model I) are from this published work, the results of this paper have not been used in any sections of this dissertation. In this project as my PhD thesis, I was responsible for programming Model I calculations in FORTRAN 2003, as well as collecting and analyzing the results of this model for the single H2-surface interactions (parts of Chapter 4), H2-H2 interactions (parts of Chapter 5), and the adsorbed H2 film Monte Carlo simulations (Chapter 6) with guidance from Dr. David Jack. I was also responsible for the literature review for this study (Chapter 1), developing models II and III vi (most parts of Chapters 2 and 3), programming the calculations and simulations in FORTRAN 2003, as well as collecting and analyzing the results of these models for the single H2-surface interaction (most parts of Chapter 4), H2-H2 interaction (most parts of Chapter 5), as well as the Monte Carlo simulations of the adsorbed H2 films (Chapters 7 and 8) with guidance from Dr. David Jack. Some parts of this dissertation (parts related to models I and II) have been presented in several conferences: 1- David B. Jack, Alireza Sadeghifar Structures formed by para- and ortho-H2 on MgO(001), The International Chemical Congress of Pacific Basin Societies (Pacifichem 2015), December 2015, Hawaii, USA (poster presentation of parts of Chapters 1-6 related to Model I). 2- David B. Jack, Alireza Sadeghifar The Influence of Molecular Rotations on the Structure and Stability of an H2 Film on MgO(001), 98th Canadian Chemistry Conference and Exhibition (CSC 2015), May 2015, Ottawa, Canada (oral presentation of parts of Chapters 1-6 related to Model I). 3- Alireza Sadeghifar, David B. Jack, The Influence of Molecular Rotational state on the Structure of H2 Films adsorbed on MgO(001), 100th Canadian Chemistry Conference and Exhibition (CSC 2017), May 2017, Toronto, Canada (oral presentation of parts of Chapters 1-7 related to models I and II). 4- Alireza Sadeghifar, David B. Jack, Formation of ortho- and para-H2 Films on MgO at low temperatures, 101st Canadian Chemistry Conference and Exhibition (CSC 2018), May 2018, Edmonton, Canada (oral presentation of parts of Chapters 1-7 related to models I and II). Besides, we are planning to submit the results of this study as several journal articles. vii Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ........................................................................................................................ vii List of Tables .............................................................................................................................. xvi List of Figures .......................................................................................................................... xxxii List of Abbreviations ................................................................................................................ xlvi Acknowledgements ................................................................................................................. xlviii Dedication ....................................................................................................................................... l Introduction ............................................................................................................... 1 1.1 General considerations ........................................................................................................ 3 1.2 Basic concepts ..................................................................................................................... 5 1.2.1 Rotational states of H2.................................................................................................. 5 1.2.2 The adsorbed H2 film structures................................................................................... 7 1.2.2.1 The unit cells ...................................................................................................... 7 1.2.2.2 The film structures .............................................................................................. 8 1.3 Field Background ................................................................................................................ 9 1.3.1 H2 on MgO ................................................................................................................. 10 1.3.1.1 Binding energies ............................................................................................... 10 1.3.1.2 Structures .......................................................................................................... 11 1.3.1.3 Other properties ................................................................................................ 13 1.3.2 H2 on NaCl ................................................................................................................. 15 viii 1.3.2.1 Binding energies ............................................................................................... 15 1.3.2.2 Structures .......................................................................................................... 17 1.3.2.3 Other properties ................................................................................................ 18 1.4 Summary and problems in the field .................................................................................. 18 1.5 Questions in this thesis and the hypothesis ....................................................................... 22 1.6 Approach ........................................................................................................................... 23 Theory....................................................................................................................... 25 2.1 The formulation of the adsorption system ........................................................................ 25 2.2 Perturbation Theory .......................................................................................................... 26 2.2.1 NDPT method ............................................................................................................ 27 2.2.2 The DPT method ........................................................................................................ 28 2.3 Metropolis Monte Carlo simulations ................................................................................ 30 Potential Models ...................................................................................................... 34 3.1 H2-surf interaction potentials ............................................................................................ 37 3.1.1 Model I (the RAP model) ........................................................................................... 41 3.1.1.1 The dispersion and repulsion interactions ........................................................ 42 3.1.1.2 The bulk interactions ........................................................................................ 43 3.1.1.3 The electrostatic interactions ............................................................................ 43 3.1.1.3.1 H2 quadrupole-ionic surface interaction .................................................. 44 3.1.1.3.2 H2 induced dipole-ionic surface interaction (the induction energy) ....... 46 3.1.2 Model II...................................................................................................................... 48 3.1.2.1 The 𝑽𝒔𝒆𝒑 case .................................................................................................. 49 3.1.2.1.1 The dispersion and repulsion interactions ............................................... 50 ix 3.1.2.1.2 The electrostatic Interactions .................................................................. 54 H2 quadrupole-ionic surface interaction ................................................................... 55 H2 induced dipole-ionic surface (induction) interaction ........................................... 60 3.1.2.2 The 𝑽𝒕𝒐𝒕 case ................................................................................................... 61 3.1.3 Model III .................................................................................................................... 65 3.2 H2-H2 interaction potentials .............................................................................................. 69 3.2.1 Model I ....................................................................................................................... 73 3.2.2 Model II...................................................................................................................... 74 Parameterization of the potential and the single H2-surface interactions ......... 78 4.1 Parametrization procedure ................................................................................................ 78 4.1.1 Model I (the RAP Model) .......................................................................................... 80 4.1.2 Model II...................................................................................................................... 82 4.2 Convergence of the potential ............................................................................................ 82 4.2.1 The Fourier series convergence ................................................................................. 84 4.2.1.1 Model I (The RAP Model) ............................................................................... 84 4.2.1.2 Model II ............................................................................................................ 84 4.2.1.2.1 H2-ionic lattice Repulsion and dispersion interactions (LJ interactions) 85 The Fourier series convergence ................................................................................ 85 The substrate layer convergence (the bulk potential) ............................................... 90 4.2.1.2.2 H2 quadrupole-ionic lattice interaction (quadrupole potential) ............... 95 4.2.1.2.3 H2 induced dipole-surface interactions (induction potential) .................. 96 4.2.2 The PT series convergence ........................................................................................ 98 4.2.2.1 The MgO substrate ......................................................................................... 102 x 4.2.2.1.1 Heli/Mg2+ (the UBE case) ..................................................................... 103 4.2.2.1.2 Para/Mg2+ (the UBE case) ..................................................................... 113 4.2.2.2 The NaCl substrate ......................................................................................... 119 4.2.2.2.1 Heli/Na+ ................................................................................................. 119 4.2.2.2.2 Para/Na+ ................................................................................................ 120 4.3 Single H2-ionic surface energies (the minimum energies) .............................................. 130 4.3.1 Model I ..................................................................................................................... 130 4.3.1.1 The MgO substrate ......................................................................................... 131 4.3.1.1.1 The UBE case ........................................................................................ 131 4.3.1.1.2 The LBE case ........................................................................................ 135 4.3.1.2 The NaCl substrate ......................................................................................... 137 4.3.2 Model II.................................................................................................................... 139 4.3.2.1 The MgO substrate ......................................................................................... 140 4.3.2.1.1 The UBE Case ....................................................................................... 140 4.3.2.1.2 The LBE Case ....................................................................................... 143 4.3.2.2 The NaCl substrate ......................................................................................... 146 4.4 Summary ......................................................................................................................... 149 H2-H2 interactions.................................................................................................. 151 5.1 Instantaneous interactions ............................................................................................... 152 5.2 RAP interactions ............................................................................................................. 154 5.2.1 Model I ..................................................................................................................... 154 5.2.2 Model II.................................................................................................................... 156 5.2.2.1 The MgO substrate ......................................................................................... 158 xi 5.2.2.2 The NaCl substrate ......................................................................................... 160 5.3 Summary ......................................................................................................................... 161 Simulation results for Model I ............................................................................. 163 6.1 H2 films over NaCl .......................................................................................................... 166 6.1.1 Monolayers of n-H2 .................................................................................................. 167 6.1.2 Monolayers of Para molecules ................................................................................. 173 6.2 H2 films over MgO .......................................................................................................... 178 6.2.1 Half coverage (coverage of 0.50) ............................................................................. 179 6.2.1.1 The UBE case ................................................................................................. 180 6.2.1.1.1 Films of n-H2 molecules ........................................................................ 180 6.2.1.1.2 Films of Para molecules ........................................................................ 194 6.2.1.2 The LBE case.................................................................................................. 198 6.2.1.2.1 Films of n-H2 molecules ........................................................................ 198 6.2.1.2.2 Films of Para molecules ........................................................................ 206 6.3 Summary ......................................................................................................................... 211 6.3.1 The NaCl results....................................................................................................... 211 6.3.2 The MgO results....................................................................................................... 212 6.3.2.1 The UBE case ................................................................................................. 212 6.3.2.2 The LBE case.................................................................................................. 213 Simulation results for Model II ............................................................................ 215 7.1 H2 films over NaCl .......................................................................................................... 215 7.1.1 Monolayers of n-H2 molecules ................................................................................ 216 7.1.2 Monolayers of Para molecules ................................................................................. 222 xii 7.2 H2 films over MgO .......................................................................................................... 228 7.2.1 Half coverage (coverage of 0.50) ............................................................................. 230 7.2.1.1 The UBE case ................................................................................................. 230 7.2.1.1.1 Films of n-H2 molecules ........................................................................ 230 7.2.1.1.2 Films of Para molecules ........................................................................ 239 7.2.1.2 The LBE case.................................................................................................. 247 7.2.1.2.1 Films of n-H2 molecules ........................................................................ 247 7.2.1.2.2 Films of Para molecules ........................................................................ 255 7.2.2 Coverage of 0.75 ...................................................................................................... 262 7.2.2.1 A mixture of Heli and Para molecules (the HP layer) .................................... 263 7.2.2.2 A mixture of Heli and Cart molecules (the HC layer) .................................... 266 7.2.2.3 A layer of only Heli molecules (the H layer) ................................................. 268 7.2.2.4 A layer of only Para molecules (the P layer) .................................................. 270 7.2.3 Coverage of 0.83 ...................................................................................................... 275 7.2.3.1 A layer of only Para molecules (the UBE case) ............................................. 276 7.2.3.2 External vibrations in a layer of Para over MgO at 0.83 coverage ................ 279 7.2.3.3 Final considerations ........................................................................................ 285 7.3 Summary ......................................................................................................................... 287 7.3.1 The NaCl results....................................................................................................... 287 7.3.2 The MgO results....................................................................................................... 288 7.3.2.1 Coverage of 0.5 .............................................................................................. 288 7.3.2.1.1 The UBE case ........................................................................................ 288 7.3.2.1.2 The LBE case ........................................................................................ 290 xiii 7.3.2.2 Coverage of 0.75 ............................................................................................ 292 7.3.2.3 Coverage of 0.83 ............................................................................................ 292 Model III and its impact on the simulation results ............................................ 295 8.1 Ortho-H2 molecule-ionic surface interactions ................................................................ 295 8.1.1 The MgO substrate (the UBE case) ......................................................................... 295 8.1.2 The NaCl substrate ................................................................................................... 304 8.2 H2 adsorbed layers........................................................................................................... 307 8.2.1 The NaCl substrate ................................................................................................... 307 8.2.1.1 A monolayer of Para molecules...................................................................... 307 8.2.1.2 A monolayer of n-H2 molecules ..................................................................... 309 8.2.2 The MgO substrate (the UBE case) ......................................................................... 310 8.2.2.1 Layers of Para molecules ................................................................................ 311 8.2.2.2 Layers of n-H2 molecules ............................................................................... 312 8.2.2.2.1 Half (0.5) coverage ................................................................................ 312 8.2.2.2.2 Coverage of 0.75 and 0.83 .................................................................... 314 8.3 Summary ......................................................................................................................... 315 8.3.1 The interaction of H2 with NaCl and MgO .............................................................. 315 8.3.2 The H2 layers over NaCl and MgO .......................................................................... 316 Concluding remarks and future work ................................................................. 318 9.1 Conclusions ..................................................................................................................... 318 9.2 Future Work .................................................................................................................... 320 9.2.1 The assumptions ....................................................................................................... 320 9.2.1.1 The accurate binding energy........................................................................... 320 xiv 9.2.1.2 The zero-point energy ..................................................................................... 321 9.2.2 The methods ............................................................................................................. 322 9.2.2.1 The NDPT and DPT methods ......................................................................... 322 9.2.2.2 MC simulations .............................................................................................. 322 Bibliography ...............................................................................................................................324 Appendix A ..................................................................................................................... 334 The integral of the product of three spherical harmonics .................................................. 334 An example for the derivation of the 𝑿𝒊 equations in Table 3-1 ..................................... 336 Derivation of the first-order 𝑬𝑸𝒆𝒍𝒔 energy .............................................................................. 342 The 𝑫𝒊𝒈 functions used in section 3.1.2.1.1 ............................................................................... 343 Derivation of the second-order dispersion and repulsion energy for Para ................ 344 Derivation of the 𝑽𝟎𝟐𝑽𝟐𝟐𝑽𝟐𝟎 term in (3-45) for the Para molecule ........................... 347 The (𝑽𝟏𝟑. 𝑽𝟑𝟑. 𝑽𝟑𝟏) terms used in Eq. (3-38) for the Heli and Cart molecules ...... 351 The 𝑽𝟎𝟐𝑽𝟐𝟐𝑽𝟐𝟐𝑽𝟐𝟎 and 𝑽𝟎𝟐𝑽𝟐𝟒𝑽𝟒𝟐𝑽𝟐𝟎 terms used in Eq. (3-40) for the Para molecules .............................................................................................................................................................. 352 The 𝑽𝟏𝟑𝑽𝟑𝟑𝑽𝟑𝟑𝑽𝟑𝟏 and 𝑽𝟏𝟑𝑽𝟑𝟓𝑽𝟓𝟑𝑽𝟑𝟏 terms used in Eq. (3-42) for the Heli molecules .............................................................................................................................................................. 353 The 𝑽𝟏𝟑𝑽𝟑𝟑𝑽𝟑𝟑𝑽𝟑𝟏 and 𝑽𝟏𝟑𝑽𝟑𝟓𝑽𝟓𝟑𝑽𝟑𝟏 terms used in Eq. (3-42) for the Cart molecules (the 𝑽𝒕𝒐𝒕 case) ............................................................................................................................ 357 The 𝑽𝟎𝟐𝑽𝟐𝟐𝑽𝟐𝟎 term in the third-order energy for the Para molecules in the 𝑽𝒕𝒐𝒕 case 360 The 𝑽𝟏𝟑𝑽𝟑𝟑𝑽𝟑𝟏 terms in the third-order energy for the Heli and Cart molecules in the 𝑽𝒕𝒐𝒕 case .................................................................................................................................................. 361 The cubic equation in Model III to calculate the first-order dispersion and repulsion energy................................................................................................................................................ 363 xv The cubic equation in Model III to calculate the first-order H2 quadrupole energy 364 The cubic equation in Model III to calculate the first-order H2 induced dipole energy 365 Appendix B ..................................................................................................................... 366 The development of the different terms of the H2-H2 interaction energy in Eq. (3-54) 366 The development of the 𝑰𝟎𝟎𝟎(𝟏), 𝑰𝟎𝟎𝟎(𝟐), 𝑰𝟐𝟎𝟐(𝟐), 𝑰𝟎𝟐𝟐(𝟏), and the 𝑺𝟐𝒎𝑱(𝒊) terms (𝒊 = 𝟏 𝒂𝒏𝒅 𝟐) for Para, Heli, and Cart in Model I .................................................................. 371 Derivation of the 𝝈𝒊𝒋 coefficients as well as the normalization factor in Eq. (3-61) 373 Derivation of the Para-Para interaction energy in Model II............................................... 375 Appendix C ..................................................................................................................... 378 The interaction parameters used in models I and II ............................................................. 378 The Fourier series convergence (Chapter 4) ............................................................................ 381 The PT series convergence for H2/MgO systems in the UBE case (section 4.2.2.1) 383 The PT series convergence for H2/MgO systems in the LBE case (section 4.2.2.1) 396 The PT series convergence for H2/NaCl systems (section 4.2.2.2) ................................. 410 Appendix D ..................................................................................................................... 413 xvi List of Tables Table 1-1 The findings and questions in the field for H2/MgO, H2/NaCl. ................................... 21 Table 3-1 The Xi functions in Eq. (3-12). As seen, all these functions only depend on the x, y, z coordinates. In these equations, i is the imaginary number, G = 2π(c/2), and Bnm =Mels An, m exp (‒ γn, m. z)fac, where fac is either 1 or 2 depending on the |n|, |m| values as explained in section 3.1.1.3.1. Mels is defined in section 0A.2.................................................... 45 Table 3-2 The Cindi functions in Eq. (3-18). As seen, all these functions only depend on the x, y, z coordinates in the system. Also, L(0) = α2 × G2, where α2 = 136π5α‖-α ⊥ and G =2π(c/2) . ....................................................................................................................................... 48 Table 3-3 The KDR(i) functions in Eq. (3-23). As seen, all these functions only depend on the x, y, z coordinates. For simplicity, in all these equations D0 . α = 6,12sD1p = 0. -1, . .. is shown by fDR . “all |n|, |m|” was explained in section 3.1.1.3.1. X and Y are the positions of the H2 molecule with respect to the origin of the coordinate system (an anion site) in the system. ........ 52 Table 3-4 The Xi terms in Eq. (3-30) for the second-order energy for the Para molecule in the Vsep case and their corresponding expressions in Eq. (3-47) in the Vtot case ............................ 65 Table 3-5 The Hijp integrals in Model III used in Eq. (2-15) for the dispersion and repulsion interactions. ................................................................................................................................... 67 Table 3-6 The Hijp integrals in Model III used in Eq. (2-15) for the H2 quadrupole electrostatic interactions. ................................................................................................................................... 68 Table 3-7 The Hijp integrals in Model III used in Eq. (2-15) for the H2 induction electrostatic interactions. ................................................................................................................................... 69 xvii Table 3-8 The H2-H2 interaction operators in Eq. (3-53). The ΥJ, mJ functions are spherical harmonics and show the orientation dependence of the potential operator. The Vijk functions show the distance dependence of the operator. These functions have been tabulated in Refs. 78,79. The ' sign in the equations shows the dependence of the function on the second H2 molecule (molecule 2). The unprimed functions show the dependence on the first molecule (molecule 1)........................................................................................................................................................ 72 Table 3-9 The amount of the I000(1), I000(1), I202(2), I022(1), and S2mJ(i) integrals (for mJ = -2, -1,0,1,2 and i = molecule 1 or 2 ) defined in section B.1 of Appendix B for different H2 species in Model I. Molecule i = 1 (or 2) can be either a Para, or a Heli, or a Cart molecule........................................................................................................................................................ 74 Table 4-1 The minimum LJ energies in the UBE case after adding each Fourier vector set to the energy. The data are presented for the Para/Mg2+ system at the height of 3.25 Å, and for the Heli/Mg2+ and Cart/Mg2+ systems at 3.24 Å. The (k,j) vector set is defined in section 4.2 of the text. All the energies are in kcal/mol. ........................................................................................... 86 Table 4-2 The LJ energies for the Heli/Mg2+ system in the UBE case after adding each vector set into the potential. The (k,j) vector set is defined in section 4.2 of the text. All the energies are in kcal/mol......................................................................................................................................... 88 Table 4-3 The LJ energies in kcal/mol for the Heli/Mg2+ system in the UBE case when the layers below the top layer are added into the potential. Here, “top layer” means that only the surface layer (p = 0 in Eq. (3-23)) was used in the calculation of the LJ energy. “down to 6th” means that a total of six layers below the top layer have been included in the calculation. .................... 93 Table 4-4 The shift in the minimum LJ energy, its height as well as the height of the zero energy for the Heli/Mg2+ system in the UBE case after adding each layer of ions into the potential. ..... 94 xviii Table 4-5 The interaction of the quadrupole moment of Heli H2 with the MgO surface in the LBE case for the Heli/Mg2+ system after adding each Fourier term into this potential. The k, j vector set is defined in section 4.2 of the text. .............................................................................. 97 Table 4-6 The second order energies and the corresponding ratios for dispersion-repulsion (DR), anisotropic induction (ind2), and the quadrupole electrostatic interaction (Qels) of a single H2 molecule with the surface for the Heli/Mg2+ system and in the UBE case. The optimum height occurs at 2.90 Å. Etot-2 in this table is the sum of all components of the energies up to second order PT corrections and is different from the Etot shown in Table 4-7...................................... 106 Table 4-7 The energy results for the Heli/Mg2+ system in the UBE case. The numbers in brackets (superscripts) indicate the order of the correction term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic H2 induced dipole-surface (induction), and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol.......................................................................................... 107 Table 4-8 The energies and PEPR quantities for the Heli/Mg2+ system in the UBE case after including the fourth-order EQels PT term into the total energy. The RI for this system is from 2.50 up to 2.90 Å. EQels(tot) is the sum of all EQels PT terms up to the fourth-order energy (shown in Table 4-7). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels correction terms in its PT series (presented in Table 4-7). EQels(5) is estimated using the assumption explained in the text. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %. ............................................................................................................................ 111 Table 4-9 The energy results for Para/Mg2+ system in the UBE case. The numbers in brackets (superscripts) indicate the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, xix isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol....................................................................................................................................... 115 Table 4-10 The energies and PEPR quantities for a system of Para/Mg2+ in the UBE case after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.60 up to 3.00 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table 4-9). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels PT correction terms in the series presented in Table 4-9. EQels(5) was estimated using the assumption explained in the text. 5th in the ratios refers to this estimated energy. Heights are in Å, the energies are in kcal/mol and the ratios are in %. ........................................................................ 117 Table 4-11 The energy results for the Heli/Na+ system. The numbers in brackets (superscripts) indicate the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol...................................................................................................................................................... 122 Table 4-12 The energies and PEPR quantities for the system of Heli/Na+ after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.30 up to 2.70 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table 4-11). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels correction terms in the PT series presented in Table 4-11. EQels(5) was estimated using the assumption explained in in section 4.2.2. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %. ................................................................................................ 124 Table 4-13 The energy results for the system of Para/Na+. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and xx Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol...................................................................................................................................................... 126 Table 4-14 The energies and PEPR quantities for the system of Para/Na+ after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.35 up to 2.75 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table 4-13). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels correction terms in its PT series presented in Table 4-13. EQels(5) was estimated using the assumption explained in in section 4.2.2.1. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %. ................................................................................................ 128 Table 4-15 The Model I H2-MgO minimum interaction energies for nine different systems over this substrate in the UBE case. Etot, ER, ED, EQels, Eind, and Ebulk are the total, repulsion, dispersion, H2 quadrupole-surface, H2 induced dipole-surface (induction), and bulk energies, respectively. ................................................................................................................................ 133 Table 4-16 The Model I MiEB and MaEB data for the H2 species over MgO in the UBE case. The MiEB and MaEB terms were defined in the text. ................................................................ 135 Table 4-17 The Model I H2-MgO minimum interaction energies for nine different systems over this substrate in the LBE case. Etot, ER, ED, EQels, Eind, and Ebulk are the total, repulsion, dispersion, H2 quadrupole-surface, H2 induced dipole-surface (induction), and bulk energies, respectively. ................................................................................................................................ 136 Table 4-18 The Model I MiEB and MaEB data for the H2 species over MgO in the LBE case. The MiEB and MaEB terms were defined in section 4.3.1.1.1. ................................................. 137 xxi Table 4-19 The Model I H2-NaCl minimum interaction energies for nine different systems over this substrate. Etot, ER, ED, EQels, Eind, and Ebulk are the total, repulsion, dispersion, H2 quadrupole-surface, H2 induced dipole-surface (induction), and bulk energies, respectively. ...................... 138 Table 4-20 The Model I MiEB and MaEB data for the H2 species over NaCl. The MiEB and MaEB terms were defined in section 4.3.1.1.1. .......................................................................... 139 Table 4-21 The Model II H2-MgO minimum interaction energies for nine different systems in the UBE case. Etot, ER, ED, EQels, and Eind are the total, repulsion, dispersion, H2 quadrupole-surface, and H2 induced dipole-surface (induction) energies, respectively. ............................................. 142 Table 4-22 The Model II MiEB and MaEB data for the H2 species over MgO in the UBE case. The MiEB and MaEB terms were defined in section 4.3.1.1.1. ................................................. 143 Table 4-23 The Model II H2-MgO minimum interaction energies for nine different systems over this substrate in the LBE case. Etot, ER, ED, EQels, and Eind are the total, repulsion, dispersion, H2 quadrupole-surface, and H2 induced dipole-surface (induction) energies, respectively. ............ 145 Table 4-24 The Model II MiEB and MaEB data for the H2 molecules over MgO in the LBE case. The MiEB and MaEB terms were defined in section 4.3.1.1.1. ................................................. 145 Table 4-25 The Model II H2-NaCl minimum interaction energies for nine different systems over this substrate. Etot, ER, ED, EQels, and Eind are the total, repulsion, dispersion, H2 quadrupole-surface, and H2 induced dipole-surface (induction) energies, respectively. ............................... 148 Table 4-26 The Model II MiEB and MaEB data for the H2 molecules over NaCl. These MiEB and MaEB terms were defined in section 4.3.1.1.1. ................................................................... 148 Table 6-1 The “actual” and “specified” ULH quantities for a Heli or Para H2 molecule over NaCl when Model I is used in the calculation of the energy. The ULH term is defined in the text...................................................................................................................................................... 167 xxii Table 6-2 Comparison of the binding energy of a Heli H2 molecule over the Na+ site, the binding energy between two Heli H2 molecules, as well as the MiEB quantity for this molecule over the NaCl surface when Model I is used in the calculations. The MiEB is defined in section 4.3.1.1.1...................................................................................................................................................... 172 Table 6-3 Energies (per molecule) of a monolayer of Heli molecules over NaCl. The quantities in brackets are either the uncertainties in the experimental data (previous studies) or the standard deviations resulted from the statistical averaging of the energies in the MC-NVT simulations (present work using Model I). ..................................................................................................... 173 Table 6-4 Comparison of the binding energy of a Para H2 molecule over the Na+ site, the MiEB quantity of this molecule over the NaCl surface, as well as the binding energy between two Para H2 molecules when Model I is used in the calculations. The MiEB has been defined in section 4.3.1.1.1....................................................................................................................................... 177 Table 6-5 The energies (per molecule) of a monolayer of Para molecules over NaCl. The quantities in brackets are either the uncertainties in the experimental data (previous studies) or the standard deviations resulted from the statistical averaging in the MC simulations (present work using Model I).................................................................................................................... 178 Table 6-6 The ULH quantities for a Para and Heli molecule over the MgO substrate and in the UBE and LBE cases when Model I is used in the calculation of the energies. The ULH term has been defined in the text (see Eq. (6-1) and the explanation underneath this equation). ............. 179 Table 6-7 The binding energy of a Heli molecule over the Mg2+ cation, the energy between two Heli molecules, as well as the MiEB of a Heli molecule over the MgO surface when Model I along with the UBE case are used in the calculations. The MiEB has been defined in section 4.3.1.1.1....................................................................................................................................... 192 xxiii Table 6-8 The energies (per molecule) of an MC-NVT simulated layer of Heli molecules over MgO at half coverage at temperatures up to the ULT quantity. The errors are the standard deviations obtained from the statistical averaging in the MC-NVT simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. ............................................................ 193 Table 6-9 The binding energy of a Para molecule over Mg2+, the binding energy between two Para molecules, as well as the MiEB of a Para molecule over the MgO surface when Model I along with the UBE case are used in the calculations. The MiEB has been defined in section 4.3.1.1.1....................................................................................................................................... 197 Table 6-10. The binding energy of a Heli molecule over Mg2+, the binding energy between two Heli molecules, as well as the MiEB of a Heli molecule over the MgO surface when Model I along with the LBE case are used in the calculations. The MiEB has been defined in section 4.3.1.1.1....................................................................................................................................... 205 Table 6-11 The energies of the MC-NVT simulated c(2×2) films of Heli molecules over MgO when Model I along with the LBE case are used in the simulations. The numbers after the ± sign are the MC-NVT simulation standard deviations for the energies. ............................................ 206 Table 6-12 The binding energy of a Para molecule over Mg2+, the binding energy between two Para molecules, as well as the MiEB of a Para molecule over the MgO surface when Model I along with the LBE case are used in the calculations. The MiEB has been defined in section 4.3.1.1.1....................................................................................................................................... 210 Table 6-13 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over NaCl. In the MC-NVT simulations, Model I was used in the calculation of the energies. The results for the energy are only shown at a specific temperature. The experimental references have been listed right below the table. .................... 211 xxiv Table 6-14 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over MgO. In the MC-NVT simulations, Model I along with the UBE case were used in the calculation of the energies. The results for the energy are only shown at a specific temperature. “NA” means that the data are not available, neither from simulations nor from experiments. ................................................................................................................. 213 Table 6-15 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over MgO. In the MC-NVT simulations, Model I along with the LBE case were used in the calculation of the energies. The results for the energy are only shown at a specific temperature. “NA” means that the data are not available, neither from simulations nor from experiments. ................................................................................................................. 214 Table 7-1 The “actual” and “specified” ULH quantities for Heli or Para H2 molecule over NaCl when Model I is used in the calculation of the energy. The ULH term has been defined in the text (see Eq. (6-1) and the explanation underneath this equation). .................................................... 216 Table 7-2 The binding energy of a Heli H2 molecule over the Na+ site, the binding energy between two Heli H2 molecules, as well as the MiEB quantity for this molecule over the NaCl surface using Model II. The MiEB has been defined in section 4.3.1.1.1. ................................. 220 Table 7-3 Energies of a monolayer of Heli molecules over NaCl. The quantities in brackets are either the uncertainties in the experimental data (previous studies) or the standard deviations resulted from the statistical averaging in the MC-NVT simulations (present work using Model II)................................................................................................................................................. 221 Table 7-4 The binding energy of a Para molecule over the Na+ site, the Para-Para binding energy, as well as the MiEB quantity for this molecule over the NaCl surface using Model II. The MiEB has been defined in section 4.3.1.1.1. ......................................................................................... 225 xxv Table 7-5 Energies of a monolayer of Para molecules over NaCl when Model II is used in the simulations. The quantities in brackets are either the uncertainties in the experimental data (previous studies) or the standard deviations resulted from the statistical averaging in the MC-NVT simulations (present work using Model II). ....................................................................... 227 Table 7-6 The ULH quantities for Para and Heli molecules over the MgO substrate for both the UBE and LBE cases when Model II is used in the calculation of the energies. ......................... 229 Table 7-7 The energy of a Heli molecule over cation, the attractive energy between two of these species, as well as the MiEB of this molecule over MgO using Model II and the UBE case. The MiEB has been defined in section 4.3.1.1.1. .............................................................................. 237 Table 7-8 The energies of MC-NVT simulated layers of Heli molecules over MgO at half coverage at temperatures up to the ULT quantity when Model I along with the UBE case were used in the calculation of the energies. The errors are the standard deviations obtained from the statistical averaging in the MC simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. ...................................................................................................................... 238 Table 7-9 The energy of a Para molecule over cation, the attractive energy between two of these species, as well as the MiEB of this molecule over MgO using Model II and the UBE case. The MiEB has been defined in section 4.3.1.1.1. .............................................................................. 245 Table 7-10 The energies of MC simulated layers of Heli molecules over MgO at half coverage at temperatures up to the ULT quantity when Model II along with the LBE case are used in the calculation of the energies. The errors are the standard deviations obtained from the statistical averaging in the MC simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. ................................................................................................................................. 246 xxvi Table 7-11 The energy of a Heli molecule when sits over cation, the attractive energy between two of these species, as well as the MiEB of this molecule over MgO using Model II and the LBE case. The MiEB has been defined in section 4.3.1.1.1. ...................................................... 253 Table 7-12 The energies of MC simulated layers of Heli molecules over MgO at half coverage at temperatures up to the ULT quantity when Model II along with the LBE case were used in the calculation of the energies. The errors are the standard deviations obtained from the statistical averaging in the MC simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. ................................................................................................................................. 254 Table 7-13 The energy of a Para molecule when sits over cation, the attractive energy between two of these species, as well as the MiEB of this molecule over MgO using Model II and the LBE case. The MiEB has been defined in section 4.3.1.1.1. ...................................................... 260 Table 7-14 The energies of MC-NVT simulated layers of Para molecules over MgO at half coverage at temperatures up to the ULT quantity when Model II along with the LBE case are used in the calculation of the energies. The errors are the standard deviations obtained from the statistical averaging in the MC-NVT simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. ......................................................................................................... 261 Table 7-15 The energies of the HP, HC, and the H layers over MgO at 0.75 coverage and at 5 K when Model II along with the UBE case are used in the calculation of the energies. The errors are the standard deviations in the MC-NVT simulation energies for each layer. ....................... 273 Table 7-16 The calculated (this work) and experimentally measured (previous studies) vibrational energies in a layer of Para molecules at both 0.75 and 0.83 coverages over the MgO surface. In the calculation of all these energies Model II along with the UBE case were used. xxvii “disp. E” in brackets refers to the “dispersive energy” that is the vibrational energy of a collection of molecules over the surface. .................................................................................... 280 Table 7-17 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over NaCl. In the MC-NVT simulations, Model II was used in the calculation of the energies. The results for the energy are only shown at a specific temperature. The experimental references have been listed right below the table. .................... 288 Table 7-18 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over MgO. In the MC-NVT simulations, Model II along with the UBE case were used in the calculation of the energies. The results for each H2 species are only shown at a specific temperature. “NA” means that the data are not available from the experiments. ................................................................................................................................ 290 Table 7-19 The summary of the MC-NVT simulation results as well as the experimental data for the layers of H2 molecules over MgO at half coverage. In the MC-NVT simulations, Model II along with the LBE case were used in the calculation of the energies. The results for each H2 species are only shown at a specific temperature. “NA” means that the data are not available from the experiments. ................................................................................................................. 291 Table 7-20 The summary of the MC-NVT simulation results as well as the experimental data for the layers of H2 molecules over MgO at the coverages of 0.75 and 0.83. In the MC-NVT simulations, Model II along with the UBE case were used in the calculation of the energies. “ND” means that the simulations were not done for that part of the study so the structures were not determined for that part. ........................................................................................................ 294 Table A4-1 The D(i)୥ functions used in Table 3-3 of section 3.1.2.1.1……………….………344 xxviii Table C1-1 The initial interaction parameters used in the Tang-Toennies potential (Eq. (3-10)) of Model I before fitting to the desired energies. These parameters were obtained from Refs. 78,79 for H2-H2 interactions and Ref.25 for ion-ion interactions. For the H2-ion interactions some mixing rules of Ref. 25 were used……………………………………………………………….379 Table C1-2 The final interaction parameters used in the Tang-Toennies potential (Eq. (3-10)) of Model I after fitting the energies to the desired values. For MgO, two separate adjustments were done, one for the UBE case and one for the LBE case. The difference in the parameters listed in this table with Table C1-1 (before the adjustment) is only in the Aij parameter. The energies presented in section 4.3 are calculated using the parameters listed in this table…………….....380 Table C1-3 The initial interaction parameters used in Eq. (3-23) (the LJ potential of Briquez et al.26 used in Model II, see also section A.4) before fitting the energies to the desired values. These parameters were obtained from fitting the LJ potential to the energy obtained from the Tang-Toennies potential in Model I…………………………………………………………....381 Table C1-4 The final interaction parameters used in Eq. (3-23) (the LJ potential of Briquez et al.26 used in Model II, see also section A.4) after fitting the energies to the desired values. The energies presented in section 4.3.2 are calculated using the parameters listed in this table………………………………………………………………………………………….…381 Table C3-1 The energy results for Heli/Saddle system in the UBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol………………………………………………………………………………...……….386 xxix Table C3-2 The energies and PEPR quantities for a system of Heli/Saddle (MgO) in the UBE case after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 3.05 up to 3.45 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table C3-1). 4th in the ratios refers to the corresponding EQels PT correction term in its PT series presented in Table C3-1. Heights are in Å, energies are in kcal/mol, and the ratios are in %..................................................................................................388 Table C3-3 The energy results for Cart/Mg2+ in the UBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic H2 induced dipole-surface (induction), and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol…………………………………………………………...391 Table C3-4 The energy results for Cart/Saddle in the UBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic H2 induced dipole-surface (induction), and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol…………………………………………………...………393 Table C3-5 The energy results for the Heli/O2- system in the UBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic H2 induced dipole-surface (induction), and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol…………………………………………….……………..395 Table C4-1 The energy results for the system of Heli/Mg2+ in the LBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. xxx ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol…………………………………………………………………..…….399 Table C4-2 The energies and PEPR quantities for Heli/Mg2+ in the LBE case after adding the fourth-order EQels PT correction term to the energy. The RI for this system is from 2.40 up to 2.80 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy. 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels PT correction terms presented in Table C4-1. EQels(5) was estimated using the assumption explained in section 4.2.2.1. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %..................................................................................................................................................401 Table C4-3 The energy results for Para/Mg2+ in the LBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol…………………………………………………………………...…………………….404 Table C4-4 The energies and PEPR quantities for Para/Mg2+ in the LBE case after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.50 up to 2.90 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table C4-3). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels correction terms in Table C4-3. EQels(5) was estimated using the assumption explained in in section 4.2.1. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %..............................................................................................................................406 xxxi Table C4-5 The energies and PEPR quantities for a system of Heli/Saddle (MgO) in the LBE case after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 3.00 Å up to 3.40 Å. EQels(tot) is the sum of all EQels PT terms up to the fourth-order PT energy. 4th in the ratios refers to the fourth order EQels correction term in its PT series. Heights are in Å, energies are in kcal/mol, and the ratios are in %.............................................409 Table C5-1 The energies and PEPR quantities for the system of Cart/Na+ after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.80 up to 3.20 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy. 4th in the ratios refers to the fourth-order EQels correction term in the corresponding PT series. Heights are in Å, energies are in kcal/mol, and the ratios are in %................................................................412 Table D-1 The calculated (this work) and experimentally measured (previous studies) vibrational energies in a layer of Para molecules at both 0.75 and 0.83 coverages over the MgO surface. In the calculation of all these energies Model II along with the LBE case were used. “disp. E” in brackets refers to the “dispersive energy” that is the vibrational energy of a collection of molecules over the surface………………………………………………………..…………….414 xxxii List of Figures Figure 1-1 The delocalization of the molecular axis around the axis of quantization. This results in a molecular-axis distribution that forms the shape of the molecule for the ground state ortho- and para-H2 molecules. .................................................................................................................. 7 Figure 1-2 The c-type (left) and p-type (right) unit cells for the adsorbed H2 films over ionic surfaces. .......................................................................................................................................... 7 Figure 1-3 The c(n×2) structures formed by H2 molecules on MgO. The H2 molecules are in two different types. The n number in the brackets shows that every nth cation along the horizontal and vertical channels is occupied by the same H2 molecule. The hexagon depicts an hcp-like structure, sometimes used to characterize both the c(4×2) and c(6×2) structures. ......................... 9 Figure 3-1 The coordinate systems for, a) the H2 molecule as a two-atom particle in the space, (the previous MC simulation studies25), and b) the H2 molecule as a single-body particle made by the distribution of the molecular axis around the axis of quantization (the present work). In this figure, the external object shown is an ionic surface. ................................................................... 35 Figure 4-1 A system of Heli/Mg2+. The blue spheres show the cations and the red spheres show the anions in the ionic crystal. The dashed yellow arrow shows the surface normal, which is perpendicular to the plane of the ionic surface. The violet molecule is the Heli molecule sitting right above the cation. The dashed line is a visual aid to show the relative position of this molecule above the surface. The cyan arrow shows the axis of quantization of the Heli particle. Also, the saddle site (not shown) is defined as the midway point between two neighboring anions or cations in the crystal surface. .................................................................................................... 81 xxxiii Figure 4-2 The LJ (dispersion-repulsion) energy for the Heli/Mg2+ system in the UBE case after adding each Fourier term into the potential. (k,j) is defined in section 4.2 of the text. ............... 87 Figure 4-3 The LJ energy for the Heli/Mg2+ system in the UBE case. The curves show how adding each layer of ions in the MgO crystal below the surface layer affects the potential. ....... 91 Figure 4-4 The interaction of the quadrupole moment of Heli H2 with the MgO surface in the UBE case for the Heli/Mg2+ system after adding each Fourier term into this potential. The (k, j) vector set is defined in section 4.2 of the text. .............................................................................. 96 Figure 4-5 The 2nd/1st PEPR for a system of Heli/Mg2+ in the UBE case for a. dispersion, repulsion and the sum of these interactions, b. anisotropic H2 induced dipole-surface (induction), and c. H2 quadrupole-surface interaction energy. ....................................................................... 105 Figure 5-1 Different configurations for pair interactions between two-atom H2 molecules. The solid arrows indicate the molecular axes in each molecule. The dashed arrows ( r ) show the vector connecting the molecules’ centers of mass. ⦿ depicts a molecular axis perpendicular to the plane of the paper. ................................................................................................................. 153 Figure 5-2 The H2-H2 interaction potentials as functions of distance r for the nine configurations shown in Figure 5-1. as well as the RAP Para-Para case. The energies for cases 6 and 8 are not shown here, because case 6 is very similar to case 1 and case 8 is similar to the Para-Para interaction. .................................................................................................................................. 154 Figure 5-3 Model I H2-H2 RAP energies between a pair of like H2 molecules. The Para-Para interaction is the reference interaction to which the other energies are compared. .................... 155 Figure 5-4 Model I H2-H2 RAP energies between a pair of unlike H2 molecules. The Para-Para interaction is the reference interaction to which the other energies are compared. .................... 156 xxxiv Figure 5-5 Model II a. Heli-Heli, b. Para-Para, and c. Heli-Para interaction energies for the wavefunctions fixed at different heights over the MgO surface in the LBE case. ..................... 159 Figure 5-6 Model II a. Heli-Heli, b. Para-Para, and c. Heli-Para interaction energies for the wavefunctions fixed at different heights over the MgO surface in the LBE case. ..................... 161 Figure 6-1 The HDFs of a monolayer of Heli molecules over NaCl at temperatures up to 20 K. At 20 K, the function becomes nonzero at the ULH quantity of 4.0 Å. ..................................... 168 Figure 6-2 The color key diagram for the simulated H2 layers over the NaCl and MgO substrates. Although, the axis of quantization is applied to both para- and ortho-H2 species, for visualization purposes, it is only used for the ortho species in the images of the simulated configurations. .. 169 Figure 6-3 Snapshots of the final configurations of a simulated monolayer of Heli molecules over NaCl using Model I and at, a. 3 K, b. 7 K, c. 14 K, and d. 19 K. The square in the subfigure a demonstrates the p(1×1) structure. The arrow in subfigure d indicates the movement of one of the Heli molecules away from atop a cation. .................................................................................... 170 Figure 6-4 Heli-Heli RDFs of the simulated monolayers of Heli molecules over NaCl at temperatures up to 19 K. ............................................................................................................. 171 Figure 6-5 The HDFs of a monolayer of Para molecules over NaCl at temperatures up to 11 K. At 11 K and at the ULH quantity of 3.90 Å the probability becomes nonzero. ......................... 174 Figure 6-6 Snapshots of the final configurations of a monolayer of Para molecules over NaCl at a. 3 K, b. 7 K, and c. 10 K. As shown by the square in subfigure a, the layer forms a p(1×1) structure at all temperatures up to 10 K. ..................................................................................... 175 Figure 6-7 The Para-Para RDFs of a monolayer of Para molecules over NaCl at 3, 7 and 10 K, respectively. ................................................................................................................................ 176 xxxv Figure 6-8 The HDFs for a simulated layer of Heli molecules over MgO at half coverage using Model I. The results at 8 K show a nonzero probability at the ULH of 4.0 Å, so the MC-NVT simulations are consistent with the experiment only up to this temperature. ............................. 181 Figure 6-9 Final configurations of a layer of Heli molecules at a. 3 K, b. 5 K, c. 6 K, and d. 7 K using Model I. The square in subfigure a indicates a c(2×2) unit cell for the H2 film. The arrows in subfigures c and d show defects in the c(2×2) structure. The number of arrows show the number of defects formed in the film at each temperature. ........................................................ 182 Figure 6-10 The Heli-Heli RDFs of a simulated layer of Heli molecules over MgO at half coverage using Model I. The larger peaks show the average positions of the molecules which are similar to the positions expected for a c(2×2) structure. The two bumps in the graphs at 6 and 7 K indicate the appearance of some defects in the c(2×2) structure. The larger bumps at 7 K compared to 6 K show the presence of more defects in the structure. ........................................ 184 Figure 6-11 The c(2×2) structure (subfigure a) as well as a 4d-c(2×2) structure (subfigure b), which is created by four defects in a c(2×2) layer. The circles in subfigure b indicates the location of each one of the defects in the layer and how they are spaced out in the layer. ........ 186 Figure 6-12 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 5 K. The defect c(2×2) layers contain 4 defects (4d-c(2×2) layer) at the temperature range of 1 to 4 K, whereas at 5 K, they only include three defects in the layer (3d-c(2×2) layer). The equations shown on each curve are only based on the data from 1 to 4 K. Also, they are the equations used for the statistical mechanics model explained in the text. The error bars in each case show the standard deviation in the energy. ................................................................................................ 188 xxxvi Figure 6-13 The HDFs for a layer of Para molecules over MgO at half coverage and at temperatures 1 to 3 K. In these simulations Model I is used along with the UBE case. At 3 K, some of the molecules gain access to the ULH quantity, so the ULT value is 3 K. ................... 194 Figure 6-14 Final configurations of the MC-NVT simulated layers of Para molecules over MgO at a. 1 K, and b. 2 K using Model I. The results show an aggregated structure, which is equivalent to a hexagonal closed pack (hcp) structure depicted by a hexagon in subfigure a. In this case, this structure is very close to a c(6×2) structure shown by the rectangle in subfigure b...................................................................................................................................................... 196 Figure 6-15 The Para-Para RDFs of a film of Para molecules over MgO at half coverage and at temperatures of 1 and 2 K. In the MC-NVT simulation of these layers, Model I is used along with the UBE case. ...................................................................................................................... 196 Figure 6-16 The HDFs of a layer of Heli molecules over MgO at half coverage. In these simulations, Model I was used along with the LBE case. The data show that 10 K is the maximum temperature up to which the coverage can be kept consistent with the experiment. . 199 Figure 6-17 Final configurations of the simulated layers of Heli H2 molecules over MgO at half coverage when Model I along with the LBE case are used in the simulations. These configurations are at a. 3 K, b. 5 K, c. 8 K, and d. 9 K, respectively. The square in subfigure a indicates the c(2×2) unit cell in the layer. The arrows in subfigure d show the location of defects in the structure............................................................................................................................. 200 Figure 6-18 The Heli-Heli RDFs of a layer of Heli H2 molecules over MgO at half coverage. In these simulations, Model I along with the LBE case were used in the calculation of the energies...................................................................................................................................................... 201 xxxvii Figure 6-19 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 8 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at the temperature range of 1 to 4 K, whereas at higher temperatures, either fewer or more defects appear in the layer. The equations shown on each curve are only based on the data from 1 to 4 K and are used for the statistical mechanics model explained in the text. The bars in each case show the standard deviation in the energy. ............................................................................................................... 203 Figure 6-20 The HDFs of a layer of Para molecules over MgO at different temperatures when the coverage is kept fixed at half. In these simulations, Model I along with the LBE case were used in the calculation of the energies. At 3 K, some of the molecules gain access to the ULH quantity and hence, 3 K is the maximum temperature up to which the MC-NVT simulation conditions are consistent with the experiment.................................................................................................... 207 Figure 6-21 The final configurations of a layer of Para molecules over MgO at half coverage and at a. 1 K, and b. 2 K. In these simulations, Model I along with the LBE case were used in the calculation of the energies. The hexagon shows an hcp structure. ............................................. 208 Figure 6-22 The Para-Para RDFs of a layer of Para molecules at half coverage when Model I along with the LBE case are used in the calculation of the interaction between the molecules in the system.................................................................................................................................... 209 Figure 7-1 The HDFs of a monolayer of Heli molecules over NaCl at temperatures up to 20 K when Model II is used in the simulations. At 20 K, the function becomes nonzero at the ULH quantity of 4.0 Å. ........................................................................................................................ 217 Figure 7-2 Snapshots of the final configurations of a simulated monolayer of Heli molecules over NaCl using Model II and at, a. 3 K, b. 7 K, c. 14 K, and d. 19 K. The square in subfigure a shows xxxviii the p(1×1) structure. The arrow in subfigure d indicates the start of the movement of Heli molecules away from atop cations. ............................................................................................. 218 Figure 7-3 Heli-Heli RDFs of the simulated monolayers of Heli molecules over NaCl at temperatures up to 19 K when Model II is used in the simulations. ........................................... 219 Figure 7-4 The HDFs of a monolayer of Heli molecules over NaCl at temperatures up to 13 K using Model II. At 13 K, the HDF becomes nonzero at the ULH quantity of 3.90 Å. ............... 223 Figure 7-5 Snapshots of the final configurations of a monolayer of Para molecules over NaCl at a. 3 K, b. 7 K, and c. 12 K when Model II is used in the simulations. As shown by the square in subfigure a, the layer forms a p(1×1) structure at all temperatures shown here. ........................ 224 Figure 7-6 The Para-Para RDFs of a simulated monolayer of Para molecules over NaCl at 3, 7 and 10 K, respectively, when Model II is used in the simulations. ............................................ 225 Figure 7-7 The HDFs for a simulated layer of Heli molecules over MgO at half coverage using Model II and the UBE case. The data at 8 K show a nonzero probability at the ULH of 3.80 Å, so the MC-NVT simulations are consistent with the experiment only up to this temperature. ...... 231 Figure 7-8 Final configurations of a layer of Heli molecules at a. 3 K, b. 5 K, c. 6 K, and d. 7 K using Model II and the UBE case. The square in subfigure a indicates a c(2×2) unit cell over the surface. The four arrows in subfigure d show the four defects formed in the c(2×2) layer. ...... 232 Figure 7-9 The Heli-Heli RDFs of a simulated layer of Heli molecules over MgO at half coverage using Model II as well as the UBE case. The larger peaks appear at positions expected for a c(2×2) structure. The two bumps in the graphs at 7 K indicate the appearance of some defects in the c(2×2) structure. ................................................................................................... 233 Figure 7-10 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 6 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at the temperature range of 1 to 4 xxxix K. At 5 K and 6 K, more defects appear in the structure and they are no longer 4d-c(2×2) layers. The equations shown on each curve are only based on the data from 1 to 4 K and are used for the statistical model explained in the text. The bars in each case show the standard deviation in the energy. ......................................................................................................................................... 236 Figure 7-11 The HDFs for a layer of Para molecules over MgO at half coverage and at temperatures 1 to 4 K when Model II is used in the simulations along with the UBE case. At 4 K, some of the molecules gain access to the ULH quantity, so 4 K is the ULT value for this layer...................................................................................................................................................... 240 Figure 7-12 Final configurations of the simulated layers of Para molecules over MgO at a. 1 K, and b. 2 K, and c. 3 K, using Model II along with the UBE case. The results show a c(2×2) structure shown by a square in subfigure a. ................................................................................ 241 Figure 7-13 The Para-Para RDFs of a layer of Para molecules at half coverage when Model II along with the UBE case are used in the calculation of the interaction between the molecules in the system.................................................................................................................................... 242 Figure 7-14 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 3.5 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at the temperature range of 1 to 2 K. At higher temperatures, more defects appear in the structure and they are no longer 4d-c(2×2) layers. The equations shown on each curve are only based on the data from 1 to 2 K and are used for the statistical model explained in the text. The bars in each case show the standard deviation in the energy. ............................................................................................................... 243 Figure 7-15 The HDFs for a simulated layer of Heli molecules over MgO at half coverage using Model II and the LBE case. The data at 10 K show a nonzero probability at the height of 3.80 Å xl (ULH), so the MC-NVT simulations are only consistent with the experiment up to this temperature. ................................................................................................................................ 247 Figure 7-16 Final configurations of a layer of Heli molecules at a. 3 K, b. 5 K, c. 8 K, and d. 9 K using Model II as well as the LBE case. The square in subfigure a indicates a c(2×2) unit cell over the surface. The arrows in subfigure d show the defects appearing in the c(2×2) structure. The number of arrows in this subfigure show the number of defects formed in the layer at 9 K...................................................................................................................................................... 248 Figure 7-17 The Heli-Heli RDFs of a simulated layer of Heli molecules over MgO at half coverage using Model II as well as the LBE case. The larger peaks appear at positions expected for a c(2×2) structure. The two bumps in the graphs at 9 K indicate the appearance of some defects in the c(2×2) structure. ................................................................................................... 249 Figure 7-18 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 8 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at the temperature range of 1 to 6 K. At higher temperatures, more defects, some of which are independent of each other, appear in the structure and they are no longer 4d-c(2×2) layers. For consistency purposes, the equations of the straight lines are extracted based on the data from 1 to 4 K. The bars in each case show the standard deviation in the energy. ................................................................................................ 250 Figure 7-19 The HDFs for a layer of Para molecules over MgO at half coverage and at temperatures 1 to 6 K when Model II along with the LBE case is used in the simulations. At 6 K, some of the molecules gain access to the ULH quantity, so 6 K is the ULT quantity for this layer...................................................................................................................................................... 255 xli Figure 7-20 Final configurations of simulated layers of Para molecules over MgO at a. 1 K, and b. 3 K, c. 4 K, and d. 5 K using Model II along with the LBE case. The c(2×2) structure is shown by a square in subfigure a. .......................................................................................................... 256 Figure 7-21 The Para-Para RDFs of a layer of Para molecules at half coverage when Model II along with the LBE case are used in the calculation of the interactions between the molecules in the system.................................................................................................................................... 257 Figure 7-22 The energies of the perfect and defect c(2×2) layers at temperatures of 1.5 to 3 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at these temperatures. At 1, 3.5 and 4 K, the correlation of the data is very high, so these data were not used in the analysis here. The bars in each case show the standard deviation in the energy obtained in the MC-NVT simulations. ................................................................................................................................. 258 Figure 7-23 a. The final configuration of an MC-NVT simulated HP layer at the coverage of 0.75 and at 5 K, and b. the Heli-Heli RDF data for this layer. In these simulations, Model II along with the UBE case were used in the calculation of the energies. The square in subfigure a indicates a c(4×2) unit cell. In subfigure b, the peaks appear at positions expected for a perfect c(4×2) layer. ................................................................................................................................ 265 Figure 7-24 a. The final configuration of an MC-NVT simulated HC layer (using Model II and UBE) at the coverage of 0.75 and at 5 K, and b. the RDF data for this layer. In subfigure b, the c(4×2) RDF graph is obtained from Figure 7-23. The square in subfigure a indicates a c(4×2) unit cell. In subfigure b, the Heli-Heli RDF peaks appear at positions expected for a c(4×2) structure....................................................................................................................................... 267 Figure 7-25 a. The final configuration of an MC-NVT simulated H layer at the coverage of 0.75 and at 5 K, and b. the RDF data for this layer. In these simulations, Model II along with the UBE xlii case were used in the calculation of the energies. In subfigure b, the c(4×2) RDF graph is obtained from Figure 7-23. ......................................................................................................... 269 Figure 7-26 The final configuration of an MC-NVT simulated P layer at the coverage of 0.75 and at 5 K, and b. the RDF data for this layer. In these simulations, Model II along with the UBE case were used in the calculation of the energies. In subfigure a, the hcp and c(6×2) structures are shown by a hexagon and a rectangle, respectively. In subfigure b, the c(4×2) RDF graph is obtained from Figure 7-23. ......................................................................................................... 271 Figure 7-27 a. The final configuration of an MC-NVT simulated layer of Para molecules at the coverage of 0.83 and at 5 K, and b. the RDF data for this layer. In these simulations, Model II along with the UBE case were used in the calculation of the energies. In subfigure b, the c(4×2) RDF is obtained from Figure 7-23. ............................................................................................. 278 Figure 7-28 Schematic diagram for the vibration of a number of Para molecules along the [110] direction. The green dashed arrow indicates the maximum vibration of the Para molecules along the [110] direction. The green solid arrow shows the vibration due to the interaction between two neighboring Para molecules. The solid wine arrows show the directions along the surface of the crystal. “‒“ is an anion site whereas “+” is a cation site in the surface. The blue circles depict the Para molecules in the adsorbed layer. ......................................................................................... 283 Figure 8-1 The minimum energy of the Heli and Cart molecules when up-to-first-order (1st) and up-to-fourth-order (4th) PT terms are used in the calculation of the total energy in the Vsep case of Model II. The data are for: a. along the anion-cation-anion channel, and b. along the saddle-cation-saddle channel of the MgO substrate. .............................................................................. 297 xliii Figure 8-2 The comparison between total energies of the Heli and Cart molecules in the Vsep and Vtot cases of Model II for the MgO substrate and along the channels of a. anion-cation-anion, and b. saddle-cation-saddle. All the energies are up-to-fourth-order (4th) PT energies. . 299 Figure 8-3 Comparison of the minimum energies of the two states of the ortho-H2 molecule in the Vsep case of Model II and the three states of this molecule (MS1, MS2, and MS3) in Model III over the MgO substrate. The energies presented in this figure are only up to first order and at positions along the anion-cation-anion channel. ......................................................................... 301 Figure 8-4 Comparison of the minimum energies of the two states of the ortho-H2 molecule in Model II (the Vsep case) and the three states of this molecule (MS1, MS2, and MS3) in Model III. The energies presented in this figure are only up to first order and at positions along the saddle-cation-saddle channel of MgO. ....................................................................................... 303 Figure 8-5 Comparison of the minimum energies of the two states of the ortho-H2 molecule in Model II (the Vsep case) and the three states of this molecule (MS1, MS2, and MS3) in Model III. The energies presented in this figure are only up to first order and at positions along the saddle-anion-saddle channel of MgO. ........................................................................................ 304 Figure 8-6 Comparison of the minimum energies of the two states of the ortho-H2 molecule in Model II (the Vsep case) and the three states of this molecule (MS1, MS2, and MS3) in Model III over the NaCl substrate. The energies presented in this figure are only up to first order and at positions along the a. anion-cation-anion channel, and b. saddle-cation-saddle channel. .......... 306 Figure 8-7 The up-to-fourth-order minimum energies of a Para molecule over NaCl along the saddle-cation-saddle channel when Model II is used in the calculation of the energy in the Vsep and Vtot cases. ............................................................................................................................ 308 xliv Figure 8-8 The up-to-fourth order minimum energies of a Para molecule over MgO along the saddle-cation-saddle channel when Model II is used in the calculation of the energy in Vsep and Vtot cases. ................................................................................................................................... 311 Figure C2-1 The LJ energy for the Para/Mg2+ system in the UBE case after adding each Fourier term into the potential. (k,j) is defined in section 4.2 of the text………………………………382 Figure C2-2 The interaction of the quadrupole moment of Para H2 with the MgO surface for the Para/Mg2+ system and in the UBE case after adding each Fourier term into this potential. The (k, j) vector set is defined in section 4.2 of the text…………………………………………….383 Figure C2-3 The interaction of the quadrupole moment of Cart H2 with the MgO surface for the Cart/Mg2+ system and in the UBE case after adding each Fourier term into this potential. The (k, j) vector set is defined in section 4.2 of the text…………………………………………….383 Figure C3-1 The 2nd/1st PEPR for a system of Cart/Mg2+ in the UBE case for a. dispersion, repulsion and the sum of these interactions, b. anisotropic H2 induced dipole-surface (induction), and c. H2 quadrupole-surface interaction energy…………………………………….…………390 Figure D-1 The minimum energy of the Heli and Cart molecules when up-to-first-order (1st) and up-to-fourth-order (4th) PT terms are used in the calculation of the total energy in the V෡ୱୣ୮ case of Model II. The data are along the saddle-anion-saddle channel of the MgO substrate……....415 Figure D-2 The comparison between total energies of the Heli and Cart molecules in the Vୱୣ୮ and V୲୭୲ cases of Model II for the MgO substrate and along the saddle-anion-saddle channel. All the energies are up-to-fourth-order (4th) PT energies……………………………..…..……….416 Figure D-3 The minimum energy of the Heli and Cart molecules when up-to-first-order (1st) and up-to-fourth-order (4th) PT terms are used in the calculation of the total energy in the V෡ୱୣ୮ case xlv of Model II. The data are for: a. along the anion-cation-anion channel, and b. along the saddle-cation-saddle channel of the NaCl substrate……………………………………………...…….417 Figure D-4 The minimum energy of the Heli and Cart molecules when up-to-first-order (1st) and up-to-fourth-order (4th) PT terms are used in the calculation of the total energy in the V෡ୱୣ୮ case of Model II. The data are along the saddle-anion-saddle channel of the NaCl substrate…...….418 Figure D-5 The comparison between total energies of the Heli and Cart molecules in the Vୱୣ୮ and V୲୭୲ cases of Model II for the NaCl substrate and along the channels of a. anion-cation-anion, and b. saddle-cation-saddle. All the energies are up-to-fourth-order (4th) PT energies…………………………………………………………………………………………419 Figure D-6 The comparison between total energies of the Heli and Cart molecules in the Vୱୣ୮ and V୲୭୲ cases of Model II for the NaCl substrate and along the saddle-anion-saddle channel. All the energies are up-to-fourth-order (4th) PT energies……………………………….…....……420 Figure D-7 Comparison of the minimum energies of the two states of the ortho-H2 molecule in Model II (the Vୱୣ୮ case) and the three states of this molecule (MS1, MS2, and MS3) in Model III over the NaCl substrate. The energies presented in this figure are only up to first order and at positions along the saddle-anion-saddle channel……………………………………………….421 xlvi List of Abbreviations hcp hexagonal close packed 2D two-dimensional 3D three-dimensional HAS Helium Atom Scattering INS Inelastic Neutron Scattering ND Neutron Diffraction MC Monte Carlo IR Infrared Spectroscopy MC-NVT NVT ensemble Monte Carlo PT Perturbation Theory NDPT Non-Degenerate Perturbation Theory DPT Degenerate Perturbation Theory MD Molecular Dynamics PBC Periodic Boundary Conditions GCMC Grand Canonical Monte Carlo RAP Rotationally Averaged Potential LJ Lennard-Jones UBE Upper Bound Energy LBE Lower Bound Energy RI Range of Interest PEPR Perturbation Energy Percentage Ratio xlvii HDF Height Distribution Function MiEB Minimum Energy Barrier MaEB Maximum Energy Barrier MC-μVT μVT ensemble Monte Carlo HD-MiEB Height Dependent Minimum Energy Barrier ULH Upper Limit Height ULT Upper Limit Temperature RDF Radial Distribution Function MS Mixed State xlviii Acknowledgements To complete this project, I have had great help and support from many people whom I am very grateful to: First off, I would like to express my deep gratitude to my supervisor, Dr. David Jack, for all his support and guidance during my PhD studies at UBC. I have learnt a lot from him during my studies and the completion of this work was not possible without him as my supervisor. Also, my sincere thanks to Dr. Gino DiLabio for all his support and guidance during my PhD studies. He, with his great knowledge and insight, has helped me a lot to grow to my fullest potential and to improve my skills and confidence in my work and in my research. I would also like to thank Dr. Isaac Li for all his support and guidance during my studies here, especially with respect to the experimental concepts and techniques related to my project. My special thanks to Hossein Khalilian for all his help and support during the tough times of my studies at UBC and also for helping me with the computer resources for most of the calculations and simulations done in this study. I would also like to thank Dr. Paul Shipley, Dr. Jennifer Robison, Dr. Sandra Mecklenburg, Dr. James Bailey, Dr. Tamara Freeman, Dr. Frederic Menard, Dr. Rebecca Tyson, Mr. Wade Klaver, Mr. Olivier Dube, Mrs. Michelle Cannon, the staff of the chemistry department as well as the DRC department, and the Colledge of Graduate Studies for all their kind help and support during my PhD studies at UBC. I also owe particular thanks to the past and present members of Dr. Gino DiLabio’s research group, specially Dr. Alberto Otero-de-la-Roza, Viki Kumar Prasad, Eduardo Romero Montalvo, and Zhipeng Pei. xlix I would also like to thank the past members of Dr. Jack’s research group who had worked on this project before me, especially Christina F. Livingston, for welcoming me to this project with open arms and for keeping me motivated until the end. I also wish to thank the past and present members of Dr. Rebecca Tyson’s research group for all the motivation and energy that they gave me over all these years. I would also like to thank Moein Ahmadipour, Ahmad Rahmzadeh, Jimit Majmudar, Joey Wang, Dr. Nima Moallemi, Dr. Mitra Tabatabaee, Dr. Faezeh Dousty, Mr. Hamed Ghannad, Dr. Ehssan Hosseini Koupaie, Dr. Shelir Ebrahimi, and Dr. Sadegh Abbasian for all their kindness and support during my studies at UBC. Special thanks are owed to my beloved parents, my sisters and my brothers, who have endlessly supported me during the challenging times of my life and helped me to never give up on my dreams. Words cannot express how grateful I am to have had them beside myself in every step along the way. Last but not least, many thanks are owed to Roja Kompany, for all her support during my PhD studies and before that, for believing in me, encouraging me and helping me to follow my dreams. I would also like to thank the kind help and support from her parents and her brothers, for helping me to keep my morale high and to never lose faith in myself. l Dedication In appreciation of their contribution to this work: to all those who make the world a better place for other people to live in. 1 Introduction The process of H2 adsorption on ionic surfaces has long been of technological importance, e.g. heterogeneous catalysis, as well as fundamental scientific interest. Experiment and theory have shown that adsorbed H2 layers on ionic surfaces go through a series of structural changes and phase transitions as a function of coverage and temperature. The details of these structures and their transformations are not fully understood. As suggested by previous studies, the key point is to understand the nature of the molecular interactions, when the quantum rotational state of the H2 molecules (para-H2 or ortho-H2) is included in the interactions, and its influence on the layers’ structure. The goal of this thesis is to achieve this understanding and determine the detailed structure of H2 layers on ionic surfaces, specifically NaCl and MgO substrates. Towards this end, a hierarchy of interaction models that depend on the quantum rotational states of the adsorbed H2 molecules are presented. These models are then used to construct interaction potentials for the H2-H2 interactions as well as the H2-ion interactions. The latter potentials are adjusted to reproduce the experimental binding energies of single adsorbed molecules. These potentials are subsequently used in Monte Carlo (MC) simulations of layers of H2 molecules at a variety of coverages and temperatures to determine the layers’ structure and stability. Having determined the structural details, the frequencies of the external vibrational modes of the layers are then calculated and compared with the experimental values. The agreement is good enough that details of the underlying dynamical mechanism of the external vibrations can be determined. To present this work, the thesis is laid out as follows. In the remainder of Chapter 1, the place of this work in the overall scheme of scientific and general knowledge is discussed in section 1.2. 2 Section 1.2 reviews some of the fundamental ideas needed for a discussion of rotating molecules as well as the basics of characterizing surface structures. The remainder of the chapter (sections 1.3 – 1.6) reviews the background literature of H2 films on ionic surfaces, identifies extant problems, and expands upon the objectives and approach taken to solve these problems. The rest of the thesis presents the solutions to these problems in the following manner. Chapter 2 provides a general background on perturbation theory, degenerate and non-degenerate, as well as an overview of the Metropolis Monte Carlo method used in all of the computer simulations. Perturbation theory was used extensively in the development of the hierarchy of interaction potentials: (models I, II, and III) where models I and II are based on non-degenerate perturbation theory, whereas III is based on degenerate perturbation theory. The interaction models I, II, and III are developed and presented in Chapter 3 and subsequently parametrized in Chapter 4. The parametrization is usually accomplished through comparison with experimental quantities, but the repulsion parameters are adjusted to ensure that the calculated heats of adsorption match the experimental values. Because the rotationally dependent molecule-molecule (H2-H2) interaction potential is new and unique, Chapter 5 is devoted to a discussion of its development, form, and properties. With the tools explained, the results of the Monte Carlo simulations of films of molecular hydrogen adsorbed on NaCl and MgO surfaces are reported. The results from using interaction models I, II, and II, are presented in Chapter 6, Chapter 7, and Chapter 8, respectively. And finally, the conclusions are given in Chapter 9. 3 1.1 General considerations Hydrogen is the most abundant molecule in the universe1 and is considered a crucial material for industrial processes2. Its high availability1, high energy density3, wide range of potential sources, and its small impact on the environment4–6, along with its light weight for mobile applications7 make it a potential future source of energy and an alternative to fossil fuels. The fact that hydrogen can be more compact when absorbed within solid systems than in the bulk liquid has made the adsorption of hydrogen on solids an attractive process in the hydrogen storage industry8, both in terms of understanding the precursor stage to absorption as well as characterizing the strength of the molecule substrate/absorbent interactions. The examination of hydrogen on ideal ionic surfaces is particularly attractive in this regard. The adsorption of H2 on ionic surfaces at low temperatures and pressures attracted particular interest after it was suggested that the two-dimensional (2D) adsorbed layers may show unusual physical properties9–12. It was also suggested that these properties are caused by quantum effects which are mostly due to the light mass of the H2 molecule. These effects are also important in space programs where conditions of low temperature and pressure are encountered1,13–15. Although experimental and theoretical studies have partly explored the energetics of adsorption and the structural properties of H2 layers above ionic surfaces at low temperatures and pressures9,10,16–21, a complete understanding of the energies, structures, and the elementary steps in the adsorption process at these thermodynamic conditions is still lacking. This includes many unclarified cases such as the in-layer and layer-by-layer adsorption mechanism22,23, film composition9,24, layer energies, structures, and stabilities, the ortho-para inter-conversion in different layers9,10,25, the film energy corrugation profiles22,23,26,27, the surface topography of the layers27, and the film’s phase states at different surface/gas temperatures and pressures9,28. 4 Understanding these systems can also help us to resolve discrepancies in the experimental findings of the layer growth mechanism at low temperatures22,23,29–31. They may also provide the basis for a more accurate description of the interconversion between H2 quantum species (para-ortho conversion) on surfaces, a phenomenon that is of great importance in the industry (outer space programs for example)1,32–34. On the other hand, the conceptual and practical simplicity of the hydrogen adsorption on ionic surfaces at low temperatures and pressures, makes them ideal systems to test theories in quantum chemistry and reduce ambiguities in physisorption processes9,22,30,35–37. In fact, a systematic theoretical study for H2 film adsorption on surfaces is extremely useful due to the lack of a general quantum mechanical theory of adsorption. In particular, one would like to describe the interactions between molecules in restricted geometries where quantum effects play a significant role in determining the properties of the system11,38,39. In addition to this logical formalism, such a study aids in theoretical interpretations of processes in the most basic quantum mechanical molecule-adsorbate systems10,30,35. It also helps in finding the drawbacks of the experimental techniques in probing the properties of interest40. Through understanding these phenomena, one can gain fundamental insight into the structural and dynamic properties of molecules in interfacial systems11,37,41; a knowledge which is also of great importance in related areas such as studying superfluid materials9,28,38, dielectrics42, heterogeneous catalysis20,43,44 and separation processes18,23,26,43,45. It should be added that experiments and theory have shown dramatic changes to the characteristics of the H2 systems in 2D films compared to three-dimensional (3D) bulk41. This was considered as a reasonable sign of observing unusual physical properties in the adsorbed H2 layers and/or sub-monolayers on the surfaces9,21,28,37,46. To study these, some physical properties 5 such as density21,28, structure factor38, and heat capacity9,21,28 of the system are measured. In fact, the key point in observing any changes compared to the bulk solid is due to the change in the inter-particle (H2-H2 or H2-molecule) interactions, which are in turn affected significantly by the substrate or even by the presence of impurities in the system20,38,41. In addition, experiment and theory have both shown the effect of the quantum rotational states of the H2 species on the physical properties of the adsorbed H2 layers47. Predictions from theoretical calculations can aid in understanding the qualitative properties of the possible phases formed over the surface, quite similar to the predictions made in the bulk (solid) H2 prior to its experimental discoveries47. 1.2 Basic concepts 1.2.1 Rotational states of H2 Because of the light mass of the H2 molecule, in the gas phase (isolated phase), the molecular axis of this molecule becomes delocalized and rotates freely in space. In quantum mechanics, this rotational motion is formulated using the rigid rotor model1,2. The solutions to this model yield probability distributions of the molecular axis of the H2 molecule around a fixed axis in the space (henceforth referred to as the axis of quantization, see Figure 1-1). These distributions are called the rotational states of the H2 molecule and are in the form of the spherical harmonic functions (see Eq. (3-3)). The rotational states of the H2 molecule in the rigid rotor model are in fact quantized and are shown with two quantum numbers: 1) the rotational quantum number, 𝐽, and 2) the magnetic quantum number, 𝑚௃, where 𝐽 = 0, 1, 2, 3, . .. and |𝑚௃| ≤ 𝐽 48–50. The values of 𝐽 ae grouped according to whether they are odd or even integers. The group of even integers are linked to the singlet spin state (para) formed by the pair of hydrogen nuclei whereas the odd integers are connected with the triplet spin state (ortho). Thus, the H2 molecules are classified as 6 para-H2 (with even 𝐽 values) or ortho-H2 molecules (with odd 𝐽 values)51. The natural mixture of the para and ortho states are referred to as normal-H2 or n-H2 which at room temperature has an ortho:para ratio of 3:1 owing to the relative degeneracies of their nuclear spin states. This ratio will persist to low temperatures provided inter-conversion between the para and ortho states does not occur. In general, the energy of these states depend only on their 𝐽 values, and not 𝑚௃ (see Eq. (3-2)) and increase as 𝐽 increases in value. This also means that to populate the higher energy levels, higher temperatures of the system are required51. Conversely, at low temperatures the para and ortho series will collapse to their lowest energy states, 𝐽 = 0 (ாబ௞ಳ = 0 K) and 𝐽 = 1 (ாభ௞ಳ = 174 K), respectively. It should be noted that, in the field of H2 adsorption on ionic surfaces, the systems of H2 over ionic surfaces have only been investigated up to 30 K9,11,27–31,41,45,52–54,16,55–64,17,65,18,20,22,23,25,26. This means that, for all intents and purposes, only the ground 𝐽 = 0 and 𝐽 =1 states are present and relevant. As is typical for work at these low temperatures, it is understood that, para- and ortho- refer to the 𝐽 = 0 and 𝐽 = 1 rotational states, respectively. The 𝐽 = 1 state is three-fold degenerate (mJ = 0, ±1) with three possible distributions of the axis of rotation of the ortho-H2 molecule with respect to the axis of quantization. The 𝑚௃ = ±1 states are called helicoptering-H2 (Heli) and the 𝑚௃ = 0 state is called cartwheeling-H2 (Cart). Figure 1-1 shows these ground rotational states of the H2 molecule. 7 Figure 1-1 The delocalization of the molecular axis around the axis of quantization. This results in a molecular-axis distribution that forms the shape of the molecule for the ground state ortho- and para-H2 molecules. 1.2.2 The adsorbed H2 film structures 1.2.2.1 The unit cells For adsorbed H2 layers over ionic surfaces, two types of the structures with rectangular symmetry are present. Based on symmetry, these structures will be classified as c-type and p-type structures66. Figure 1-2 illustrates these structures. The c-type unit cell is a structure in which the molecules at the corners of the unit cell are the same as the molecule at the center. In the p-type structure the molecule at the center is different. The case where there is no molecule at the center of the unit cell is also called a p-type structure66. Figure 1-2 The c-type (left) and p-type (right) unit cells for the adsorbed H2 films over ionic surfaces. 8 1.2.2.2 The film structures Figure 1-3 demonstrates the c(n×2) structures formed by H2 molecules over the MgO substrate10,20,58. Two different types of the H2 molecules are shown in the legend of this figure. In all of these structures, the numbers in the brackets (for example, n) shows that the nth same interacting site, along either the horizontal or vertical channel, is occupied by the same-type adsorbed molecule in the unit cell. For example, in a (4×2) structure, the number 4 shows that every 4th cation along the horizontal channel is occupied by the same H2 molecule. The number 2 shows that every 2nd cation along the vertical channel is occupied by the same H2 molecule58. In Figure 1-3, a hexagonal close packed (hcp) like structure is also shown by a green hexagon superimposed on both c(4×2) and c(6×2) structures. The term “hcp” is sometimes used as an alternative name for both the c(4×2) and c(6×2) structures. 9 Figure 1-3 The c(n×2) structures formed by H2 molecules on MgO. The H2 molecules are in two different types. The n number in the brackets shows that every nth cation along the horizontal and vertical channels is occupied by the same H2 molecule. The hexagon depicts an hcp-like structure, sometimes used to characterize both the c(4×2) and c(6×2) structures. 1.3 Field Background Studies of H2 films on ionic surfaces are mostly based on the adsorption (physisorption) of H2 on MgO (H2/MgO) and on NaCl (H2/NaCl). There are also a few studies about the adsorption of H2 on the LiF substrate19,53–55, but they are not discussed in this dissertation. For the H2/MgO and H2/NaCl systems, the main discussion will be about the binding energies of the H2 molecules and 10 films to these substrates as well as the structures of the H2 films over the surface. Also, the external vibrations observed for these adsorbed H2 films as well as some other properties that have been investigated in previous studies on H2/MgO and H2/NaCl systems will be discussed. 1.3.1 H2 on MgO 1.3.1.1 Binding energies The first energy (‒26.3 (±0.3) meV or ‒0.607 (±0.007) kcal/mol) reported for the binding energy of an H2 molecule over MgO was in a review paper by Vidali et al.67. This binding energy was based on a private communication with Vilches, but it is not known how this data was measured. It is also not known what the rotational state of the H2 molecule was. This binding energy was also used in the theoretical studies by Vidali and Karimi24,40 to calculate some parameters in their classical potential model for the interaction between a single H2 molecule and the MgO surface. In this model, the interaction between the quadrupole moment of the H2 with the surface was not included in the potential. In order to fit the binding energy to the experimental data, they adjusted the electric charge of the ionic surface (q=1.51) in their potential model. They also adjusted the long-range dispersion coefficient (C6) in their calculations. Years later, Toennies and Traeger37 reported the energy of a layer of H2 molecules over MgO at 8.88 K. This energy was ‒28.8 meV (‒0.66421 kcal/mol), which seems to be based on another study in Vilches’ group28. No information was given for the rotational states of the H2 molecules in the layer. A later Monte Carlo (MC) simulation study25 by using classical potential models on layers of adsorbed H2 molecules over MgO, used this energy as the binding energy of an H2 molecule to the MgO surface. These simulations allow the molecules to move freely over the 11 surface subject to a surface potential. The results of this study were qualitatively in agreement with the experiment. In another study10, the binding energy of an H2 molecule over MgO was reported as ‒20 to ‒25 meV (‒0.46125 to ‒0.57657 kcal/mol), which was based on a private communication with Bruch. Larerse et al.20 used this energy to fit the binding energy of a para-H2 molecule to the MgO surface in their semi-empirical model. For the repulsion potentials they used atom-atom interactions. Their dispersion interaction potential is a sum of only H2-O2- interactions and the H2-Mg2+ dispersions were neglected in the calculation of this energy. No information regarding the details of the equations used in their potential model for the electrostatic interactions were presented. Using their potential model, they could reproduce the experimental data (the binding energy of Ref. 10) only when the surface ionic charge of the MgO surface was adjusted from q=2 to 1. 1.3.1.2 Structures The studies on the structures of hydrogen molecules on MgO started by a Neutron Diffraction (ND) experiment on films of adsorbed D2 molecules over MgO58. Based on the experimental data, the authors suggested the layer formed a series of c(n×2) structures which underwent a sequence of structural phase transitions: c(2×2) → c(4×2) → c(6×2), (see Figure 1-3). that were a function of coverage and temperature58. Later, Helium Atom Scattering (HAS) measurements10,37 of adsorbed n-H2 films on MgO showed sequential c(n×2) patterns similar to what had previously been shown by ND on D2/MgO58. For para-H2, however, at half coverage (θ = 0.5), evidence of islands of c(2×2) alongside disordered patches was observed or of other structures (maybe c(4×2) or c(6×2)) were 12 suggested. At higher coverages (0.75 and 0.83) there were indications of a transformation to other (n×2) structures, occurring in islands or as multiple small domains, but no definitive assignment could be made as to the values of “n” or whether the surface unit cell symmetry was c-type or p-type68. Subsequent MC simulations using classical potential models provided insight into the layer structures and stabilities of H2 25 and D2 62,63 films on MgO. The calculated data were in good agreement with the experiments10,58 in terms of coverage and stability. There was, however, a discrepancy in terms of symmetry. The simulations yielded p(n×2) rather than c(n×2) structures (for n-H2 layers) 25,62,63. This was resolved by showing that the barrier to molecular rotational motion was lower than the rotational zero-point motion of the molecule itself, thus, delocalizing the molecular axis and converting the p-type to c-type structures. So, although these classical simulations elucidated the layer structure, they did not directly take into account the quantum mechanical rotational state of the molecules, but instead reconciled the quantum nature post simulation. But this reasoning was based on a very simple model that could only be applied to cases where the H2 molecule sits over the cation or the anion site. Another limitation in their quantum model was the simplifying assumption that the height of the H2 molecule over cation/anion had to be kept fixed and was equal to the value obtained in their classical (potential) model calculations. Nonetheless, these MC studies25,62,63 provided some valuable details of the structures of H2 films over MgO at different coverages. Using ND measurements, Larese and coworkers11 examined the structural patterns of D2 films on MgO. In the sub-monolayer regime, they observed a c(2×2) structure at half coverage, but at higher coverages up to monolayer, the structure was suggested to be an hcp structure (see Figure 1-3). It was also suggested that this hcp layer could be a sequence of p(4×2) and p(6×2) 13 structures. This was in contrast with the previous experiment58 that had suggested c(4×2) or c(6×2) structures for D2/MgO at these coverages. Also, in a later paper41, Larese and coworkers mentioned observing a sequence of c(2×2), p(4×2), and p(6×2) structures for H2/MgO at different coverages. In these experiments, there is no information about the rotational state of the H2 molecules in the layer. Besides, no experimental data was presented for these structures. Nonetheless, their structural data at high coverages (p(4×2) and p(6×2)) is in contrast with the experimental results10,37 that reported c(4×2) and c(6×2) structures for layers of n-H2 molecules but may be consistent with the results for para-H2 where mixtures of (n×2) structures were suggested58. 1.3.1.3 Other properties The first work on exploring the superfluid properties of H2/MgO systems was done in Vilches’ group. In a search for these layers over the surface, they conducted heat capacity and adsorption-isotherm measurements during layer by layer growth of H2 above the MgO surface. By considering the critical and triple-point temperatures, the phase states of the grown layers up to the sixth layer were determined and the criteria for superfluidity of the layers were discussed 21,28. The triple point of the layers was found to be higher than that expected for a superfluid phase and it was concluded that superfluidity was not present28. In another search for superfluidity in adsorbed layers of H2, Vilches9 studied the phase diagrams of the pure and mixed hydrogen isotopes on MgO. Whereas no evidence of the superfluidity in the formed layers was reported, his findings showed a lower triple point for the second adsorbed layer of a mixture of H2 and D2 than that for the second layer of the pure H2 adsorbed bilayer9. 14 In a different study10,37 on the vibrations of the molecules in the adsorbed layers, HAS measurements were used. Three vibrational modes for the layers at high coverages (either hcp or c(4×2) or c(6×2)) were reported. Although these vibrations were suggested to be due to the vertical (S-mode) and horizontal (T-mode) vibrations of the molecules, they could not be accurately assigned to the molecules in the layer since the composition and identities of the molecules could not be determined10,37. Bruch27 tried to reproduce these frequencies by using vibrational frequency calculations. He extrapolated the parameters in his model using the data from similar systems of Ar/graphite and Kr/graphite. To do this extrapolation, he used the simple classical potential model proposed by Vidali and Karimi24,40 (discussed in section 1.3.1.1). Using his vibrational frequency model, Bruch could only assign one of the experimentally observed vibrations to the oscillation of a single H2 molecule perpendicular to the surface. He proposed that the other experimentally observed vibrations can be further clarified by the corrugation energy profiles as well as the topography of the layers that are obtained by potential models that include the H2 molecule’s quantum rotational states in the interactions. On the other hand, the dynamical response of the adsorbed H2 films on MgO was studied in a combined Inelastic Neutron Scattering (INS) and ND experiment11. From an examination of the para- to ortho-H2 transition peaks (at 7.35 meV or 0.16951 kcal/mol) in the INS spectra for the adsorbate layers, they concluded that from the third layer up the adsorbed H2 films show bulk solid-like behaviour indicating the weak impact of the surface electric field on the molecules in these layers. Quite recently in Larese and Hinde’s research group, Rouvelas52 studied the effect of the surface electric field on the rotational energies of the single adsorbed H2 molecules at different rotational levels. These calculations were mainly to investigate the effect of the polar (𝜃) and 15 azimuthal (𝜑) angle dependence of the H2-surface potential on the degeneracy of the different rotational states of H2 molecules in the adsorbed phase. Their potential model is a simple two-term model, which is based on the polar and azimuthal angles of the H2 molecule. The first term only includes the polar (𝜃) angle, whereas the second term includes both the polar (𝜃) and azimuthal (𝜑) angles. To each term is associated a parameter that is systematically changed in the calculations to examine how each term (angle) in the H2-surface potential model affects the rotational energies of the H2 molecules in the adsorbed phase. 1.3.2 H2 on NaCl 1.3.2.1 Binding energies The first study for the adsorption of H2 above an ionic surface (NaCl) was by Folman and Kozirovski56. Using adsorption isotherms, they obtained the binding energy for an H2 molecule to the NaCl surface as ‒1.3 kcal/mol (or ‒56.4 meV). To help interpret their experimental data, they used classical calculations to determine the binding energy as well as the most favorable interacting sites in the surface for a single H2 molecule over the NaCl surface. The strongest binding energy (‒1.5 kcal/mol or ‒65.04 meV) of an adsorbed H2 molecule was found to be above a cation with the molecular axis oriented parallel to the surface and aligned towards neighbouring chlorine ions. In a later study57, they showed that the discrepancy between their calculated results (binding energies for example) and the experimental data can be reduced significantly by including the distortion of the surface. Using adsorption isotherm measurements, Dai and Ewing16 determined the heat of adsorption of hydrogen isotopes on NaCl. Their experimental energy was reported as ‒0.7 (±0.2) kcal/mol (or ‒31 (±9) meV) for a para-H2 molecule over the surface. They also calculated this energy 16 using a semi-empirical statistical mechanics model based on their measured isotherms. Their calculated binding energy is ‒0.9 (±0.2) kcal/mol (or ‒39 (±9) meV) for a para-H2 molecule over the surface. Since, in their calculations, the interacting sites in the NaCl substrate are treated as a combination of defect and smooth-face sites, their lower calculated binding energy compared their experimental data was concluded to be due to the defect sites in NaCl16. Heidberg and co-workers45 used IR measurements to determine the desorption energies of the para- and ortho-H2 molecules above NaCl. Their measured binding energies were ‒0.7 (±0.2) kcal/mol (or ‒31 (±9) meV) for para-H2 and ‒0.9 (±0.2) kcal/mol (or ‒38 (±9) meV) for ortho-H2. Based on the difference between these energies, the separation of the ortho- and para-H2 molecules by desorption was proposed45. Briquez et al.26 calculated the binding energies of single para- and ortho-H2 molecules on the NaCl surface using quantum calculations. Their potential model includes all the dispersion, repulsion, and the electrostatic interactions as well as the internal vibrations and zero-point motion potentials. But it is not known how much each one of these terms contribute to the total binding energy of the H2 molecule to the NaCl surface. Nonetheless, the adsorption energies for the H2 molecule above the NaCl surface were calculated as ‒0.65 kcal/mol (or ‒28.2 meV) for para-H2 and ‒0.81 kcal/mol (or ‒35.2 meV) for ortho-H2. Although their quantum mechanical energies exceed the binding energies of the preceding classical calculations56, they are in good agreement with the experimental results of Refs.16,45. A later experimental study by Traeger and Toennies22 measured stronger binding energies for the para- and ortho-H2 molecules over NaCl compared to the experimental and theoretical results of Refs.16,26,45. This new study measured binding energies of ‒0.85 (±0.11) kcal/mol (or ‒37 (±5) meV) for para-H2 and ‒0.92 (±0.11) kcal/mol (or ‒40 (±5) meV) for ortho-H2. 17 1.3.2.2 Structures The structures for the H2 molecules over NaCl was first predicted by Ewing16 based on an earlier study69 in his research group. In this structure, all the H2 molecules sit over the cations (Na+) in the surface (in the form of a p(1×1) structure). In a later study65, Ewing and his coworkers suggested a second adsorption site (Cl-) in the NaCl substrate as well. These interacting sites were suggested to be occupied after all the Na+ sites so that a second hydrogen adlayer could form65. Later experimental studies22,23,26,37 confirmed these structures and concluded that the monolayer para- and ortho-H2 molecules are in the form of a p(1×1) structure. Classical MC simulation studies of Dawoud et al.25,31 on H2/NaCl confirmed these structures. They also showed orientational preference of the H2 molecules over the cation sites, which was in agreement with the experiments29. These studies could also resolve some other ambiguities in the experimental data. For example, in the study by Grunwald and Ewing29, a denser monolayer of H2 molecules above the NaCl surface was proposed with molecules adsorbed on both Na+ and Cl- sites29. Although, initially, Heidberg et al.23 argued that steric hindrance ruled out the simultaneous adsorption above anions and cations, but the mentioned classical MC simulations25,31 showed that this effect is avoided due to the difference in the height and energies of molecules adsorbed on the two different sites (Na+ and Cl-). This could not be explained experimentally due to the restrictions on the operational temperatures and pressures (coverage) as well as the large amounts of uncertainties associated with the data in each of these studies25. 18 1.3.2.3 Other properties Besides energies and structures, there are also some other properties of H2/NaCl systems that have been investigated by experiments. The most relevant one is probably the growth mechanism of the adsorbed H2 molecules over NaCl. Heidberg and co-workers23 are the first group who tried to explain this mechanism for the adsorption of n-H2 (with 75% ortho- and 25% para-H2) layers above the NaCl surface. To do that, they calculated the sticking coefficients of the ortho- and para-H2 molecules on NaCl. The preferential adsorption of the para-H2 molecules at low temperatures and coverages was shown to be due to the larger sticking coefficients for these species. This was concluded that is an indication of kinetic control (rather than thermodynamic control). That is, the para molecules adsorb more readily and quickly that the ortho molecules even though ortho molecules have a stronger binding energy. These coefficients were approximated by fitting the experimental data to the Langmuir Model and using the extended Langmuir equations23. At higher temperatures the para-H2 molecules in the layer are replaced by more strongly bound ortho molecules and this was explained to be because of the larger binding energies of the ortho-H2 species to the surface. They concluded that this phenomenon is a sign of thermodynamic control (not kinetic control). In a later HAS study22, the adsorbed para/ortho ratio was shown to be strongly dependent on the applied H2 gas pressure, a parameter that was not been considered in Ref. 23. Therefore, it was suggested that the mechanism must be more complicated than that proposed in Ref. 23. 1.4 Summary and problems in the field In the experimental studies mentioned above, the exact positions and quantum rotational identities of the H2 molecules in the adsorbed phase is not known. This information is crucial in 19 answering the questions and ambiguities remaining in the field for the past 17 years. By using classical and quantum mechanical calculations for a single molecule, only the identities, orientations, and positions of the molecules right above the cations were predicted. It was proposed that since the helicoptering-H2 (Heli) molecules have the highest binding energy when they sit above cations and since the observed peaks in the diffraction patterns correspond to the distance between the neighboring cations in the crystal, the observed patterns must be due to the adsorption of Heli molecules above the cations. Using quantum mechanical calculations for a single molecule26, the heights of para- and ortho-H2 right above the cations and anions in NaCl were obtained. These heights were 2.57 Å for the para- and helicoptering-H2 molecules over the Mg2+ site. More details of the adsorbed layer at different coverages and temperatures were obtained using classical MC simulations25,31, but since the adsorbed molecules were treated as classical objects, their quantum identities (rotational state) remained undetermined. These calculations also showed that the experimental data cannot be completely reproduced if quantum effects are not considered in the simulations. In particular, the symmetry of the simulated layer was not in accord with the experimental findings. Their subsequent quantum mechanical calculations25,61,63 outside the simulations resolved this problem by taking into account the quantum mechanical delocalization of the molecular axis. However, these calculations were performed for only a single molecule above each of the adsorption sites (cation, anion, and saddle). This two-stage approach does not provide a way of including quantum effects in the interaction between molecules. Thus, the quantum rotational identities of the molecules in the adsorbed layer remained unspecified and there remains uncertainty as to the details of the structure such as the location and height of the molecules and thermal stabilities of the sub-monolayer H2 films. 20 In the experimental search for superfluidity9,28, layers of n-H2 on MgO were examined. These studies showed that the first two layers were solid, but without having access to the structural details of the upper layers the interpretation of the experimental data for the assignment of physical properties and phase changes could not be made, blocking the search for superfluidity. Thus, knowing the structural details of multiple layers of H2 would aid in the search for evidence of superfluidity in hydrogen films. As discussed in section 1.3.1.1 in the description of Larese et al.’s work20, the adjustment of ion charge at the solid surface to fit experimental data may not be the correct way to calculate the binding energies. Moreover, HAS experiments10 on H2/MgO confirmed the unit cell symmetry found in ND studies58 on D2/MgO. But subsequent experiments on D2 20 and then H2 41 using ND resulted in symmetries slightly different from HAS, leading to ambiguity in the experimental record. Besides structure the HAS experiments were also able to measure the external vibrational energies of the H2 molecules within the film and with respect to the surfaces. A full understanding of the structures as well as the external vibrational modes has yet to be achieved. As noted in Bruch’s discussion27 of these modes in section 1.3.1.3, to fully explain the observed frequencies the energy topography and corrugation of the layers must be developed through accurate quantum potential models. A reference key for the studies in the field on the H2/MgO and H2/NaCl systems is listed in Table 1-1. 21 Table 1-1 The findings and questions in the field for H2/MgO, H2/NaCl. MgO NaCl Binding Energies Experimental Single Molecule ortho-H2 ? ? para-H2 ? ? Monolayer n-H2 ?  22,23,26,37 para-H2 ?  16,22,23,26,37 H2 ?  56 Sub-layer n-H2 ? ? para-H2 ? ? Theoretical (quantum) Single Molecule ortho-H2 ?  26 para-H2 ?  26 Patterns Experimental  10,11,20,37,41,58  16,22,23,37,65 Theoretical Classical  25  25 Quantum ? ? Adsorbate Vibration Experimental  37  37 Theoretical ? ? Potential Models Theory Classical Single Molecule  24,25,40  24,25,40,56 Layer  25  25 Quantum Single Molecule ?  26 Layer ? ? 22 All of the experimental methods have limitations, drawbacks, and ambiguities of which some of the most important are given in the following list: 1- The cryogenic temperatures are difficult to achieve29. 2- The recorded temperatures and pressures might not be the actual system temperatures10. 3- Large uncertainties in all experimental data obtained so far makes it difficult to compare with simulation data25,37. 4- The rotational states of individual H2 molecules cannot be determined in the experiments. One consequence of this inaccessibility is that the external vibration data from HAS cannot be assigned to the H2 species in the adsorbed film10. 5- The positions of the H2 molecules above the surface cannot be determined10. 6- It is very difficult to maintain a clean surface during the experiment29. Computer simulations can help resolve the ambiguities and overcome the limitations above. 1.5 Questions in this thesis and the hypothesis As shown in Table 1-1, there are still many open questions in the field, some of which have been left unanswered. Although some efforts have been made so far to interpret the observed experimental results, without a thorough and accurate potential models that include the quantum rotational states these explanations may not be reliable. The main purpose of this research is to answer these questions by developing appropriate potential models and using them in computer simulations. To do so, one should think of answering the most important and basic questions in the field systematically: 1- How does the structure and composition (para:ortho ratio) of monolayer and sub-monolayer films vary with temperature and coverage? 23 2- What are the possible phases and phase transitions in the monolayer and sub-monolayer films? Are they restricted to a specific composition? What about the temperature of the layer? 3- How do the adsorption energies of the layers differ with changing temperature and coverage? 4- What are the frequencies of the external vibrations of para and ortho-H2 molecules above the surface? How do these depend on the structure and composition? 5- What are the desorption temperatures for each layer? Therefore, the central hypothesis of the proposed research may be stated as follows: The energetics, structure, and stability of the H2 films on ionic surfaces depend on the quantum mechanical rotational state of the individual H2 molecules. It is felt that only by exploring the validity of this hypothesis will a comprehensive understanding of H2 films on ionic surfaces be gained. 1.6 Approach Although classical theoretical studies coupled with single-molecule quantum mechanical calculations of the adsorbed H2 films provided some new information about the structure of these layers25,31,61,63,64, there are still many open questions in the field. To answer these questions computer simulations must be used to obtain the structural details and properties of the adsorbed quantum H2 layers. The accuracy of the results depends strongly on the potential models that are used. Until now, there was no model of H2-H2 and H2-substrate interactions that incorporated quantum mechanical rotations that was suitable for computer simulations. One of the primary goals of this thesis is to fully develop such a potential. A simple form of this potential was used 24 to successfully reproduce some of the experimental data of the H2/MgO system70 (for more detailed information, see also Chapter 3). This potential is developed more fully into a hierarchy of potentials of increasing accuracy. After developing this hierarchy, the developed potential models are validated to assure that they can reproduce experimentally observed energies and structures. To do so, the adsorption of a single molecule above the surface is checked and either compared with or fitted to the experimental data based on the limitations in the calculations (see Chapter 4). These potential models are then used in MC computer simulations to simulate the whole layer. Starting from low coverages, the structures and stabilities of the layer are examined, and information gathered about the rotational distributions, binding energies, corrugation profiles, and topography of each of the observed phases within the layers. This approach and techniques will be applied to systems of H2/MgO and H2/NaCl. 25 Theory 2.1 The formulation of the adsorption system In this thesis, the end goal is to use NVT ensemble Monte Carlo (MC-NVT) simulations70,71 to study layers of H2 molecules on ionic surfaces (see also section 2.3). In all these simulations, the heart of the algorithms is the potential model that governs the interactions between the molecules in the system 72–75. In our adsorption system, the potential models are used to calculate the interactions between H2 molecules and these molecules and the ionic surface. These interaction potentials must account for and include the rotational quantum state of the H2 molecule in the (see section 3.1). Thus, the adsorption system in our study is partially a quantum mechanical system and to develop the interaction potential the first step is to solve the Schrödinger equation48,49: 𝐻෡𝛹௡ = 𝐸𝛹௡ (2-1) where the Hamiltonian operator 𝐻෡ = 𝐾෡ + 𝑉෠ contains contributions from the kinetic energy operator (𝐾෡), which is related to the dynamics of the particles, and the potential energy operator (𝑉෠ ), which is related to the interactions between the particles in the system. These operators are defined in a similar way as in classical mechanics with the difference that they are now operators instead of observables48,50. Solving an eigenvalue equation like the Schrödinger equation yields the energy (𝐸) as the observable eigenvalue of the system as well as the eigenfunction, 𝛹௡, also known as “wavefunction” or state function. The subscript “n” is typically a set of integers that specifies the quantum mechanical state. According to the first postulate of quantum mechanics, the wavefunction completely specifies the state of the system48. The square of this function gives 26 the probability density of finding the particle in a particular location and/or orientation in space49. Usually only approximate solutions and energies can be obtained. In the next section, an important method to approximately solve the Schrödinger equation is presented. 2.2 Perturbation Theory Although for some quantum mechanics systems the Schrödinger equation can be solved exactly, for many of them, finding the exact analytical solution to this equation for the energies and wavefunctions is impractical or impossible. Instead, approximate methods are used to find the energies and wavefunctions of these systems. One of these methods is called the Perturbation Theory (PT). In this method, the Hamiltonian of the system is divided into two parts; an “unperturbed” reference system that can be solved exactly and the remainder of the Hamiltonian that is regarded as the perturbation. The total Hamiltonian is then written as49,50: 𝐻෡ = 𝐻෡௨ + 𝐻෡௣ (2-2) where 𝐻෡௨ is the Hamiltonian for the unperturbed reference system, and 𝐻෡௣ is the Hamiltonian for the perturbed system. The Schrödinger equation then becomes49,50: ൫𝐻෡௨ + 𝐻෡௣൯𝛹௡ = 𝐸𝛹௡ (2-3) The perturbation approach applies a series of consecutive perturbation corrections to both the energy and wavefunction of the unperturbed reference system. In this series, the reference system is the zeroth-order perturbation correction and the whole system is described through the addition of higher-order corrections. The energy and the wavefunction of the whole system are then defined as the sum of these correction terms as follows48: 𝛹௡ = 𝛹௡(଴) + 𝛹௡(ଵ) + 𝛹௡(ଶ) + ⋯ (2-4) 27 𝐸௡ = 𝐸௡(଴) + 𝐸௡(ଵ) + 𝐸௡(ଶ) + ⋯ (2-5) where 𝐸௡ and 𝛹௡ are in turn the energy and wavefunction of state 𝑛 in the perturbed system. 𝐸௡(୫) and 𝛹௡(୫) are the mth-order correction terms to the energy and wavefunction, respectively. These correction terms usually get smaller with increasing “m” and the series is truncated when the corrections become negligible. The PT calculations are usually formulated in two different methods. If none of the eigenstates of the reference (unperturbed) system has the same energy, then the PT method used in the calculations is the Non-Degenerate PT (NDPT) method48–50. If some of the eigenstates of the reference system have the same energy (degenerate states), then the correct PT method is the Degenerate PT (DPT) method48–50. In this case, the perturbation potential removes (completely or partially) the degeneracy in the eigenstates of the reference system and replaces them with an eigenstate which is a linear combination of the unperturbed eigenstates. More information about these two methods can be found elsewhere48–50. Here, only the final equations for the PT correction terms in the energy and wavefunctions for each of these two methods will be presented. 2.2.1 NDPT method In this method, the corrections to the energy and wavefunction are calculated based on the reference state 𝛹௡(଴). The first-order correction to the energy is48–50,76: 𝐸௡(ଵ) = 𝑉௡௡ (2-6) where 𝑛 denotes the 𝛹௡(଴) eigenstate of the reference system and 𝑉௡௡ =< 𝛹௡(଴)ห𝐻෡௣ห𝛹௡(଴) >= න 𝛹௡(଴) · 𝐻෡௣ · 𝛹௡(଴) 𝑑𝑞 (2-7) 28 where 𝑞 shows that the integration is taken over the independent variables of the system. The second-order correction to the energy is48–50,76: 𝐸௡(ଶ) = ෍𝑉௡௠ 𝑉௠௡𝐸௡௠௠ஷ௡ (2-8) where 𝑛 and 𝑚 denote the 𝛹௡(଴) and 𝛹௠(଴) eigenstates of the reference system, and 𝐸௡௠ =𝐸௡(଴) − 𝐸௠(଴) in which 𝐸௡(଴) and 𝐸௡(଴) are the corresponding unperturbed energies for these wavefunctions. Following Szabo76, the third-order correction to the energy is: 𝐸௡(ଷ) = ෍ ෍𝑉௡௠𝑉௠௞𝑉௞௡𝐸௡௞ 𝐸௡௠௠ஷ௡௞ஷ௡− 𝑉௡௡ ෍|𝑉௡௜|ଶ𝐸௡௜ଶ௜ஷ௡ (2-9) and the fourth-order correction is76: 𝐸௡(ସ) = ෍ ෍ ෍𝑉௡௠ 𝑉௠௞ 𝑉௞௜ 𝑉௜௡𝐸௡௜ 𝐸௡௞ 𝐸௡௠௜ஷ௡௞ஷ௡௠ஷ௡− ෍ ෍|𝑉௡௠|ଶ𝐸௡௠ଶ.|𝑉௡௜|ଶ𝐸௡௜௜ஷ௡௠ஷ௡− 2 𝑉௡௡ ෍ ෍𝑉௡௠ 𝑉௠௞ 𝑉௞௡𝐸௡௞ଶ 𝐸௡௠௞ஷ௡௠ஷ௡+ 𝑉௡௡ଶ ෍|𝑉௡௠|ଶ𝐸௡௠ଷ௠ஷ௡ (2-10) where 𝑘 and 𝑖 are some other eigenstates of the unperturbed system. Following the PT procedure49,76, the first-order correction to the wavefunction is: 𝛹௡(ଵ) = ෍𝑉௠௡𝐸௡௠ · 𝛹௡(଴)௠ஷ௡ (2-11) 2.2.2 The DPT method In this method, the reference wavefunction for state 𝑛 is considered to be a linear combination of all degenerate states in the unperturbed system: 𝛹௡(଴) = ෍ a௜ · 𝛹௜(଴)௜ (2-12) 29 For example, for a system with three degenerate states 𝛹௡(଴) = aଵ · 𝛹ଵ(଴) + aଶ · 𝛹ଶ(଴) + aଷ · 𝛹ଷ(଴) (2-13) Following the PT procedure to solve the Schrödinger equation, a system of linear equations is obtained as follows: ቌ𝐻෡ଵଵ௣ 𝐻෡ଵଶ௣ 𝐻෡ଵଷ௣𝐻෡ଶଵ௣ 𝐻෡ଶଶ௣ 𝐻෡ଶଷ௣𝐻෡ଷଵ௣ 𝐻෡ଷଶ௣ 𝐻෡ଷଷ௣ቍ · ൭aଵaଶaଷ൱ = 𝐸௣ · ൭aଵaଶaଷ൱ (2-14) where 𝐻෡௜௝௣ =< 𝛹௜(଴)ห𝐻෡௣ ห𝛹௝(଴) > with 𝑖, 𝑗 =1, 2, 3. 𝐸௣ is the first-order correction to the energy. In this system of equations, the unknowns are the coefficients a௜ and the energy 𝐸௣. In order to get non-trivial answers for this system of equations, the determinant of the matrix (𝐻෡௜௝௣ ‒ 𝐸௣𝛿௜௝) must be zero. This will result in the following cubic equation in 𝐸௣: 𝐸௣ଷ − ൫𝐻෡ଵଵ௣ + 𝐻෡ଶଶ௣ + 𝐻෡ଷଷ௣ ൯. 𝐸௣ଶ + ൫𝐻෡ଵଵ௣ · 𝐻෡ଶଶ௣ + 𝐻෡ଵଵ௣ · 𝐻෡ଷଷ௣ + 𝐻෡ଶଶ௣ · 𝐻෡ଷଷ௣ −𝐻෡ଶଷ௣ · 𝐻෡ଷଶ௣ − 𝐻෡ଵଶ௣ · 𝐻෡ଶଵ௣ − 𝐻෡ଵଷ௣ · 𝐻෡ଷଵ௣ ൯ · 𝐸௣ − ൫𝐻෡ଵଵ௣ · 𝐻෡ଶଶ௣ · 𝐻෡ଷଷ௣ − 𝐻෡ଵଵ௣ · 𝐻෡ଶଷ௣ ·𝐻෡ଷଶ௣ − 𝐻෡ଵଶ௣ · 𝐻෡ଶଵ௣ · 𝐻෡ଷଷ௣ + 𝐻෡ଵଶ௣ · 𝐻෡ଶଷ௣ · 𝐻෡ଷଵ௣ + 𝐻෡ଵଷ௣ · 𝐻෡ଶଵ௣ · 𝐻෡ଷଶ௣ − 𝐻෡ଵଷ௣ · 𝐻෡ଶଶ௣ ·𝐻෡ଷଵ௣ ൯ = 0 (2-15) By solving this cubic equation, three energies are obtained for 𝐸௣, and the degeneracy may be lifted (completely or partially). After finding the energy 𝐸௣, Eq. (2-14) can be used to calculate the coefficients a௜ to obtain the wavefunction in Eq. (2-13). Then this wavefunction is used as the reference wavefunction in the PT procedure to calculate the higher order wavefunction and energies50. 30 2.3 Metropolis Monte Carlo simulations The Metropolis Monte Carlo or more conventionally MC simulation is a class of a molecular simulation procedure that is used to study the physical and chemical properties of a thermodynamic system74,75. In general, a molecular simulation, is an algorithm for the evolution of a system from near-equilibrium conditions to complete equilibrium, after which the properties of the system can be determined74. In these algorithms, the movements and orientations of the molecules in the system as well as the interactions between these molecules are followed at both near-equilibrium and equilibrium conditions72. Once equilibrium is reached the algorithm should generate a large number of ideally unrelated configurations to allow statistical sampling of the properties of interest to occur. This is done by moving the particles in the system. The simplest way to do this is to follow the molecular motion using the laws of classical mechanics. This procedure is known as molecular dynamics (MD) simulations73,75. If the molecules in the system are selected and moved at random with no reference to their velocity, then the procedure is called an MC simulation72,74,75. In terms of generating statistically independent configurations, the MC method is more efficient than MD. In order to calculate the properties of the system under specific conditions, the problem needs to be simplified. This is done through putting some constraints on the thermodynamic variables of the system in order to make it consistent with the experiments. This set of constrained variables that specifies a thermodynamic region is called an “ensemble”. The selection of these variables is based on statistical mechanics considerations required to calculate different properties of the systems under different thermodynamic conditions77. One of the main ensembles used in MC simulations is the NVT ensemble (MC-NVT simulation). In this method, the number of particles (molecules), N, the volume, V, as well as the 31 temperature of the system, T, are kept constant during the simulation. All of the simulations presented in this thesis use the NVT ensemble. These MC-NVT simulations were run using the following conditions and steps72–75,77: 1- The system is considered as a 3D box. The number of molecules, the volume of the box, and the temperature in this system are specified based on the experimental conditions. 2- Periodic Boundary Conditions (PBC) are used in the directions parallel to the surface plane to eliminate edge effects. The simulations presented in the thesis were either a 12 × 12 or 24 × 24 patch (in terms of the surface unit cells) of the top layer of the ionic surface. 3- The solid surface, as represented by the molecule-surface potential, provides the lower boundary of the simulation box and lies along the surface plane. The upper boundary is the height at which the minimum barrier against the lateral motion of the molecule becomes zero. 4- To initialize the simulation, the size of the simulation box, the identities of the molecules, and their initial positions and rotational states are specified as suggested by previous experimental (ND and HAS) and theoretical work. 5- The simulation is done in cycles. Each cycle is done by translating (or rotating) one of the randomly selected particles in the system. 6- At the end of each cycle, the probability of accepting the movement of the particle is assessed according to an acceptance criterion, which is based on the laws of statistical mechanics. For MC-NVT simulations the acceptance criterion is determined by the Boltzmann factor and satisfies microscopic reversibility. An accepted move leads to a new configuration for the system, which when repeated a sufficient number of times will lead to a new configuration that is statistically independent of the initial configuration. This process is repeated over and 32 over until the end of the simulations, generating a Markov chain of configurations which may then be analyzed statistically. 7- The simulation has two parts. The first part is to bring the system form initial near-equilibrium conditions to equilibrium and is called the equilibrium run. Equilibrium is reached when the energy of the system becomes constant (with small fluctuations around the average value). After reaching equilibrium, the data collection phase (production run) may commence where the properties of the system are calculated by averaging them over the Markov chain of cycles that are generated from the equilibrium configuration. The number of the H2 molecules in the simulation box is determined based on the observed experimental coverage. As stated above, PBC is used only in the lateral (x-y) directions. Perpendicular to the surface (z direction), the only limit that is placed on the motion of the molecule (only in Model II) is that it cannot get any closer than 2.40 Å to the surface (see section 4.2.2) or any further than the specified height based on the molecule-surface interactions. For the H2-H2 interaction in the layer, a cut-off radius of 10.0 Å is used, which is consistent with the original potential model of Diep and Johnson78,79. The target acceptance rate for the movements of the molecules is chosen to be 0.45 and the amplitudes of the movements are constantly adjusted during the equilibration run to achieve this rate. Once the production run starts, however, the movement amplitudes are fixed in order to preserve microscopic reversibility. The number of the cycles used during the equilibration and production runs vary based on the coverage of the layer as well as the temperature of the system. This number ranges from 80 million total cycles at low coverages and temperatures up to 240 million cycles at higher temperatures and coverages. To ensure that the data sets used for the calculation of the properties 33 of the system are statistically independent, correlation functions as well as the statistical inefficiency (parameter) were calculated and analyzed during the data collection75,80. This analysis led to sampling rates (number of cycles in the Markov chain) that ensured reliable uncorrelated data that can be safely compared with the experiments. As an alternative to the NVT (Canonical) ensemble, the μVT or Grand Canonical ensemble81,82 may also be used to simulate adsorption processes. In Grand Canonical Monte Carlo (GCMC) simulations the number of particles is allowed to change but its conjugate variable, the chemical potential µ, is fixed. When applied to adsorption processes, GCMC simulations allow for a change in coverage of the surface. This is a valuable tool when looking at phase stability and transformations related to changes in coverage but makes the determination of surface structure more difficult and hence is not used in this study, although it may be useful for future work. 34 Potential Models In Chapter 2, the formalism of the potential models used in this study was introduced. In fact, these interaction potentials are used to calculate the interaction energies between the H2 molecules in the adsorbed layer (H2-H2) as well as with the ionic surface (H2-surf) in the adsorption system. In these models, the usual way is to view the H2 molecule as a two-atom molecule rotating in the space. Figure 3-1a shows an example of a two-atom coordinate system when the H2 molecules is in the presence of an ionic surface. This coordinate system was used in the previous simulation studies on the adsorption of H2 molecules on ionic surfaces25,31,59–61,63. The problem with this representation is that, it cannot be used to differentiate between H2 molecules in different rotational states (para- or ortho-H2). To overcome this problem, a new way, in which the H2 molecule is represented by a single-body particle in the space, was adapted (see for example Figure 3-1b for the H2-surf interactions). Then in order to differentiate between the rotational states of H2, the molecular shape of the H2 species in each of these states is made by the distribution of the molecular axis around the axis of quantization (Figure 1-1 and Figure 3-1b for the H2-surf interactions). In fact, it is this distribution that interacts with the other particles in the system. For the H2 molecule, these distributions in the gas phase (isolated system) can be obtained from the wavefunctions of a system of an isolated H2 molecule in the space48,49. This is a very well-known system in quantum mechanics called the rigid rotor system and the Schrödinger equation for the system is: 𝐾෡ுమ𝛹௡ = 𝐸௡𝛹௡ (3-1) 35 where 𝐾෡ுమ is the rotational kinetic energy operator. The solution of this equation for the energy of state 𝑛 = 𝐽, 𝑚௃ of the system is: 𝐸௡ = 𝐸௃,௠಻ = 𝐵𝐽(𝐽 + 1) (3-2) where 𝐽 is the rotational quantum number, 𝑚௃ is the magnetic quantum number (J= 0, 1, 2, … and mJ= -J, …, J), and B is the rotational constant of the rigid rotor. (a) (b) Figure 3-1 The coordinate systems for, a) the H2 molecule as a two-atom particle in the space, (the previous MC simulation studies25), and b) the H2 molecule as a single-body particle made by the distribution of the molecular axis around the axis of quantization (the present work). In this figure, the external object shown is an ionic surface. The solutions for the wavefunctions are, in fact, the spherical harmonics are shown for state 𝑛 as follows: 𝛹௡ = ϒ௡(𝜃, 𝜑) = ϒ௃,௠಻(𝜃, 𝜑) (3-3) 36 Where the ϒ௃,௠಻ are the spherical harmonics that depend on the polar and azimuthal angles, 𝜃 and 𝜑 (see subfigure (a) of Figure 3-1 for an H2 molecule in the presence of an external object, for example, the ionic surface). In the adsorption system or in the presence of another molecule, however, the H2 molecule is no longer isolated and is affected by the external force fields from these particles. The Schrödinger equation for the rotating H2 in the presence of this external potential is: ൫𝐾෡ுమ + 𝑉෠௘௫௧൯𝜑௡ = 𝐸௡𝜑௡ (3-4) where 𝜑௡ is the wavefunction of the H2 molecule in the presence of the external electric field, and 𝑉෠௘௫௧ is the interaction potential between the H2 molecule and other objects in the system. In fact, in our adsorption system, 𝑉෠௘௫௧, can be either 𝑉෠ுమି௦௨௥ for the H2-surf interactions (section 3.1) or 𝑉෠ுమିுమ for the H2-H2 interactions (section 3.2). In this thesis, a hierarchy of three models of increasing complexity have been developed for the H2-surf interactions, along with accompanying models for the H2-H2 interactions. Model I employs first-order perturbation theory and simply averages the H2 orientation, according to the molecule’s quantum mechanical rotational state and more or. It is also referred to as the Rotationally Average Potential (RAP) model. The rotational averaging is performed on both the H2-surf and the H2-H2 interactions. This model is used in the simulations in the first phase of this study (see Chapter 6 for the results of these simulations). Since the adsorption system is complicated, the problem is simplified by making some assumptions in the formulation of the system (Model I). Then, based on the successes and failures of these models in reproducing the experimental data, the model is improved step by step by removing the assumptions and extending the order of the perturbative expansions. These improvements, as applied to the H2-37 surf and H2-H2 interactions, form the basis of Model II used in the simulations in the second phase of our study (see Chapter 6 for the results). Model III is used for an examination of the mixing of ortho-H2 states induced by the H2-surf interactions. This model employs degenerate perturbation theory but does not include H2-H2 interactions. The results of this model for the interaction between a single ortho-H2 molecule and the ionic surface are presented in Chapter 8. In the following paragraphs, first the H2-surf interactions will be presented in great detail for each one of models I, II, and III, after which the H2-H2 interactions are presented for models I and II. 3.1 H2-surf interaction potentials For these interactions, the 𝑉෠௘௫௧ in Eq. (3-4) is replaced by the interaction potential between the H2 molecule and the ionic surface, 𝑉෠ுమିௌ௨ . This potential consists of the following components26,53,54: 𝑉෠ுమି௦௨௥௙ = 𝑉෠ுమି௦௨௥௙௥௘௣ + 𝑉෠ுమି௦௨௥௙ௗ௜௦௣ + 𝑉෠ுమି௦௨௥௙ொ + 𝑉෠ுమି௦௨௥௙௜௡ௗ (3-5) where 𝑉෠ுమି௦௨௥௥௘௣ and 𝑉෠ுమି௦௨௥௙ௗ௜௦௣ are the repulsion and dispersion interactions of the H2 molecule with the ionic surface, 𝑉෠ுమି௦௨௥௙ொ is the interaction between the quadrupole moment (Q) of the H2 molecule and the ionic surface, and 𝑉෠ுమି௦௨௥௜௡ௗ is the interaction between the induced dipole moment of H2 and the ionic surface. These last two terms are usually referred to as the electrostatic interactions, i.e., 𝑉෠ுమି௦௨௥௙௘௟௘௖௧௥௢௦௧௔௧௜௖ = 𝑉෠ுమି௦௨௥௙ொ + 𝑉෠ுమି௦௨௥௙௜௡ௗ . All the components in Eq. (3-5) have been modelled and used in the past in the studies on the adsorption of H2 molecules on ionic surfaces of NaCl 23,26 and LiF19,53–55. 38 To solve Eq. (3-4), the PT method explained in Chapter 2 is used. By comparing Eq. (3-4) and Eq. (2-2) 𝐾෡ுమ is identified as the operator of the unperturbed system (𝐻෡௨) and 𝑉෠ுమି௦௨௥௙ as the perturbation operator (𝐻෡௣). This means that the interaction with the surface can be considered as a perturbation of the system. Following the PT procedure, the energies can be calculated and the wavefunctions for the para- and ortho-H2 molecules adsorbed on the ionic surfaces (MgO and NaCl). But based on the equations given in Chapter 2 (Eqs. (2-7) to (2-11)), using the PT formalism requires solving integrals in the following form: < ϒ௄,௠಼ห𝑉෠ுమି௦௨௥௙หϒ௃,௠಻ >= න න ϒ௄,௠಼∗ · 𝑉෡ுమି௦௨௥௙ · ϒ௃,௠಻గ଴ 𝑠𝑖𝑛 𝜃 𝑑𝜃 𝑑𝜑ଶగ଴ (3-6) In order to evaluate these integrals, the 𝑉෠ுమି௦௨௥௙ equations are first written in terms of the products of spatial functions and spherical harmonics as follows: 𝑉෠ுమି௦௨௥௙ = ෍ 𝑓௅,௠ಽ(𝑥, 𝑦, 𝑧) · ϒ௅,௠ಽ௅,௠ಽ (3-7) where 𝑓௅,௠ಽ(𝑥, 𝑦, 𝑧) is a complex function of the position of the molecule. The 𝑌௅,௠ಽ functions carry that information on the orientation of the molecule. The decomposition of the potential into spatial and orientational components allows Eq. (3-6) to be written as: < ϒ௄,௠಼ห∑ 𝑓௅,௠ಽ(𝑥, 𝑦, 𝑧) · ϒ௅,௠ಽ௅,௠ಽ หϒ௃,௠಻ > = ∑ ቀ𝑓௅,௠ಽ(𝑥, 𝑦, 𝑧) × <௅,௠ಽϒ௄,௠಼หϒ௅,௠ಽหϒ௃,௠಻ >ቁ (3-8) Section A.1 shows how the integrals over the product of spherical harmonics in Eq. (3-8) are evaluated. It should be noted that this same kind of reasoning will hold true for the H2-H2 interactions as well (section 3.2). 39 As stated earlier, for the H2-surf interactions, four different terms of Eq. (3-5) must be considered. These terms are, in fact, the different types of interactions between an H2 molecule and the ionic crystal. Using the surface potential operator and the PT method, three different models were developed for the H2-ionic crystal interactions. These models differ according to the assumptions used in their formulation. Nonetheless, there are also some assumptions that are common among all these three models. Here is a list of these common assumptions: 1- Static surface: The surface and the entire substrate are considered to be rigid. The interaction between H2 and the surface ions is weak compared to the ion-ion interactions in the ionic crystal. Therefore, the distortion of the surface is small and can be neglected in the simulations. Studies confirmed that these modifications in the surface are minor and do not play as crucial a role as other factors10,83. Also, in all significant theoretical studies in the field, the same assumption was used19,23,25,26,84 and good agreement with experimental data was reported. On the other hand, the small distortions in the surface do not strongly affect the positions of the H2 molecules. In the present work, only the overall structure of the H2 layer is studied and these small deformations of the surface are not considered. 2- Rigid rotor approximation (fixed bond length): At the relevant experimental temperatures (5–20 K or energies of 0.43-1.7 meV), the H2 molecule is in its ground vibrational state since the lowest vibrational excitation has an energy of 496.6 meV (5763 K)85 and hence, cannot be excited to the next vibrational state37,84. In the formulation of our potential models for the H2-H2 potential, the model of Diep and Johnson78,79 is used. This model was calculated for the ground vibrational state with a fixed bond length78,79,85. 3- Ground rotational states of ortho- and para-H2: Only lowest rotational states (𝐽 = 0,1) are present at such low temperatures. Therefore in our adsorption system, only low-lying rotational 40 states (𝐽 = 0 or ground para-H2 and 𝐽 = 1 or ground ortho-H2) are populated16,44,47,65. For ground para species, to go to the next rotational energy level, 𝐽 = 0→ 2, one would need an energy of around 6. 𝐵ுమ which is around 44.1 meV or 512 K, where 𝐵 is the molecular rational constant. For ortho molecules the transition, 𝐽 = 1→ 3, needs 10. 𝐵ுమ which is around 74.4 meV or 852.8 K. This is a substantially large energy that is not accessible at the thermal energies available in our system, based on the probabilities calculated from equilibrium Boltzmann distribution. Also, for all these rotational states of the H2 molecule, the axis of quantization is always assumed to be perpendicular to the surface plane. 4- No para-ortho interconversion: Experiments11,20,41 have shown that the ground state spin flip transition, 𝐽 = 0→ 1, energy for H2 above MgO is around 11.25 meV or 130 K which is too large an energy and is not accessible at the temperatures studied in this work. Furthermore, the magnetic impurities1 required for ortho-para interconversion are not present. 5- Implicit inclusion of the zero-point energy: At low temperatures, including the zero-point energy of the molecules in the interactions is crucial to obtaining accurate molecule-surface binding energies. In order to include these interactions explicitly in the calculations, the accurate quantum mechanical binding energy of the H2 molecule with the ionic surface is required26,27. Due to lack of such accurate potential energies, the simplest way is to fit the energy to the experimental data. This means that the zero-point energy of the H2 molecules in the layer is implicitly incorporated into the energy calculations. As will be discussed in section 4.1, the fitting procedure is done only over a specific interacting site in the surface. Then it is assumed that the dispersion and repulsion parameters that are obtained this way, accurately account for the zero-point energies at all positions over the surface. 41 Once again, the assumptions discussed above are used in the formulation of all the potential models presented in this study (models I, II, and III). Additional assumptions specific to each of these models are presented in the following sections where models I, II, and III are developed. 3.1.1 Model I (the RAP model) In the Rotationally Averaged Potential model, the interaction potentials are simply averaged over the distribution of the orientations of the H2 molecular axis as follows 𝐸ுమି௦௨௥௙(ଵ) =< ϒ௄,௠಼ห𝑉෠ுమି௦௨௥௙หϒ௄,௠಼ > (3-9) where ϒ௄,௠಼ is the wavefunction of the isolated H2 molecule (rigid rotor) in the gas phase. Note that this energy is the second term in the perturbation series as denoted by the superscript (1) (see Eq. (2-5)). The effects of averaging the components of the surface potential vary since the different types of interactions in Eq. (3-5) have different representations. For instance, in model I the dispersion and repulsion interactions are assumed to be isotropic and centered on a single interaction site (the center of mass). Consequently, the repulsion and dispersion energies are independent of the rotational states of the H2 molecule. The electrostatic interactions, however, depend on the rotational state of the molecule. This arises because the interaction is due to the coupling of the molecular quadrupole to the electric field gradient of the ionic substrate. The orientation of the quadrupole in space will affect its interaction energy which will then be dependent on the rotational state of the molecule. Another important assumption in Model I is the constancy of rotational state. That is, no matter where a molecule migrates to on the surface its rotational state will remain the same. Thus, there is no interconversion between para and ortho states, as expected, as well as no 42 interconversion between helicoptering and cartwheeling states. In the following sections, the development of RAP for each of the components of the surface potential (see Eq. (3-5)) are presented. It should be mentioned that in Model I, only the mJ= +1 state of the Heli molecule is used in the calculations as the mJ= ‒1 will yield the same result. 3.1.1.1 The dispersion and repulsion interactions In this model, the dispersion and repulsion interactions between an H2 molecule and the cations or anions of the ionic crystal are modelled by the pairwise potential of Tang and Toennies 86,87 (a more sophisticated and extended version of the Buckingham potential): 𝑉൫𝑟௜௝൯ = 𝐴௜௝ exp൫−𝐵௜௝ · 𝑟௜௝൯ − ෍ ൭𝑓ଶ௡൫𝑟௜௝൯𝐶ଶ௡௜௝൫𝑟௜௝൯ଶ௡൱ହ௡ୀଷ (3-10) In the Tang-Toennies potential, the repulsion is modelled by an exponential term with two parameters: 𝐴௜௝ (the strength constant) and 𝐵௜௝ (the decay constant) where the subscripts i and j denote the individual molecules/ions. The dispersion part of Eq. (3-10) includes the instantaneous dipole-dipole (the 𝐶଺௜௝ term), dipole-quadrupole (the 𝐶଼௜௝ term) and quadrupole-quadrupole plus dipole-octupole (the 𝐶ଵ଴௜௝ term) dispersion interactions88. The 𝑓ଶ௡൫𝑟௜௝൯ coefficients are damping functions that remove the singularities in the dispersion potentials at short distances of 𝑟௜௝. The complete information about these interactions as well as the parameters used in these potentials can be found in Ref. 25. In Chapter 4, the parametrization procedures used in Model I will be fully discussed, and the parameters used in this potential will be presented. 43 As stated earlier, in Eq. (3-10) one important assumption is that, the dispersion and repulsion interactions are isotropic. The impact of this assumption will be discussed in more detail in section 4.3.2. Another important note about Eq. (3-10) is that it is only used for the top layer of ions. Since the repulsion potential decays exponentially, the underlying layers of ions (bulk) only contribute to the dispersion energies in the total potential. In Model I, the dispersion interactions from these underlying layers are modelled separately as the bulk potential. 3.1.1.2 The bulk interactions The long-range dispersion interactions between the H2 molecule and the ions in the crystal below the top layer is modelled using an averaged potential based on only the 𝐶଺ dispersion terms of the potential as follows: 𝑉஻௨௟௞(𝑧) = −2𝜋൫𝐶଺ுమିୡୟ୲୧୭୬ + 𝐶଺ுమିୟ୬୧୭୬൯3𝑐ଷ ቀ𝑧 + 𝑐2ቁଷ (3-11) where 𝑧 is the distance of the H2 molecule above the surface plane of the crystal, and 𝑐 is the lattice constant of the ionic crystal (𝑐 = 4.20 Å for MgO and 5.64 Å for NaCl). Again, due to the dispersion interaction of H2 molecule being treated isotropically, the bulk potential does not depend on the rotational state of the H2 molecule. 3.1.1.3 The electrostatic interactions These interactions are divided into two parts; one part is the interaction of the permanent quadrupole moment of H2 with the ionic crystal and the second part is the interaction of the induced dipole of this molecule with the ionic crystal. Again, both of these interactions are 44 determined by Eq. (3-9), the second term in the NDPT series. The implicit assumption regarding the electrostatic interactions is that higher-order energies in the NDPT series are small. 3.1.1.3.1 H2 quadrupole-ionic surface interaction For these interactions the operator (𝑉෠ுమି௦௨௥௙ொ ) is from the model proposed by Hill89 (see section A.2), which has already been used in the studies of the adsorption of H2 on LiF19,53,54. Following the form of Eq. (3-7), a new version of Hill’s potential is developed as follows: 𝑉෠ுమି௦௨௥௙ொ 𝑜𝑟 𝑉෠ொ௘௟௦= 𝑋଴ (Υଶଶഥ + Υଶଶ) + 𝑖𝑋ଵ (Υଶଶഥ − Υଶଶ) + 𝑋ଶ (Υଶଵഥ − Υଶଵ)+ 𝑖𝑋ଷ (Υଶଵഥ + Υଶଵ) + 𝑋ସ Υଶ଴ (3-12) where the 𝑋௜‘s are functions of (𝑥, 𝑦, 𝑧) coordinates and are shown in Table 3-1. In this table, the summation index “𝑎𝑙𝑙 |𝑛|, |𝑚|”, means that each term in the summation is the sum of all {+n,+m}, {+n,–m}, {–n,+m}, and {–n,–m} Fourier terms (or vectors). These Fourier terms model the periodicity of the surface potential53,54. Also, 𝑓௔௖ in the 𝑋௜ equations is always equal to 1, except when either n or m in the “𝑎𝑙𝑙 |𝑛|, |𝑚|” set is zero. In that case, 𝑓௔௖ is equal to 2. Section A.2 in Appendix B shows a step-by-step example of how the 𝑋௜ functions in Table 3-1 are developed for each “𝑎𝑙𝑙 |𝑛|, |𝑚|” set. The variables 𝑀, 𝐴௡,௠, and 𝛾௡,௠ in Table 3-1 were defined in Refs. 53,54 (also see section A.2). In Chapter 4 and in section 4.2.1.2.2, the convergence of the 𝐸ொ௘௟௦ potential energy will be discussed based on the number of Fourier terms that are used in the potential. 45 Table 3-1 The 𝑿𝒊 functions in Eq. (3-12). As seen, all these functions only depend on the (𝒙, 𝒚, 𝒛) coordinates. In these equations, 𝒊 is the imaginary number, 𝑮 = 𝟐𝝅(𝒄/√𝟐), and 𝑩𝒏𝒎 =𝑴𝒆𝒍𝒔 𝑨𝒏,𝒎 𝒆𝒙𝒑 (‒𝜸𝒏,𝒎.𝒛)𝒇𝒂𝒄, where 𝒇𝒂𝒄 is either 1 or 2 depending on the |𝒏|, |𝒎| values as explained in section 3.1.1.3.1. 𝑴𝒆𝒍𝒔 is defined in section 0A.2. 𝑋଴ = ෍ 𝐵௡௠ × ቆ2 ·𝑛ଶ + (𝑚. 𝑖)ଶ𝑛ଶ + 𝑚ଶ · [cos 𝐺(𝑛𝑥 + 𝑚𝑦) + cos 𝐺(𝑛𝑥 − 𝑚𝑦)]ቇ௔௟௟ |௡|,|௠| 𝑋ଵ = ෍ 𝐵௡௠ × ቀ2 ·𝑛. 𝑚𝑛ଶ + 𝑚ଶ· 2𝑖 · [cos 𝐺(𝑛𝑥 + 𝑚𝑦) − cos 𝐺(𝑛𝑥 − 𝑚𝑦)]ቁ௔௟௟ |௡|,|௠| 𝑋ଶ = ෍ 𝐵௡௠ × ൬2𝑖 · 𝑛√𝑛ଶ + 𝑚ଶ· 2𝑖 · [sin 𝐺(𝑛𝑥 + 𝑚𝑦) + sin 𝐺(𝑛𝑥 − 𝑚𝑦)]൰௔௟௟ |௡|,|௠| 𝑋ଷ = ෍ 𝐵௡௠ × ൬2𝑖 ·𝑚𝑖√𝑛ଶ + 𝑚ଶ· 2𝑖 · [sin 𝐺(𝑛𝑥 + 𝑚𝑦) − sin 𝐺(𝑛𝑥 − 𝑚𝑦)]൰௔௟௟ |௡|,|௠| 𝑋ସ = ෍ 𝐵௡௠ × ൫−2 √6 · [cos 𝐺(𝑛𝑥 + 𝑚𝑦) − cos 𝐺(𝑛𝑥 − 𝑚𝑦)]൯௔௟௟ |௡|,|௠| By using Eq. (3-12) in Eq. (3-9), the H2-surf equations in Model I can be developed for all the three different rotational states of H2; para-H2 (Para), Heli, and Cart. These potentials are: for Para: 𝐸ொ௘௟௦ౌ౗౨౗(ଵ) = 0 (3-13) for Heli: 𝐸ொ௘௟௦ౄ౛ౢ౟(ଵ) = −0.126 · 𝑋ସ (3-14) for Cart: 𝐸ொ௘௟௦ి౗౨౪(ଵ) = 0.252 · 𝑋ସ (3-15) Section A.3 gives an example for the derivation of these energies. 46 3.1.1.3.2 H2 induced dipole-ionic surface interaction (the induction energy) This interaction is divided into two parts; the isotropic (𝑉෠ுమି௦௨௥௙௜௡ௗ଴ ) and the anisotropic induction interactions (𝑉෠ுమି௦௨௥௜௡ௗଶ ). For both of these operators, the equations proposed by Kroes and Mowrey53,54 are used. For The isotropic operator (𝑉෠ுమି௦௨௥௙௜௡ௗ଴ ), their equation is simple and does not depend on the rotational state of the molecule. It has the following form: 𝑉෠ுమି௦௨௥௙௜௡ௗ଴ 𝑜𝑟 𝑉෠௜௡ௗ଴= −64 𝜋ଶ𝛼(଴)𝑒ଶ𝑐ସ × exp(−2 · 𝐺 · 𝑧) × ൤11 + exp(−2𝜋)൨ଶ × [1 + cos(𝐺 · 𝑥) cos(𝐺 · 𝑦)] (3-16) where 𝛼(଴) is the isotropic polarizability given by 𝛼(଴) = ଵଷ(𝛼‖ + 2𝛼ୄ). with 𝛼‖ and 𝛼ୄ the parallel and perpendicular polarizabilities, respectively. The symbol 𝑒 denotes the electric charge of the ions, 𝑐 is the lattice constant of the ionic crystal, and 𝐺 is defined in Table 3-1. As seen in Eq. (3-16), the 𝑉෠ுమି௦௨௥௙௜௡ௗ଴ (𝑜𝑟 𝑉෠௜௡ௗ଴) operator only depends on the position (𝑥, 𝑦, 𝑧) of the molecule over the surface (but not its rotational state). For simplicity throughout this thesis, this operator is shown by 𝑊(0). Now, if this operator is used in Eq. (3-9), the first-order 𝐸௜௡ௗ଴ energies can be obtained: 𝐸௜௡ௗ଴(ଵ) = 𝑊(0) (3-17) It should be mentioned that, this energy is the same for the Para, Heli, and Cart molecules. Unlike 𝑉෠ுమି௦௨௥௙௜௡ௗ଴ , the anisotropic operator (𝑉෠ுమି௦௨௥௙௜௡ௗଶ ), depends on both the position of the molecule as well as its rotational state53,54. Once again, by following Eq. (3-7), a new version of Kores and Morey’s potential (Eq. (3-16)) is developed as follows: 47 𝑉෠ுమି௦௨௥௙௜௡ௗଶ 𝑜𝑟 𝑉෠௜௡ௗଶ= 𝐶௜௡ௗబ(𝑧)· ൛𝐶௜௡ௗభ(𝑥, 𝑦) · Υଶ଴ + 𝐶௜௡ௗమ(𝑥, 𝑦) · (Υଶଶഥ + Υଶଶ)+ 𝐶௜௡ௗయ(𝑥, 𝑦) · 𝑖 · (Υଶଶഥ − Υଶଶ) + 𝐶௜௡ௗర(𝑥, 𝑦) · 𝑖 · (Υଶଵഥ+ Υଶଵ) + 𝐶௜௡ௗఱ(𝑥, 𝑦) · (Υଶଵഥ − Υଶଵ)ൟ (3-18) where 𝐶௜௡ௗ೔’s are shown in Table 3-2. It should be mentioned that to develop Eq. (3-18), only the Fourier terms up to (1,1) have been used in the potential. In deriving these equations, a similar procedure to what presented in Section A.2 was used. In section 3.1.2.1.2, the truncation of the Fourier terms in the series will be explained in detail based on the convergence of the potential for this type of the interaction energy. If the operator of Eq. (3-18) is used in Eq. (3-9), the first-order energies, 𝐸௜௡ௗଶ(ଵ) , will be obtained for the Para, Cart, and Heli molecules as follows: for Para: 𝐸௜௡ௗଶౌ౗౨౗(ଵ) = 0 (3-19) for Heli: 𝐸௜௡ௗଶౄ౛ౢ౟(ଵ) = ‒ 0.126 𝐶௜௡ௗబ · 𝐶௜௡ௗభ (3-20) for Cart: 𝐸௜௡ௗଶి౗౨౪(ଵ) = +0.252 𝐶௜௡ௗబ · 𝐶௜௡ௗభ (3-21) The procedure for the derivation of these equations is the same as the example shown in section A.3. In Chapter 4 and in section 4.2.1.2.3, the convergence of this potential will be discussed based on the number of Fourier terms that are used in the series. 48 Table 3-2 The 𝑪𝒊𝒏𝒅𝒊 functions in Eq. (3-18). As seen, all these functions only depend on the (𝒙, 𝒚, 𝒛) coordinates in the system. Also, 𝑳(𝟎) = 𝜶(𝟐) × 𝑮𝟐, where 𝜶(𝟐) = 𝟏𝟑ට𝟔𝝅𝟓൫𝜶‖ − 𝜶ୄ൯ and 𝑮 =𝟐𝝅(𝒄/√𝟐) . 𝐶௜௡ௗబ(𝑧) = 𝐿(0) · 𝐴଴ଵଶ 𝑟଴ଵସ · exp(−2 𝛾଴ଵ 𝑧) 𝐶௜௡ௗభ(𝑥, 𝑦) = 2 ቌඨ23− ඨ83ቍ · cos (2 · 𝐺 · 𝑦) − ቌඨ23+ ඨ83ቍ· cos (2 · 𝐺 · 𝑦) − ඨ83· 2 · cos 𝐺(𝑥 + 𝑦) − ඨ83· 2 · cos 𝐺(𝑥 − 𝑦) 𝐶௜௡ௗమ(𝑥, 𝑦) = −cos (2 · 𝐺 · 𝑦) + cos (2 · 𝐺 · 𝑥) 𝐶௜௡ௗయ(𝑥, 𝑦) = +2 cos 𝐺(𝑥 + 𝑦) − 2 cos 𝐺(𝑥 − 𝑦) 𝐶௜௡ௗర(𝑥, 𝑦) = −2 sin(2 · 𝐺 · 𝑦) − 2 sin 𝐺(𝑥 + 𝑦) + 2 sin 𝐺(𝑥 − 𝑦) 𝐶௜௡ௗఱ(𝑥, 𝑦) = −2 sin(2 · 𝐺 · 𝑥) − 2 sin 𝐺(𝑥 + 𝑦) − 2 sin 𝐺(𝑥 − 𝑦) 3.1.2 Model II In this model, all the components of the potential in Eq. (3-5) are treated anisotropically and thus become dependent on the H2 molecule’s rotational state. In particular, the repulsion-dispersion potential is based on Lennard-Jones potentials centered on atoms rather than the Tang-Toennies potential centered on molecules. Thus, all components of the surface potential in model II depend on the molecule’s rotational state and that the rotational states are constant. The major difference between the two models is the number of terms kept in the perturbation series calculation of the energy. Whereas Model I truncates at first-order corrections to create RAP, 49 Model II goes up to second order for dispersion and repulsion interactions and up to fourth order for electrostatic interactions to get a converged energy. To carry out these higher order calculations it is necessary to consider both Heli states (mJ =+1 and mJ =-1). In order to facilitate the calculation of the higher order contributions it was assumed that cross-terms of the component interactions are small e.g. terms containing ( 𝑉෠ுమି௦௨௥௙௥௘௣ିௗ௜௦௣)·( 𝑉෠ுమି௦௨௥௙ொ ), compared with diagonal terms, and set to zero. This case is called the 𝑉෠௦௘௣ case of Model II. In 0, the effect of this assumption on the energies will be investigated in more detail (see Figure 8-2) where the results of the 𝑉෠௦௘௣ case will be compared with the results of the quite-recently developed case in which all those cross terms are also included in the energy. This latter case is referred to as the 𝑉෠௧௢௧ case of Model II. As will be seen in 0, in the end the results are almost identical. 3.1.2.1 The 𝑽෡𝒔𝒆𝒑 case In this case, the total energy for each correction term in the PT series is the sum of the individual energies. For example, based on Eq. (2-8), the second-order correction to the energy is: 𝐸௡(ଶ) = ෍< 𝑛(଴)ቚ𝑉෠ுమି௦௨௥௙௥௘௣ିௗ௜௦௣ቚ𝑚(଴) >< 𝑚(଴)ቚ𝑉෠ுమି௦௨௥௙௥௘௣ିௗ௜௦௣ቚ𝑛(଴) >𝐸௡௠௠ஷ௡+ ෍< 𝑛(଴)ቚ𝑉෠ுమି௦௨௥௙ொ ቚ𝑚(଴) >< 𝑚(଴)ቚ𝑉෠ுమି௦௨௥௙ொ ቚ𝑛(଴) >𝐸௡௠௠ஷ௡+ ෍< 𝑛(଴)ห𝑉෠ுమି௦௨௥௙௜௡ௗ଴ ห𝑚(଴) >< 𝑚(଴)ห𝑉෠ுమି௦௨௜௡ௗ ห𝑛(଴) >𝐸௡௠௠ஷ௡+ ෍< 𝑛(଴)ห𝑉෠ுమି௦௨௥௙௜௡ௗଶ ห𝑚(଴) >< 𝑚(଴)ห𝑉෠ுమି௦௨௥௙௜௡ௗଶ ห𝑛(଴) >𝐸௡௠௠ஷ௡ (3-22) 50 As seen in Eq. (3-22), each operator is used independently from the other operators in the calculation of the second-order correction to the energy (using the NDPT method, compare it with Eq. (3-46)). The same method is used in the calculation of the higher-order energies as well. In the following paragraphs, it will be discussed how the NDPT energy correction terms are developed for each component of the potential interactions as shown in Eq. (3-5). 3.1.2.1.1 The dispersion and repulsion interactions To calculate these interactions in Model II, the anisotropic potential of Briquez et al. 26,90,91 is used. This potential was developed in the form of a Lennard-Jones (LJ) potential and incorporates the anisotropy of the H2 molecule was incorporated into the equation. Therefore, unlike the Tang-Toennies potential in Model I (Eq. (3-10)), the Briquez’ potential does depend on the rotational state of the H2 molecule. Following Eq. (3-7), a new version of this potential is developed as follows: 𝑉෠ுమି௦௨௥௙௥௘௣ିௗ௜௦௣ 𝑜𝑟 𝑉෠஽ோ= 𝐷(0) . ෍ ෍ 𝐷(1) ෍ {𝐷(2)௣ୀ଴,ିଵ,...௦ఈୀ଺,ଵଶ· [𝐾஽ோ(0) Υ଴଴ + 𝐾஽ோ(1) Υଶ଴ + 𝐾஽ோ(2)(Υଶଶഥ + Υଶଶ)+ 𝑖 𝐾஽ோ(3)(Υଶଶഥ − Υଶଶ) + 𝑖 𝐾஽ோ(4)(Υଶଵഥ + Υଶଵ)+ 𝐾஽ோ(5)(Υଶଵഥ − Υଶଵ)]} (3-23) where 𝐾஽ோ(𝑖) terms (i = 1,2, .., 5) are listed in Table 3-3 and are functions of surface coordinates x and y. Their developments follows the same procedure as the example shown in section A.1 of Appendix A. All the 𝐷(𝑖)௚ functions are listed in Table A4-1 in section A.4. The summation indices 𝑝, 𝛼 , and 𝑠 are also explained in section A.4. 51 In order to calculate the first-order dispersion and repulsion energies, 𝐸஽ோ(ଵ), for the Para, Cart, and Heli molecules, Eq. (3-23) is used in Eq. (3-9) and the following equation are obtained: for Para: 𝐸஽ோౌ౗౨౗(ଵ) = 𝑓஽ோ 𝐾஽ோ(0) × 0.282 (3-24) for Heli: 𝐸஽ோౄ౛ౢ౟(ଵ) = 𝑓஽ோ 𝐾஽ோ(0) × 0.282 − 𝑓஽ோ 𝐾஽ோ(1) × 0.126 (3-25) and for Cart: 𝐸஽ோి౗౨౪(ଵ) = 𝑓஽ோ 𝐾஽ோ(0) × 0.282 + 𝑓஽ோ 𝐾஽ோ(1) × 0.252 (3-26) where 𝑓஽ோ is used instead of 𝐷(0) · ∑ ∑ 𝐷(1) ∑௣ୀ଴.ିଵ,... ௦ఈୀ଺,ଵଶ in the equations to simplify the notation. The procedure for the development of these equations is the same as that shown in section A.3. In Chapter 4 and in section 4.2.1.2.1, the convergence of this potential will be discussed in detail based on the number of the Fourier terms used in the PT series. 52 Table 3-3 The 𝑲𝑫𝑹(𝒊) functions in Eq. (3-23). As seen, all these functions only depend on the (𝒙, 𝒚, 𝒛) coordinates. For simplicity, in all these equations 𝑫(𝟎) . ∑ ∑ 𝑫(𝟏) ∑𝒑ୀ𝟎.ି𝟏,... 𝒔𝜶ୀ𝟔,𝟏𝟐 is shown by 𝒇𝑫𝑹 . “𝒂𝒍𝒍 |𝒏|, |𝒎|” was explained in section 3.1.1.3.1. 𝑿 and 𝒀 are the positions of the H2 molecule with respect to the origin of the coordinate system (an anion site) in the system. 𝐾஽ோ(0) = 𝑓஽ோ ෍ ൫2 · 𝐷(5)௚ · [cos 𝐺(𝑛𝑋 + 𝑚𝑌) + cos 𝐺(𝑛𝑋 + 𝑚𝑌)]൯௔௟௟ |௡|,|௠| 𝐾஽ோ(1) = 𝑓஽ோ ෍ ൫2 · 𝐷(6)௚ · 𝐷(7)௚ · [cos 𝐺(𝑛𝑋 + 𝑚𝑌) + cos 𝐺(𝑛𝑋 − 𝑚𝑌)]൯௔௟௟ |௡|,|௠| 𝐾஽ோ(2) = 𝑓஽ோ ෍ ൫−2 · (𝑛ଶ + (𝑚𝑖)ଶ) × 𝐷(6)௚ · 𝐷(9)௚ ௔௟௟ |௡|,|௠| · [cos 𝐺(𝑛𝑋 − 𝑚𝑌) + cos 𝐺(𝑛𝑋 − 𝑚𝑌)]൯ 𝐾஽ோ(3) = 𝑓஽ோ ෍ ൫−2 × 2 · 𝑛 · 𝑚𝑖 × 𝐷(6)௚ · 𝐷(9)௚ ௔௟௟ |௡|,|௠| · [cos 𝐺(𝑛𝑋 + 𝑚𝑌) − cos 𝐺(𝑛𝑋 − 𝑚𝑌)]൯ 𝐾஽ோ(4) = 𝑓஽ோ ෍ ൫−𝑚 · 2𝑖 · 𝐷(6)௚ · 𝐷(8)௚ · [sin 𝐺(𝑛𝑋 + 𝑚𝑌) − sin 𝐺(𝑛𝑋 − 𝑚𝑌)]൯௔௟௟ |௡|,|௠| 𝐾஽ோ(5) = 𝑓஽ோ ෍ ൫−2 · 𝑛 · 𝐷(6)௚ · 𝐷(8)௚ · [sin 𝐺(𝑛𝑋 + 𝑚𝑌) + sin 𝐺(𝑛𝑋 − 𝑚𝑌)]൯௔௟௟ |௡|,|௠| As discussed in section 3.1.2, higher-order terms are also included in the calculation of the energies in Model II. The higher-order energies can be calculated using Eq. (2-8), Eq. (2-9) and Eq. (2-10). The development of these equations is based on (common) PT procedures explained in sections A.5 and A.6. These procedures are very long specially for the third- and fourth-order energies and hence, throughout the rest of this thesis, only one sample for the development of each of the second-, third-, and fourth-order energies is shown in Appendix A. Starting with the second-order energy, the equations for this energy are as follows: for Para: 53 𝐸஽ோౌ౗౨౗(ଶ) =(0.282)ଶ𝐸଴ଶ × ቄ2 × ቂ൫𝑓஽ோ 𝐾஽ோ(2)൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(3)൯ଶቃ+ 2 × ቂ൫𝑓஽ோ 𝐾஽ோ(4)൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(5)൯ଶቃ+ ൫𝑓஽ோ 𝐾஽ோ(1)൯ଶቅ (3-27) where 𝐸௃ is the rotational energy of the H2 molecule in state 𝐽 (see Eq. (3-2)). for Heli: 𝐸஽ோౄ౛ౢ౟(ଶ) =1𝐸ଵଷ× ቄ+ ቂ൫𝑓஽ோ 𝐾஽ோ(2)൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(3)൯ଶቃ × (8.25 × 10ିଶ)ଶ+ ቂ൫𝑓஽ோ 𝐾஽ோ(4)൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(5)൯ଶቃ × 0.143ଶ+ ൫𝑓஽ோ 𝐾஽ோ(1)൯ଶ× 0.202ଶ+ ቂ൫𝑓஽ோ 𝐾஽ோ(4)൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(5)൯ଶቃ × 0.261ଶ+ ቂ൫𝑓஽ோ 𝐾஽ோ(2)൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(3)൯ଶቃ × 0.319ଶቅ (3-28) and for Cart: 𝐸஽ோి౗౨౪(ଶ) =1𝐸ଵଷ× ቄ+ 2 × ቂ൫𝑓஽ோ 𝐾஽ோ(2)൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(3)൯ଶቃ × 0.184ଶ+ 2 × ቂ൫𝑓஽ோ 𝐾஽ோ(4)൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(5)൯ଶቃ × 0.233ଶ+ ൫𝑓஽ோ 𝐾஽ோ(1)൯ଶ× 0.247ଶቅ (3-29) As state earlier, the same (PT) procedure is used to develop each one of the above second-order equations for the energy. In section A.5, it was shown how Eq. (3-27) is developed. 54 It should be mentioned that, based on the convergence analysis of the dispersion and repulsion energies (𝐸஽ோ, see section 4.2.1.2.1), these interactions converge after including the second-order correction in the energy and hence, they were truncated at second order for all the rotational states of the H2 molecule (Para, Heli, and Cart). This means that, the third- and fourth-order dispersion and repulsion energy equations have not been developed in this study. It should also be emphasized again that unlike Model I, in Model II the bulk potential is not formulated separately from the dispersion and repulsion interactions. In fact, in the LJ potential of Eq. (3-23), the layers below the surface layer (𝑝 < 0) evaluated explicitly and represent the bulk potential in the H2-surf potential energy. 3.1.2.1.2 The electrostatic Interactions The electrostatic interactions are unchanged from Model I, i.e., Eq. (3-12) and Eq. (3-16) and Eq. (3-18) except for the number of Fourier terms used in the 𝑉෠ுమି௦௨௥௙ொ operator. But, the number of Fourier terms used in the 𝑉෠ுమି௦௨௥௙௜௡ௗ operator is the same in models I and II. The convergence of these potentials with respect to their Fourier series will be discussed in detail in sections 4.2.1.2.2 and 4.2.1.2.3. The primary difference between models I and II in the calculation of the electrostatic energy is that Model I only uses the first-order correction whereas Model II will use higher-order corrections. The convergence of the electrostatic energy with respect to the number of terms in the PT series will be discussed in great detail in sections 4.2.2.1 and 4.2.2.24.2.2.1. In the following sections, the calculation of the electrostatic energies using NDPT is discussed. 55 H2 quadrupole-ionic surface interaction In section 3.1.1.3.1, all first-order correction terms for the interaction between the quadrupole moment of H2 and the ionic surface (𝐸ொ௘௟௦ ಹమ(ଵ) ) were presented (Eq. (3-13), Eq. (3-14), and Eq. (3-15)). In Model I, it was assumed that going up to first-order energy is enough and higher-order terms are negligible. As will be seen in 0 (Figure 8-1), this assumption seems to be reasonable for the ortho molecules (due to the higher-order terms cancelling one another), but as will be seen in Chapter 4, it is not reasonable for the Para molecules. Nonetheless, based on the analyses done in section 4.2.2, up-to-fourth-order correction terms were used in the calculation of the 𝐸ொ௘௟௦ energy regardless of the rotational state of the H2 molecules. Using the 𝑉෠ுమି௦௨௥௙ொ operator (Eq. (3-13)) in Eq. (2-8), and following the procedure for the derivation shown in section A.5, the second-order 𝐸ொ௘௟௦ energy for the Para molecule is obtained as follows: 𝐸ொ௘௟௦ౌ౗౨౗(ଶ) =(0.282)ଶ𝐸଴ଶ× {2𝑋଴ଶ + 2𝑋ଵଶ + 2𝑋ଶଶ + 2𝑋ଷଶ + 𝑋ସଶ} (3-30) where the 𝑋௜’s are the functions shown in Table 3-1. 𝐸଴ଶ is 𝐸௡௠ shown in Eq. (2-8) and is calculated using Eq. (3-2) for 𝐸௡ when 𝐽 is 0 (for state 𝑛) and 2 (for state 𝑚). Using a similar procedure to the example shown in section A.5, the second-order energies for the Heli and Cart molecules are: for Heli: 𝐸ொ௘௟௦ౄ౛ౢ౟(ଶ) =1𝐸ଵଷ× {(𝑋଴ଶ + 𝑋ଵଶ) × (8.25 × 10ିଶ)ଶ + (𝑋ଶଶ + 𝑋ଷଶ) × 0.143ଶ+ 𝑋ସଶ × 0.202ଶ + (𝑋ଶଶ + 𝑋ଷଶ) × 0.261ଶ+ (𝑋଴ଶ + 𝑋ଵଶ) × 0.319ଶ} (3-31) 56 and for Cart: 𝐸ொ௘௟௦ి౗౨౪(ଶ) =1𝐸ଵଷ× {+2 × (𝑋଴ଶ + 𝑋ଵଶ) × 0.184ଶ + 2 × (𝑋ଷଶ + 𝑋ଶଶ) × 0.233ଶ+ 𝑋ସଶ × 0.247ଶ} (3-32) As will be discussed in Chapter 4, for this type of interactions, going up to fourth-order NDPT correction energies is required. Using the 𝑉෠ுమି௦௨௥௙ொ operator (Eq. (3-13)) in Eq. (2-9), the third-order 𝐸ொ௘௟௦ energy can be calculated. The formula for the third-order energy in the NDPT procedure (Eq. (2-9)) for the interaction of the quadrupole moment of Para with the ionic crystal has the following form: 𝐸ொ௘௟௦ౌ౗౨౗(ଷ) = +𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴𝐸଴ଶ𝐸଴ଶ+𝑉଴ଶ𝑉ଶସ𝑉ସ଴𝐸଴ସ𝐸଴ଶ+ . . . +𝑉଴ସ𝑉ସସ𝑉ସ଴𝐸଴ସ𝐸଴ସ+ . . . −𝑉଴଴ ቈ|𝑉଴ଶ|ଶ𝐸଴ଶଶ+|𝑉଴ସ|ଶ𝐸଴ସଶ+ . . . ቉ (3-33) where, similar to Eq. (A5-1), all the 𝑉଴ସ, 𝑉ସ଴, 𝑉଴଺, 𝑉଺଴, . .. integrals vanish to zero, which means that all the terms shown by “…” in Eq. (3-33) vanish to zero. 𝑉଴଴ can be replaced by 𝐸ொ௘௟௦ౌ౗౨౗(ଵ) and |௏బమ|మாబమ by 𝐸ொ௘௟௦ౌ౗౨౗(ଶ) to get: 𝐸ொ௘௟௦ౌ౗౨౗(ଷ) = +𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴𝐸଴ଶଶ− 𝐸ொ௘௟௦ౌ౗౨౗(ଵ) 𝐸ொ௘௟௦ುೌೝೌ(ଶ)𝐸଴ଶ (3-34) For a Para molecule 𝐸ொ௘௟௦ುೌೝೌ(ଵ) = 0, thus the third-order energy simplifies to: 𝐸ொ௘௟௦ౌ౗౨౗(ଷ) =𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴𝐸଴ଶଶ (3-35) 57 Therefore, to evaluate the third-order energy, the 𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴ term must be evaluated. The step-by-step procedure for evaluating this term is presented in section A.6 in Appendix A. Substituting this term from section A.6 into Eq. (3-35) results in the third-order energy for the Para molecule in the following form: 𝐸ொ௘௟௦ౌ౗౨౗(ଷ) =(0.282)ଶ𝐸଴ଶଶ× {−(𝑋଴ଶ + 𝑋ଵଶ) · 𝑋ସ × 6 × 0.180 + (𝑋ଶଶ + 𝑋ଷଶ)· 𝑋ସ × 4 × 9.011 × 10ିଶ + 𝑋ସଷ × 0.180+ [𝑋଴(𝑋ଶଶ − 𝑋ଷଶ) − 2 · 𝑋ଵ𝑋ଶ𝑋ଷ] × 2 × 0.220 × 3} (3-36) For the Heli and Cart molecules, Eq. (2-9) is also used for the evaluation of the third-order energy. This equation has the following general form for both of these molecules: 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଷ)= +𝑉ଵଷ𝑉ଷଷ𝑉ଷଵ𝐸ଵଷ𝐸ଵଷ+𝑉ଵଷ𝑉ଷହ𝑉ହଵ𝐸ଵହ𝐸ଵଷ+ . . . +𝑉ଵହ𝑉ହହ𝑉ହଵ𝐸ଵହ𝐸ହଵ+ . . . −𝑉ଵଵ ቈ|𝑉ଵଷ|ଶ𝐸ଵଷଶ+|𝑉ଵହ|ଶ𝐸ଵହଶ+ . . . ቉ (3-37) where, similar to Eq. (A5-1) from Appendix A, all the 𝑉ଵହ, 𝑉ହଵ, 𝑉ଵ଻, 𝑉଻ଵ, . .. integrals vanish to zero, which means that all the terms shown by “…” in Eq. (3-37) vanish to zero. Also, 𝑉ଵଵ can be replaced by 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଵ) and |௏భయ|మாభయ by 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଶ) to get: 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଷ) = +𝑉ଵଷ𝑉ଷଷ𝑉ଷଵ𝐸ଵଷ𝐸ଵଷ− 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଵ)𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଶ)𝐸ଵଷ (3-38) 58 Thus, only the 𝑉ଵଷ𝑉ଷଷ𝑉ଷଵ term must be evaluated. To derive the final form of these equations for Heli and Cart molecules, a similar procedure to the example shown in section A.6 in Appendix A is used. These final equations for the Heli and Cart molecules are presented in section A.7 in Appendix A. For the fourth-order energies, Eq. (2-10) is used. The fourth-order energy for the Para molecule is: 𝐸ொ௘௟௦ౌ౗౨౗(ସ) = +𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴𝐸଴ଶ𝐸଴ଶ𝐸଴ଶ+𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴𝐸଴ଶ𝐸଴ସ𝐸଴ଶ+ . . . +𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴𝐸଴ଶ𝐸଴ସ𝐸଴ଶ+ . . . − ቈ|𝑉଴ଶ|ଶ𝐸଴ଶଶ.|𝑉଴ଶ|ଶ𝐸଴ଶଶ+|𝑉଴ଶ|ଶ𝐸଴ଶଶ·|𝑉଴ସ|ଶ𝐸଴ସଶ+ . . . +|𝑉଴ସ|ଶ𝐸଴ସଶ·|𝑉଴ଶ|ଶ𝐸଴ଶ+|𝑉଴ସ|ଶ𝐸଴ସଶ.|𝑉଴ସ|ଶ𝐸଴ସ+ . . . ቉ − 2· 𝑉଴଴ ቈ𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴𝐸଴ଶଶ 𝐸଴ଶ+𝑉଴ଶ𝑉ଶସ𝑉ସ଴𝐸଴ସଶ 𝐸଴ଶ+ . . . +𝑉଴ସ𝑉ସଶ𝑉ଶ଴𝐸଴ଶଶ 𝐸଴ସ+ . . . ቉+ 𝑉଴଴ଶ ቈ|𝑉଴ଶ|ଶ𝐸଴ଶଷ+|𝑉଴ସ|ଶ𝐸଴ସଷ+ . . . ቉ (3-39) Similar to Eq. (A5-1) in Appendix A, all the 𝑉଴ସ, 𝑉ସ଴, 𝑉଴଺, 𝑉଺଴, . .. integrals vanish to zero, which means that all the terms shown by “…” in Eq. (3-39) vanish to zero. Also, 𝑉଴଴ can be replaced by 𝐸ொ௘௟௦ౌ౗౨౗(ଵ) and |௏బమ|మாబమ by 𝐸ொ௘௟௦ౌ౗౨(ଶ) to get: 𝐸ொ௘௟௦ౌ౗౨౗(ସ) = +𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴𝐸଴ଶ𝐸଴ଶ𝐸଴ଶ+𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴𝐸଴ଶ𝐸଴ସ𝐸଴ଶ−ቀ𝐸ொ௘௟௦ౌ౗౨౗(ଶ) ቁଶ𝐸଴ଶ− 2· 𝐸ொ௘௟௦ౌ౗౨౗(ଵ) ቈ𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴𝐸଴ଶଷ቉ + ቀ𝐸ொ௘௟௦ౌ౗౨౗(ଵ) ቁଶ൥𝐸ொ௘௟௦ౌ౗౨౗(ଶ)𝐸଴ଶଷ൩ (3-40) 59 In Eq. (3-40), all the terms have already been developed except 𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴ and 𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴. The procedure to derive these equations is similar to the example presented in section A.6. The final equations of the 𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴ and 𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴ terms are presented in section A.8 in Appendix A. Similar to the Para molecules, equations for the ortho-H2 Heli and Cart molecules can be derived for the fourth-order energies in the NDPT series as follows: 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ସ)= +𝑉ଵଷ𝑉ଷଷ𝑉ଷଷ𝑉ଷଵ𝐸ଵଷ 𝐸ଵଷ 𝐸ଵଷ+𝑉ଵଷ𝑉ଷଷ𝑉ଷହ𝑉ହଵ𝐸ଵହ 𝐸ଵଷ 𝐸ଵଷ+ . . . +𝑉ଵଷ𝑉ଷହ𝑉ହଷ𝑉ଷଵ𝐸ଵଷ 𝐸ଵହ 𝐸ଵଷ+ . . . − ቈ|𝑉ଵଷ|ଶ𝐸ଵଷଶ.|𝑉ଵଷ|ଶ𝐸ଵଷଶ+|𝑉ଵଷ|ଶ𝐸ଵଷଶ·|𝑉ଵହ|ଶ𝐸ଵହ+ . . . +|𝑉ଵହ|ଶ𝐸ଵହଶ·|𝑉ଵଷ|ଶ𝐸ଵଷ+|𝑉ଵହ|ଶ𝐸ଵହଶ.|𝑉ଵହ|ଶ𝐸ଵହ+ . . . ቉ − 2· 𝑉ଵଵ ቈ𝑉ଵଷ𝑉ଷଷ𝑉ଷଵ𝐸ଵଷଶ 𝐸ଵଷ+𝑉ଵଷ𝑉ଷହ𝑉ହଵ𝐸ଵହଶ 𝐸ଵଷ+ . . . +𝑉ଵହ𝑉ହଷ𝑉ଷଵ𝐸ଵଷଶ 𝐸ଵହ+ . . . ቉+ 𝑉ଵଵଶ ቈ|𝑉ଵଷ|ଶ𝐸ଵଷଷ+|𝑉ଵହ|ଶ𝐸ଵହଷ+ . . . ቉ (3-41) where, similar to Eq. (A5-1) in Appendix A, all the 𝑉ଵହ, 𝑉ହଵ, 𝑉ଵ଻, 𝑉଻ଵ, . .. integrals vanish to zero, which means that all the terms shown by “…” in Eq. (3-41) vanish to zero. Also, 𝑉ଵଵ can be replaced by 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଵ) and |௏భయ|మாభయ by 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଶ) to get: 60 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ସ)= +𝑉ଵଷ𝑉ଷଷ𝑉ଷଷ𝑉ଷଵ𝐸ଵଷଷ+𝑉ଵଷ𝑉ଷହ𝑉ହଷ𝑉ଷଵ𝐸ଵଷଶ 𝐸ଵହ−ቀ𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଶ) ቁଶ𝐸ଵଷ− 2 · 𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଵ) ቈ𝑉ଵଷ𝑉ଷଷ𝑉ଷଵ𝐸ଵଷଷ቉+ ቀ𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଵ) ቁଶ቎𝐸ொ௘௟௦ౄ౛ౢ౟ (౥౨ ి౗౨౪)(ଶ)𝐸ଵଷଷ቏ (3-42) In this equation, all the terms have already been evaluated except 𝑉ଵଷ𝑉ଷଷ𝑉ଷଷ𝑉ଷଵ and 𝑉ଵଷ𝑉ଷହ𝑉ହଷ𝑉ଷଵ. The final form of these terms for the Heli and Cart molecules are presented in sections A.9 and A.10 in Appendix A, respectively. H2 induced dipole-ionic surface (induction) interaction As stated earlier in section 3.1.2.1.2, for these interactions, the same operators as Model I are used (Eq. (3-16) for 𝑉෠ுమି௦௨௥௜௡ௗ଴ and Eq. (3-18) for 𝑉෠ுమି௦௨௥௙௜௡ௗଶ ). In section 3.1.1.3.2, the first-order energies of these interactions were presented for both the anisotropic and isotropic cases. As will be discussed in section 4.2.2, going up to second-order corrections to the isotropic (𝐸௜௡ௗ଴) and anisotropic (𝐸௜௡ௗଶ) energies is enough for all H2 species (Para, Heli, and Cart). This was done by using operators 𝑉෠ுమି௦௨௜௡ௗ଴ and 𝑉෠ுమି௦௜௡ௗଶ in Eq. (2-8). But, for all H2 species, the second-order isotropic energies are zero (and even the third-order energies). For the anisotropic energy (𝐸௜௡ௗଶ), however, they are non-zero and the second-order energies are as follows: for Para: 61 𝐸௜௡ௗଶౌ౗౨౗(ଶ) =(0.282)ଶ𝐸଴ଶ× 𝐶௜௡ௗబଶ· ൛ 2 · (𝐶௜௡ௗଶଶ + 𝐶௜௡ௗଷଶ ) + 2 · (𝐶௜௡ௗସଶ + 𝐶௜௡ௗହଶ ) + 𝐶௜௡ௗభ ଶ ൟ (3-43) for Heli: 𝐸௜௡ௗଶౄ౛ౢ౟(ଶ) =1𝐸ଵଷ× 𝐶௜௡ௗబଶ· ൛ +(𝐶௜௡ௗଶଶ + 𝐶௜௡ௗଷଶ ) × [+(8.25 × 10ିଶ)ଶ + 0.319ଶ]+ (𝐶௜௡ௗସଶ + 𝐶௜௡ௗହଶ ) × [+0.143ଶ + 0.261ଶ]+ 𝐶௜௡ௗభଶ × 0.202ଶ ൟ (3-44) and for Cart: 𝐸௜௡ௗଶి౗౨౪(ଶ) =1𝐸ଵଷ× 𝐶௜௡ௗ଴ଶ· { +2 · (𝐶௜௡ௗଶଶ + 𝐶௜௡ௗଷଶ ) × 0.184ଶ+ 2. (𝐶௜௡ௗସଶ + 𝐶௜௡ௗହଶ ) × 0.233ଶ + 𝐶௜௡ௗଵଶ × 0.247ଶ } (3-45) where 𝐶௜௡ௗ೔’s are shown in Table 3-2. The procedure used to develop these equations is outlined by the example presented in section A.1 in Appendix A. 3.1.2.2 The 𝑽෡𝒕𝒐𝒕 case Unlike the 𝑉෠௦௘௣ case of Model II, in the 𝑉෠௧௢௧ case, the total energy for each correction term in the PT series is obtained using the 𝑉෠௧௢௧ operator. For example, the second order correction to the energy based on Eq. (2-8) is: 62 𝐸௡(ଶ) = ෍1𝐸௡௠ ௠ஷ௡·< 𝑛(଴)ቚ𝑉෠ுమି௦௨௥௙௥௘௣ିௗ௜௦௣ + 𝑉෠ுమି௦௨௥௙ொ + 𝑉෠ுమି௦௨௥௙௜௡ௗ଴ + 𝑉෠ுమି௦௨௥௙௜௡ௗଶ ቚ𝑚(଴) > .< 𝑚(଴)ቚ𝑉෠ுమି௦௨௥௙௥௘௣ିௗ௜௦௣ + 𝑉෠ுమି௦௨௥௙ொ + 𝑉෠ுమି௦௨௥௙௜௡ௗ଴ + 𝑉෠ுమି௦௨௥௙௜௡ௗଶ ቚ𝑛(଴) > (3-46) In this equation, unlike Eq. (3-22), all types of interactions are used in each integral to calculate the second-order energy in the NDPT series. The same method is used for higher-order correction terms as well. Here, the only assumption is that, the contribution from cross terms including the isotropic induction energies were ignored in the calculation of the fourth-order energies (in the 𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴ term for Para and in the 𝑉ଵଷ𝑉ଷଷ𝑉ଷଷ𝑉ଷଵ term for Heli and Cart). In fact, based on the data shown in 0, this assumption is a reasonable assumption. In fact, in Chapter 8 it will be shown that the total energy will not change even if all the cross terms in energy (including the isotropic induction interactions) are included in the first-, second-, and third-order energies as well as the 𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴ and 𝑉ଵଷ𝑉ଷହ𝑉ହଷ𝑉ଷଵ terms. In the following paragraphs, it is presented present how the NDPT correction terms are developed in the 𝑉෠௧௢௧ case. It should be noted again that, the first-order correction terms in the NDPT series will be the same in the 𝑉෠௦௘௣ and 𝑉෠௧௢௧ cases. Therefore, in the following sections only the second- and higher-order energies in the NDPT series will be presented. In addition, the equations for the Para, Heli and Cart molecules in the 𝑉෠௧௢௧ case follow the same formalism as in the 𝑉෠௦௘௣ case. Using Eq. (3-46), the second-order energies for the Para, Heli and Cart molecules are as follows for Para: 63 𝐸௧௢௧ౌ౗౨౗(ଶ) =(0.282)ଶ𝐸଴ଶ× ቄ2 · ൫𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൯ଶ+ 2. ൫𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൯ଶ + 2. ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ଶ+ 2. ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ቅ (3-47) for Heli: 𝐸௧௢௧ౄ౛ౢ౟(ଶ) =1𝐸ଵଷ× { + ቂ൫𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൯ଶቃ× [+(8.25 × 10ିଶ)ଶ + 0.319ଶ]+ ቂ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ଶቃ× [+0.143ଶ + 0.261ଶ]+ ቂ൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ଶቃ × 0.202ଶ } (3-48) and for Cart: 64 𝐸௧௢௧ి౗౨౪(ଶ) =1𝐸ଵଷ× ቄ+2. ቂ൫𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൯ଶቃ × 0.181ଶ+ 2. ቂ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ଶቃ× 0.233ଶ+൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ଶ× 0.247ଶቅ (3-49) where 𝑋௜, 𝐶௜௡ௗ೔, and 𝐾஽ோ(𝑖) functions were defined in Table 3-1, Table 3-2, and Table 3-3, respectively. For the third-order energy for Para Eq. (3-35) is used. The only term that must be evaluated here, is the 𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴ term. The procedure used to develop this term is similar to the example shown in section A.6 of Appendix A, but the final equation of the 𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴ term in the 𝑉෠௧௢௧ case is presented in section A.11 of Appendix A. Similarly, for the Heli and Cart molecules, the only term that must be evaluated in Eq. (3-38) is the 𝑉ଵଷ. 𝑉ଷଷ. 𝑉ଷଵ term which are presented for these molecules in section A.12 of Appendix A. For the fourth-order energy for Para, Eq. (3-40) is used. In this equation, the only terms that must be evaluated are the 𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴ and 𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴ terms. As stated earlier, in the 𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴ term for the Para molecule, it is assumed that the isotropic induction energy does not contribute to the total energy. Nonetheless, this contribution in the 𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴ term is zero. Since in the second-order energy, this contribution is zero as well, the analogy between Eq. (3-47) and Eq. (3-30) can be used in the second-order energies to develop the 𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴ and 𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴ terms. The comparison shows that Eq. (3-47) is in fact Eq. (3-30) with its 𝑋௜ terms replaced by new expressions including the contributions from the anisotropic induction and the 65 dispersion and repulsion terms. These terms in both equations Eq. (3-47) and Eq. (3-30) are listed in Table 3-4. In fact, this table shows that the 𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴ and 𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴ terms in the 𝑉෠௧௢௧ case can be simply developed by replacing the 𝑋௜’s in the 𝑉଴ଶ𝑉ଶଶ𝑉ଶଶ𝑉ଶ଴ and 𝑉଴ଶ𝑉ଶସ𝑉ସଶ𝑉ଶ଴ terms presented in section A.8 of Appendix A (the 𝑉෠௦௘௣ equations) by the corresponding expressions shown in this table. The same procedure can be used for the 𝑉ଵଷ𝑉ଷଷ𝑉ଷଷ𝑉ଷଵ and 𝑉ଵଷ𝑉ଷହ𝑉ହଷ𝑉ଷଵ terms for the Heli and Cart molecules in the 𝑉෠௧௢௧ case based on the equations presented in sections A.9 and A.10 of Appendix A. Table 3-4 The 𝑿𝒊 terms in Eq. (3-30) for the second-order energy for the Para molecule in the 𝑽෡𝒔𝒆𝒑 case and their corresponding expressions in Eq. (3-47) in the 𝑽෡𝒕𝒐𝒕 case The 𝑿𝒊 term in Eq. (3-30) (the 𝑽෡𝒔௘௣ Case) The corresponding expression in Eq. (3-47) (the 𝑽෡𝒕𝒐𝒕 Case) 𝑋଴ 𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ 𝑋ଵ 𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ 𝑋ଶ 𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ 𝑋ଷ 𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర 𝑋ସ 𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ 3.1.3 Model III In this model, the DPT formulation (section 2.2.2) is used for the interaction of an ortho-H2 molecule with the ionic surface. In the isolated case (unperturbed system), the J=1 ortho-H2 66 molecule has three degenerate eigenstates; the mJ=±1 Heli states in models I and II, and the mJ=0 Cart state. Therefore, the ortho-H2 molecule is a linear combination of these three states of the unperturbed system (Eq. (2-13)). This means that, in Model III, the simplifying assumption that the ortho-H2 molecule is always either a Heli or a Cart molecule is not used. In Model III, equations have been developed for the energies only up to the first-order correction in the PT series. Based on the DPT formulation in section 2.2.2, in order to find these energies, Eq. (2-15) must be solved. This equation is cubic in 𝐸௣. In order to solve this equation, the first step is to find each one of the 𝐻෡௜௝௣ integrals in the equation. This step was done separately for each component of the interaction potentials in Eq. (3-5). Therefore in Model III, the assumption is that each component of the energy in Eq. (3-5) is calculated separately and then the total energy is the sum of these individual components (similar to the 𝑉෠௦௘௣ case of Model II). Table 3-5 shows the 𝐻෡௜௝௣ integrals for the dispersion and repulsion interactions in Model III. In these terms, the subscript 1 refers to the mJ=+1 Heli state (which means that 𝛹ଵ(଴) in Eq. (2-13) is the Υଵଵ function). Also, subscript 2 refers to the mJ=‒1 Heli state, and subscript 3 refers to the mJ=0 Cart state. Also, 𝑡ℎ𝑒 𝐾஽ோ(𝑖) functions were defined in Table 3-3. 67 Table 3-5 The 𝑯෡ 𝒊𝒋𝒑 integrals in Model III used in Eq. (2-15) for the dispersion and repulsion interactions. 𝐻෡ଵଵವೃ௣ = −0.126 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0) 𝐻෡ଶଶವೃ௣ = −0.126 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0) 𝐻෡ଵଶವೃ௣ = +0.309 [𝑓஽ோ 𝐾஽ோ(2) + 𝑖 𝑓஽ோ 𝐾஽ோ(3)] 𝐻෡ଶଵವೃ௣ = −0.309 [𝑓஽ோ 𝐾஽ோ(2) + 𝑖 𝑓஽ோ 𝐾஽ோ(3)] 𝐻෡ଵଷವೃ௣ = +0.218 [𝑓஽ோ 𝐾஽ோ(5) + 𝑖 𝑓஽ோ 𝐾஽ோ(4)] 𝐻෡ଷଵವೃ௣ = −0.218 [−𝑓஽ோ 𝐾஽ோ(5) + 𝑖 𝑓஽ோ 𝐾஽ோ(4)] 𝐻෡ଶଷವೃ௣ = +0.218 [−𝑓஽ோ 𝐾஽ோ(5) + 𝑖 𝑓஽ோ 𝐾஽ோ(4)] 𝐻෡ଷଶವೃ௣ = − 0.218 [𝑓஽ோ 𝐾஽ோ(5) + 𝑖 𝑓஽ோ 𝐾஽ோ(4)] 𝐻෡ଷଷವೃ௣ = +0.252 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0) By using the 𝐻෡௜௝௣ terms (presented in Table 3-5) in Eq. (2-15), the cubic equation for the first-order dispersion and repulsion energies in Model III is obtained. This equation is presented in section A.13 of Appendix A. Solving this equation will result in three different first-order energies for the ortho-H2 molecules over the ionic surface. These energies will be discussed in great detail in 0. Table 3-6 shows the 𝐻෡௜௝௣ integrals for the interaction between the quadrupole moment of the H2 molecule and the ionic surface in Model III. In these terms, (similar to Table 3-5) the subscript 1 refers to the mJ=+1 Heli state (which means that 𝛹ଵ(଴) in Eq. (2-13) is the Υଵଵ function), subscript 2 refers to the mJ=‒1 Heli state, and subscript 3 refers to the mJ=0 Cart state. Also, the 𝑋௜ functions were defined in Table 3-1. The cubic equation for the first-order energy of the quadrupole moment of H2 with the surface in Model III is shown in section A.14 of 68 Appendix A. Solving this cubic equation results in three different first-order energies for the ortho-H2 molecules. These energies will be discussed in great detail in 0. Table 3-6 The 𝑯෡ 𝒊𝒋𝒑 integrals in Model III used in Eq. (2-15) for the H2 quadrupole electrostatic interactions. 𝐻෡ଵଵೂ೐೗ೞ௣ = −0.126 · 𝑋ସ 𝐻෡ଶଶೂ೐೗ೞ௣ = −0.126 · 𝑋ସ 𝐻෡ଵଶೂ೐೗ೞ௣ = −0.309 · (𝑋଴ + 𝑖𝑋ଵ) 𝐻෡ଶଵೂ೐೗ೞ௣ = −0.309 · (𝑋଴ − 𝑖𝑋ଵ) 𝐻෡ଵଷೂ೐೗ೞ௣ = +0.218 · (𝑋ଶ + 𝑖𝑋ଷ) 𝐻෡ଷଵೂ೐೗ೞ௣ = −0.218 · (−𝑋ଶ + 𝑖𝑋ଷ) 𝐻෡ଶଷೂ೐೗ೞ௣ = +0.218 · (−𝑋ଶ + 𝑖𝑋ଷ) 𝐻෡ଷଶೂ೐೗ೞ௣ = −0.218 · (𝑋ଶ + 𝑖𝑋ଷ) 𝐻෡ଷଷೂ೐೗ೞ௣ = +0.252 · 𝑋ସ For the isotropic induction energy, only the 𝐻෡௜௜௣ integrals are non-zero. This is because the operator for this interaction (𝑉෠௜௡ௗ଴) does not include any spherical harmonic functions (see Eq. (3-16)). Therefore, all 𝐻෡௜௝௣ integrals can be written as 𝑊(0) × 𝛿௜௝, in which 𝑊(0) is in fact 𝑉෠௜௡ௗ଴ and 𝛿௜௝ is the Kronecker delta, i.e., 𝛿௜௝ = ቄ01 𝑖 ≠ 𝑗𝑖 = 𝑗 . Therefore, the 𝐻෡௜௜௣ integrals are all the same and as follows: 𝐻෡ଵଵೂ೐೗ೞ௣ = 𝐻෡ଶଶೂ೐೗ೞ௣ = 𝐻෡ଷଷೂ೐೗ೞ௣ = 𝑊(0) (3-50) Thus, the cubic equation for the first-order energy of the isotropic induction energy is: 𝐸ொ௘௟௦௣ ଷ − 3 × 𝑊(0). 𝐸ொ௘௟௦௣ ଶ + 3 × 𝑊ଶ(0). 𝐸ொ௘௟௦௣ − 𝑊ଷ(0) = 0 (3-51) Solving the above equation gives the first-order energies for the isotropic H2 induction interaction. For the anisotropic interaction, the 𝐻෡௜௝௣ integrals are presented in Table 3-7. In these 69 terms (similar to Table 3-5) the subscript 1 refers to the mJ=+1 Heli state (which means that 𝛹ଵ(଴) in Eq. (2-13) is the Υଵଵ function), subscript 2 refers to the mJ=‒1 Heli state, and subscript 3 refers to the mJ=0 Cart state. Also, the 𝐶௜௡ௗ೔ functions were defined in Table 3-2. Section A.15 of Appendix A shows the cubic equation for the first-order energy of the anisotropic H2 induced dipole with the surface in Model III. Table 3-7 The 𝑯෡ 𝒊𝒋𝒑 integrals in Model III used in Eq. (2-15) for the H2 induction electrostatic interactions. 𝐻෡ଵଵ೔೙೏మ௣ = −0.126 · 𝐶௜௡ௗబ𝐶௜௡ௗభ 𝐻෡ଶଶ೔೙೏మ௣ = −0.126 · 𝐶௜௡ௗబ𝐶௜௡ௗభ 𝐻෡ଵଶ೔೙೏మ௣ = −0.309 · 𝐶௜௡ௗబ · ൫𝐶௜௡ௗమ + 𝑖𝐶௜௡ௗయ൯ 𝐻෡ଶଵ೔೙೏మ௣ = −0.309 · 𝐶௜௡ௗబ · ൫𝐶௜௡ௗమ − 𝑖𝐶௜௡ௗయ൯ 𝐻෡ଵଷ೔೙೏మ௣ = +0.218 · 𝐶௜௡ௗబ · ൫𝐶௜௡ௗఱ + 𝑖𝐶௜௡ௗర൯ 𝐻෡ଷଵ೔೙೏మ௣ = −0.218 · 𝐶௜௡ௗబ · ൫−𝐶௜௡ௗఱ + 𝑖𝐶௜௡ௗర൯ 𝐻෡ଶଷ೔೙೏మ௣ = +0.218 · 𝐶௜௡ௗబ · ൫−𝐶௜௡ௗఱ + 𝑖𝐶௜௡ௗర൯ 𝐻෡ଷଶ೔೙೏మ௣ = −0.218 · 𝐶௜௡ௗబ · ൫𝐶௜௡ௗఱ + 𝑖𝐶௜௡ௗర൯ 𝐻෡ଷଷ೔೙೏మ௣ = +0.252 · 𝐶௜௡ௗబ · 𝐶௜௡ௗభ 3.2 H2-H2 interaction potentials For these interactions the common assumption is using the pairwise approximation for all interaction potentials. All theoretical studies in the field utilized pairwise (between two different 70 H2 molecules) interaction potentials19,26,27,54,55. While the contribution of the three-body (between three different H2 molecules) potentials in the interaction energies is small (for example, for H2-H2 it is less than 5%)92, including these terms in the potential increases the computational cost dramatically73,75. For example, for a system of N molecules the pairwise interactions only include ே(ேିଵ)ଶ (~𝑁ଶ) distinct pairs of interaction, whereas the three-body case includes ே(ேିଵ)(ேିଶ)଺ (~𝑁ଷ) pairs72. As stated at the beginning of this chapter, the H2-H2 interactions are developed separately in two different models, Model I and Model II. In both of these models, the 𝑉෠௘௫௧ in Eq. (3-4) is replaced by the interaction potential between two H2 molecules, (𝑉෠ுమିுమ). Then the PT method up to first order is used to calculate the interaction between two H2 molecules, denoted by “1” and “2”. Using Eq. (2-7), the first-order perturbation energy is: 𝐸ுమିுమ =< 𝛹௄,௠಼ · 𝛹௅,௠ಽᇱห𝑉෠ுమିுమห𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ > (3-52) where 𝛹௄,௠಼ and 𝛹௅,௠ಽᇱ are the wavefunctions of the H2 molecules 1 and 2, respectively. Here, the first assumption is that going up to first-order energy in the NDPT series is enough in the calculation of the H2-H2 energy, i.e., 𝐸ுమିுమ = 𝐸ுమିுమ(ଵ) . It is also assumed that 𝛹௡(଴) in Eq. (2-7) is 𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ, the product of rotational wavefunctions of the two independent molecules (the orbital approximation). For 𝑉෠ுమିுమ in Eq. (3-52), the potential by Diep and Johnson78,79 is used. This potential is expressed in terms of the distance between these two molecules (𝑟) as well as their orientations. The orientations are expressed in terms of spherical coordinates (𝜃ଵ, 𝜑ଵ) and (𝜃ଶ, 𝜑ଶ) where the vector connecting the centers of mass of molecules “1” and “2” defines the polar axis. This anisotropic potential has the following terms: 71 𝑉෠ுమିுమ = 𝑉෠଴଴଴ + 𝑉෠ଶ଴ଶ + 𝑉෠଴ଶଶ + 𝑉෠ଶଶସ (3-53) where the individual parts of this operator, i.e., 𝑉෠଴଴଴, 𝑉෠ଶ଴ଶ, 𝑉෠଴ଶଶ, 𝑉෠ଶଶସ are presented in Table 3-8. In these operators, the orientation dependence of the molecule is included in the spherical harmonic functions, Υ(𝜃, 𝜑), whereas the spatial dependence is carried in the 𝑉௜௝௞(𝑟) functions (see Table 3-8). The dependence of the 𝑉௜௝௞(𝑟) functions on the distance between the two H2 molecules (𝑟) has been tabulated in Ref. 78,79 (Diep and Johnson’s model). To calculate the H2-H2 interaction, first, Eq. (3-52) is broken down into four terms as follows, which is because of the form of the 𝑉෠ுమିுమ operator in Eq. (3-53): 𝐸ுమିுమ = 𝐸଴଴଴ + 𝐸ଶ଴ଶ + 𝐸଴ଶଶ + 𝐸ଶଶସ (3-54) where 𝐸௜௝௞ =< 𝛹௄,௠಼ · 𝛹௅,௠ಽᇱห𝑉෠௜௝௞ห𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ > (3-55) and where the 𝑖, 𝑗, 𝑎𝑛𝑑 𝑘 indices are the same as the indices in the 𝑉௜௝௞(𝑟) functions in Table 3-8. Section B.1 of Appendix B shows how each one of the terms in Eq. (3-54) are calculated based on Eq. (3-55) and the wavefunctions of the H2 molecule 1 (𝛹௄,௠಼) and 2 (𝛹௅,௠ಽᇱ) in the H2-H2 configuration. In section B.1, it is also shown that to calculate 𝐸ுమିுమ in Eq. (3-54), first 𝐸଴଴଴, 𝐸ଶ଴ଶ, 𝐸଴ଶଶ, and 𝐸ଶଶସ terms must be calculated using Eq. (B1-1), (B1-4), (B1-9), and (B1-10) (see section B.1). Then it is shown that, to calculate these 𝐸௜௝௞ energies, first the integrals 𝐼଴଴଴(1), 𝐼଴଴଴(2), 𝐼ଶ଴ଶ(2), 𝐼଴ଶଶ(1), and 𝑆ଶ௠಻(𝑖) (for 𝑚௃ = −2, −1,0,1,2 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2) must be evaluated. Each one of these integrals is defined in section B.1 of Appendix B for the H2 molecules 1 and 2 in the H2-H2 configuration (molecule 1 or 2 can be either Para, or Heli, or Cart). It is important to emphasize that the “1” and “2” notation in brackets nominally designates the identity of the molecule in the H2-H2 configuration, but implicitly they are specifying the 72 rotational state of the molecules. The values of each of these integrals will in principle change upon changing the molecules’ rotational states. The same considerations hold for the 𝑆ଶ௠಻(𝑖) integrals, where 𝑖 = 1 𝑜𝑟 2 denotes molecule 1 or 2. Table 3-8 The H2-H2 interaction operators in Eq. (3-53). The 𝚼𝑱,𝒎𝑱 functions are spherical harmonics and show the orientation dependence of the potential operator. The 𝑽𝒊𝒋𝒌 functions show the distance dependence of the operator. These functions have been tabulated in Refs. 78,79. The ′ sign in the equations shows the dependence of the function on the second H2 molecule (molecule 2). The unprimed functions show the dependence on the first molecule (molecule 1). 𝑉෠଴଴଴ = 𝑉଴଴଴ 𝑉෠ଶ଴ଶ = 𝑉ଶ଴ଶ · √5𝜋 · ቌඨ32(Υଶଶഥ + Υଶଶ) − Υଶ଴ቍ 𝑉෠଴ଶଶ = 𝑉଴ଶଶ · √5𝜋 · ቌඨ32(Υଶଶഥᇱ + Υଶଶᇱ) − Υଶ଴ᇱቍ 𝑉෠ଶଶସ = 𝑉ଶଶସ · ൬454√70൰ · ൬2𝜋15൰· ቐ+39 · (Υଶଶഥ · Υଶଶഥᇱ + Υଶଶ · Υଶଶᇱ) − (Υଶଶഥ · Υଶଶᇱ + Υଶଶ · Υଶଶഥᇱ)− 28. (Υଶଵഥ · Υଶଵഥᇱ + Υଶଵ · Υଶଵᇱ) + 4 · (Υଶଵഥ · Υଶଵᇱ + Υଶଵ · Υଶଵഥᇱ)− 10ඨ32. [Υଶ଴ᇱ · (Υଶଶഥ + Υଶଶ) + Υଶ଴ · (Υଶଶഥᇱ + Υଶଶᇱ)] + 18 · Υଶ଴ · Υଶ଴ᇱቑ 73 In the following sections, the formalism of the H2-H2 interaction energy is discussed in models I and II based on the wavefunctions of the H2 molecules in each model. It should be mentioned that Model III only considers the interaction of a single ortho-H2 molecule with the ionic surface and is not used in the simulation of a layer of H2 molecules over the ionic surface. As a result, there is no H2-H2 interaction potential developed for model III. 3.2.1 Model I In this model, it is assumed that the ionic surface does not have any impact on the H2 molecule’s rotational state, even when this molecule sits very close to the surface. In other words, the rotational state (wavefunction) of the H2 molecule in the perturbed system (adsorbed phase) is the same as its rotational state in the unperturbed system (isolated phase). This means that 𝛹௄,௠಼ = Υ௅,௠ಽ in Eq. (3-55), where Υ௅,௠ಽ is the spherical harmonic function (ϒ଴଴ for Para, ϒଵଵ or Υଵଵഥ for Heli, and ϒଵ଴ for Cart, see Eq. (3-3)). As explained in the previous section and in section B.1, in order to calculate the interaction between different H2 species, the integrals 𝐼଴଴଴(1), 𝐼଴଴଴(2), 𝐼ଶ଴ଶ(2), 𝐼଴ଶଶ(1), and 𝑆ଶ௠಻(𝑖) (for 𝑚௃ = −2, −1,0,1,2 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2) defined in section B.1 must be calculated. Section B.2 of Appendix B shows how each one of these integrals are evaluated for the Para, Heli, and Cart molecules in Model I. Table 3-9 summarizes these calculated integrals in Model I. 74 Table 3-9 The amount of the 𝑰𝟎𝟎𝟎(𝟏), 𝑰𝟎𝟎𝟎(𝟏), 𝑰𝟐𝟎𝟐(𝟐), 𝑰𝟎𝟐𝟐(𝟏), and 𝑺𝟐𝒎𝑱(𝒊) integrals (for 𝒎𝑱 = −𝟐, −𝟏, 𝟎, 𝟏, 𝟐 and 𝒊 = 𝒎𝒐𝒍𝒆𝒄𝒖𝒍𝒆 𝟏 𝒐𝒓 𝟐 ) defined in section B.1 of Appendix B for different H2 species in Model I. Molecule 𝒊 = 𝟏 (or 𝟐) can be either a Para, or a Heli, or a Cart molecule. 𝒕𝒚𝒑𝒆 𝒐𝒇 𝒎𝒐𝒍𝒆𝒄𝒖𝒆 𝟏 𝒐𝒓 𝟐 → Para Heli Cart 𝑰𝟎𝟎𝟎(𝒊) 1 1 1 𝑰𝟐𝟎𝟐(𝟐) 1 1 1 𝑰𝟎𝟐𝟐(𝟏) 1 1 1 𝑺𝟐𝒎𝑱(𝒊) 𝒎𝑱 = −𝟐, −𝟏, 𝟏, 𝟐 0 0 0 𝑺𝟐𝟎(𝒊) 0 ‒0.126 +0.252 By using the integrals listed in Table 3-9, the H2-H2 interaction energy between any pairs of the H2 molecules in the system can be calculated using Eq. (3-54) and by calculating 𝐸଴଴଴, 𝐸ଶ଴ଶ, 𝐸଴ଶଶ, and 𝐸ଶଶସ using Eq. (B1-1), (B1-4), (B1-9), and (B1-10) in section B.1 of Appendix B. Here only two examples of these interactions in Model I are given. For the Para-Para the energy is: 𝐸୔ୟ୰ୟି୔ୟ୰ୟ = 𝑉଴଴଴ (3-56) and for Heli-Heli, it is: 𝐸ୌୣ୪୧ିୌୣ୪୧ = 𝑉଴଴଴ + 𝑉ଶ଴ଶ × √5𝜋 × 0.126 + 𝑉଴ଶଶ × √5𝜋 × 0.126+ 𝑉ଶଶସ × ൬454√70൰ · ൬2𝜋15൰ × 18 × 0.126ଶ (3-57) 3.2.2 Model II Whereas Model I assumed the molecular rotational states were unchanged, Model II includes the first-order correction to the rotational state by the surface. This corrected wavefunction is 75 then used in Eq. (3-52) to calculate the interaction energy between two H2 molecules (in Model II). Based on Eq. (3-52), the first step is to find the wavefunction of the Para, Heli, and Cart molecules in Model II. In general, this wavefunction in the adsorbed (perturbed) system has the following form: 𝛹௄,௠಼ = ෍ ෍ ൭෍ 𝑎௅,௠ಽ(௜)ஶ௜ୀଵ൱ · 𝛹௅,௠ಽ(଴)௅௠ಽୀି௅ஶ௅ୀ଴ (3-58) where 𝛹௄,௠಼ is the total wavefunction of the H2 molecule in the adsorbed phase (perturbed system), 𝛹௅,௠ಽ(଴) is the wavefunction of this molecule in the isolated system (which is Υ௅,௠ಽ), and 𝑎௅,௠ಽ(௜) is the 𝑖th-order correction to the coefficient for each 𝛹௅,௠ಽ(଴) term in the NDPT series. In Model II, it is assumed that only the first-order correction to the coefficients are important and the contributions from the higher-order corrections in the NDPT series are negligible. The impact of this assumption on the H2-H2 interaction energy will be discussed in detail in section 5.2.2. Based on these assumptions, Eq. (3-58) becomes: 𝛹௄,௠಼ = ෍ ෍ 𝑎௅,௠ಽ(ଵ) · Υ௅,௠ಽ௅௠ಽୀି௅ஶ௅ୀ଴ (3-59) where according to the NDPT formalism: 𝑎௅,௠ಽ(ଵ) = ଵாಿ(బ)ିாಽ(బ) < 𝛹௅,௠ಽ(଴) ห𝑉෠ுమି௦௨௥௙ห𝛹ே,௠ಿ(଴) > , 𝑁 ≠ 𝐿 (3-60) Also, 𝐸ே(଴) is the rotational energy of the Para, or Heli, or Cart molecule in the reference state (which is associated with 𝛹ே,௠ಿ(଴) ), 𝐸௅(଴) is the rotational energy of the molecule at a higher-energy state (the excited state which is associated with 𝛹௅,௠ಽ(଴) ), and 𝑉෠ுమି௦௨௥௙ is the H2-surface potential operator in Eq. (3-5). 76 As seen in Eq. (3-60), the coefficient of each one of the 𝛹௅,௠ಽ(଴) states (excited state) is inversely proportional to the difference between its rotational energy, 𝐸௅(଴), and the rotational energy of the reference state, 𝐸ே(଴). This means that the higher the energy of the excited state (𝐸௅(଴)), the smaller its contribution to the total wavefunction, 𝑎௅,௠ಽ(ଵ) . Based on these contributions, the wavefunction of the Para molecule in the adsorbed phase is assumed to include only the following terms: 𝛹୔ୟ୰ୟ = 𝑁୔ୟ୰ୟ. (𝜎଴଴Υ଴଴ + 𝜎ଶଶഥΥଶଶഥ + 𝜎ଶଵഥΥଶଵഥ + 𝜎ଶ଴Υଶ଴ + 𝜎ଶଵΥଶଵ + 𝜎ଶଶΥଶଶ) (3-61) where 𝑁୔ୟ୰ୟ is the normalization factor. Also, 𝜎଴଴ = 1, which means that the reference state (wavefunction) (𝛹ே,௠ಿ(଴) ) for the Para molecule in Eq. (3-60) is Υ଴଴. Section B.3 shows how the equations for the 𝜎௜௝ coefficients are developed using Eq. (3-60) and based on this reference wavefunction. In addition to these coefficients, in section B.3, the derivation of 𝑁୔ୟ୰ୟ is also presented. The next molecule discussed here is the Heli molecule. For this molecule, the wavefunction in the perturbed system is: 𝛹ୌୣ୪୧ = 𝑁ୌୣ୪୧. (𝛼ଵଵഥΥଵଵഥ + 𝛼ଵଵΥଵଵ + 𝛼ଷଷഥΥଷଷഥ + 𝛼ଷଶഥΥଷଶഥ + 𝛼ଷଵഥΥଷଵഥ + 𝛼ଷ଴Υଷ଴+ 𝛼ଷଵΥଷଵ + 𝛼ଷଶΥଷଶ + 𝛼ଷଷΥଷଷ) (3-62) where 𝑁ୌୣ୪୧ is the normalization factor. Since both Υଵଵഥ and Υଵଵ are both the wavefunctions for a Heli molecule, for this molecule, it is assumed that each contributes ଵଶ to the total wavefunction in the adsorbed system, i.e., 𝛼ଵଵഥ = 𝛼ଵଵ =ଵଶ. This means that the reference wavefunction (𝛹ே,௠ಿ(଴) ) for the Heli molecule in Eq. (3-60) is assumed to be ଵଶ(Υଵଵഥ + Υଵଵ). The development of the 𝛼௜௝ 77 coefficients in Eq. (3-62) based on this reference wavefunction follows the same procedure as presented in section B.3. The next molecule is a Cart molecule. For this Cart molecule, the wavefunction in the perturbed system is: 𝛹େୟ୰୲ = 𝑁େୟ୰୲. (𝛽ଵ଴Υଵ଴ + 𝛽ଷଷഥΥଷଷഥ + 𝛽ଷଶഥΥଷଶഥ + 𝛽ଷଵഥΥଷଵഥ + 𝛽ଷ଴Υଷ଴ + 𝛽ଷଵΥଷଵ+ 𝛽ଷଶΥଷଶ + 𝛽ଷଷΥଷଷ) (3-63) where 𝑁େୟ୰୲ is the normalization factor. Also, 𝛽ଵ଴ = 1, which means that the reference wavefunction (𝛹ே,௠ಿ(଴) ) for the Cart molecule in Eq. (3-60) is Υଵ଴. The development of the 𝛽௜௝ coefficients in Eq. (3-63) based on this reference wavefunction follows the same procedure as presented in section B.3. Then after evaluating these coefficients, the obtained wavefunctions (Eq. (3-61), Eq. (3-62), and Eq. (3-63)) are used in the calculation of the 𝐼଴଴଴(1), 𝐼଴଴଴(2), 𝐼ଶ଴ଶ(2), 𝐼଴ଶଶ(1), and 𝑆ଶ௠಻(𝑖) (for 𝑚௃ = −2, −1,0,1,2 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2) integrals presented in section B.1 of Appendix B. By using these integrals in Eq. (B1-1), (B1-4), (B1-9), and (B1-10) (of section B.1) the 𝐸଴଴଴, 𝐸ଶ଴ଶ, 𝐸଴ଶଶ, and 𝐸ଶଶସ terms can be calculated. Then these energies are used in Eq. (3-54) to calculate the H2-H2 interaction energy between any pairs of the H2 molecules in Model II (Para-Para, Heli-Para, Heli-Cart, etc.). Section B.4 shows how the Para-Para interaction energy is developed in Model II following the steps explained above. 78 Parameterization of the potential and the single H2-surface interactions 4.1 Parametrization procedure Having developed the potential models, the next steps are to parametrize and validate these models. This is done by comparing the single molecule binding energies as calculated by models I, II and III with the previous experimental and theoretical data. Note that for Model II all the data presented in this chapter are from the 𝑉෠௦௘௣ case (see section 3.1.2.1). Because the potentials are expressed in terms of perturbation expansions and Fourier series, their convergence is also tested to fully validate the models. After passing the validation stage, these models are then ready to be used in the MC simulations of a layer of H2 molecules on ionic surfaces. Since the ion-molecule repulsion and dispersion parameters available in the literature are of limited accuracy, these parameters may be adjusted. This adjustment is done by fitting the binding energy of a single H2 molecule to the experimental data. Although the fitting procedures used here are slightly different for models I and II (the procedure for Model III is the same as Model II), the reference values for the energies are the same. For MgO, there is only one literature value available67 for the binding energy of an H2 molecule to the surface. Although the experiments cannot determine the rotational state of the molecule or where it sits on the surface, previous experimental and theoretical studies10,17,20,22,23,26,37,53,54,56 suggest that the strongest binding energy of an H2 molecule to the MgO surface occurs when a helicopter-like H2 molecule sits right above a cation (this specific system is denoted by “Heli/Mg2+” throughout this thesis). The parameter fitting and validation 79 are therefore performed assuming the measured binding energies refer to the Heli/Mg2+ system (henceforth is referred to as the reference system). For NaCl, however, there are some theoretical values26 available for the binding energies of the Heli/Na+ and Para/Na+ systems and hence, both of these systems are used simultaneously in the fitting procedure. Since there is some uncertainty in the experimental data (see section 1.3.2.1), the energies obtained in models I and II (or III) must fall within the acceptable range. The fitting procedures differ between Model I and Model II (or III). This arises due to the nature of the potentials used. The Tang-Toennies potential used in model I uses two parameters to characterize repulsion and independent parameters to control the dispersion strength. In contrast, the Lennard-Jones potential used in model II has only two linked parameters to control the repulsion and dispersion combined. So, for Model I only the repulsion parameters are adjusted, whereas in Model II (or III) both repulsion and dispersion parameters are adjusted to get the desired value for the energy. Another important notion here is that, unlike the NaCl case, the literature value67 for the binding energy of an H2 molecule to the MgO surface is not reliable (see section 1.3.1.1). Therefore, for this substrate, beside the literature value of Ref. 67, one slightly stronger binding energy (a lower negative energy) is also considered as the binding energy for the reference system of Heli/Mg2+. The former is called the Upper Bound Energy (UBE) case, and the latter the Lower Bound Energy (LBE) case. In fact, the value used in the LBE case is the binding energy of a layer of H2 molecules to the surface 37 (see section 1.3.1.1). This energy seems to be a lower limit for the binding energy. It was decided to use this lower value because it not only helps in a thorough and careful comparison of our simulation results with the experimental data 80 (to validate the models used in this work), but as will be seen in Chapter 7, it also helps to resolve the ambiguities in the experiments. It should be added that, in the fitting procedure of the UBE and the LBE cases for MgO, the only difference between the parameters is in the repulsion parameter. In fact, the fitting procedure explained above is implemented first for the UBE case. Then in the LBE case, the dispersion parameter is kept the same (as the one in the UBE case), but the repulsion parameter is re-adjusted. Therefore, throughout the rest of this dissertation, both of the UBE and LBE cases are discussed wherever the H2/MgO results are presented. In the next section, the fitting procedures used in models I and II will be discussed in more detail. As mentioned earlier, the procedure for Model III is the same as Model II and is not discussed here. 4.1.1 Model I (the RAP Model) In Model I, the repulsion and dispersion interactions are formulated by the Tang-Toennies potential (Eq. (3-10)). As stated before, only the repulsion parameters are treated as adjustable variables in this model. In fact, it is assumed that the dispersion parameters that are obtained from the literature25 are accurate enough for the calculation of the energy (see Table C1-1 in section C.1 of Appendix C). Figure 4-1 illustrates the reference system of Heli/Mg2+ in the fitting procedure of the energy of an H2 molecule on the MgO substrate (for NaCl, both Heli/Na+ and Para/Na+ systems are used simultaneously). As explained in Chapter 3, the axis of quantization is always assumed to be fixed and parallel to the surface normal. 81 Figure 4-1 A system of Heli/Mg2+. The blue spheres show the cations and the red spheres show the anions in the ionic crystal. The dashed yellow arrow shows the surface normal, which is perpendicular to the plane of the ionic surface. The violet molecule is the Heli molecule sitting right above the cation. The dashed line is a visual aid to show the relative position of this molecule above the surface. The cyan arrow shows the axis of quantization of the Heli particle. Also, the saddle site (not shown) is defined as the midway point between two neighboring anions or cations in the crystal surface. In Model I, the adjusted parameter is, more precisely, the pre-exponential factor (𝐴௜௝) in the repulsion part of the Tang-Toennies potential (Eq. (3-10)). Note that there are in fact two parameters, one from each molecule-ion pair: H2-Mg2+ and H2-O2-. The initial dispersion and repulsion parameters in this potential are obtained from a previous study on the molecular simulation of H2 layers on the ionic surfaces25 (see Table C1-1 in section C.1 of Appendix C). Then only the 𝐴௜௝ parameter of this set of data is adjusted in the fitting procedure. Once this adjusted parameter was determined, the entire set of parameters was fixed and used in the calculation of the interaction between all H2 species with the ionic surface, no matter where the 82 molecule sits above the surface. The final parameters used in the calculation of energies in Model I are listed in Table C1-2 in section C.1 of Appendix C. Also, the results for the interaction of an H2 molecule with the ionic surface in Model I are presented in section 4.3.1. 4.1.2 Model II In Model II, both the C6 (dispersion) and C12 (repulsion) parameters in the LJ potential of Eq. (3-23) are adjusted in the fitting procedure. In this procedure, initial values for the C6 and C12 LJ parameters are obtained by fitting the LJ energy of Model II to the adjusted Tang-Toennies energy in Model I (see Table C1-3 in section C.1 of Appendix C). Then, after including the higher-order PT energies into the total energy of Model II, the C6 and C12 parameters are re-adjusted by fitting the total energy to the reference values for the energies (for example the UBE and LBE cases). Then, similar to Model I, these adjusted parameters are used to calculate the interaction of the H2 molecule with the surface, regardless of the position and rotational state of this molecule. The final parameters used in the calculation of energy in Model II are listed in Table C1-4 in section C.1 of Appendix C. Also, the energies between an H2 molecule with the ionic surface obtained in Model II are presented in section 4.3.2. 4.2 Convergence of the potential In Chapter 3, it was shown that the 𝑉෠ுమି௦௨௥௙ operators in Model I (only the electrostatic operators in Eq. (3-5)) and Model II (all operators of Eq. (3-5)) include Fourier terms. It was explained that the Fourier terms are shown by the {𝑛, 𝑚} terms (or vectors) in the interaction model. These vectors are added to the core potential, which is the {𝑛, 𝑚}={0,0} Fourier term (or vector) in the potential model.93 Usually in the calculations, the Fourier terms are added into the 83 potential as a set of vectors rather than as individual vectors91,93. The order of these vector sets is: (0,0), (0,1), (1,1), (0,2), (1,2), (2,2), (0,3), (1,3), 𝑒𝑡𝑐91,93. In this series, including a given vector set, (𝑘, 𝑗) with 𝑘 ≤ 𝑗, means that, the Fourier vectors {±𝑘, ±𝑗} and {±𝑗, ±𝑘} are also added to the potential. As an example, the (0,2) vector set, means that all {0,0}, {0,±1}, {±1,0}, {±1,±1}, {0,±2}, and {±2,0} vectors have been included in the calculation of the potential91,93. In principle, to get a well-behaved potential, the sum of the Fourier terms in the series should converge to a constant value. This convergence is usually obtained at a finite number of terms, say, at vector set (𝑛,𝑚), after which adding more terms does not change the magnitude of the potential. On the other hand, to get the desired binding energy for the H2 molecule, the PT corrections to the energy should also converge to the desired energy. This convergence is also achieved after the inclusion of a specific number of terms in the PT series. Therefore, in this chapter, two convergence analyses must be done on the potentials that are used in models I and II (or III); 1) the convergence of the Fourier terms in the potential operators, and 2) the convergence of the energies in the PT series. In this section, these two analyses are discussed in great detail. It should be added that, the operators that are used in models I, II and III (Eq. (3-5)) were generally developed for the adsorption of H2 on ionic surfaces and hence, they do not depend on the type of the substrate or even the interaction parameters that are used here. Therefore, in this section, the convergence analyses mentioned above are only done for the MgO substrate and just for the UBE case. 84 4.2.1 The Fourier series convergence The convergence of the Fourier series largely depends on the type of the potential used in Eq. (3-5). As discussed in Chapter 3, in model I, the Fourier series were only used in the electrostatic potentials in this equation. In Model II, however, they were used in all types of potentials shown in Eq. (3-5). In this section, the convergence of the Fourier series is discussed based on the type of the potentials used in each model. 4.2.1.1 Model I (The RAP Model) In this model and for both components of the electrostatic potentials, the argument by Kroes and Mowrey53,54 is followed. According to these researchers, for the electrostatic potential operators in Eq. (3-12) and Eq. (3-16), only terms with |𝑛| + |𝑚| ≤ 1 in the Fourier series are important in the calculation of the energy. This is equivalent to terms up to the vector set of (0,1) in Eq. (3-12) and Eq. (3-16). In Chapter 3, it was discussed that in both models I and II (or III), the same potential operators are used for the electrostatic interactions (𝑉෠ுమି௦௨௥௙ொ and 𝑉෠ுమି௦௨௥௙௜௡ௗ operators in Eq. (3-5)). Since for these interactions, more Fourier terms have been used in Model II, the Fourier series of Model I is a subset of II. Thus, in this section, only the convergence analysis of these interactions in Model II is discussed. 4.2.1.2 Model II In Model II, the Fourier series are used to formulate all components of the potential in Eq. (3-5). This means that, unlike Model I, these series are also used in the dispersion and repulsion potentials. As mentioned earlier, the convergence of this series is usually achieved at a specific vector set at which the series is truncated. In order to monitor this convergence, in all the tables 85 and figures in this chapter, the dependence of the results on these vector sets is shown. In the following paragraphs, the convergence of the energy will be discussed in more detail for each component of the interaction potential. 4.2.1.2.1 H2-ionic lattice Repulsion and dispersion interactions (LJ interactions) In Model I, the energy contribution from the bulk of the ionic crystal was modelled separately (Eq. (3-11)) from the top layer of the ionic substrate. In Model II, however, the LJ potential of Briquez et al.26 (Eq. (3-23)) was developed in a way that the layers below the top layer are added explicitly in the potential (𝑝 < 0 in Eq. (3-23)) eliminating the need for a separate bulk contribution. Therefore, in Model II, the dispersion and repulsion potentials and the bulk potential are calculated using the same equation (note that in Eq. (3-5) the bulk potential is a part of 𝑉෠ுమି௦௨௥௥௘௣ and 𝑉෠ுమି௦௨௥௙ௗ௜௦௣ operators). Here the question is how many of the substrate layers must be included into the potential to get convergence to a reasonable binding energy for an H2 molecule? Thus, for the dispersion and repulsion operators in Model II, two convergence analyses must be performed; 1) the Fourier series convergence, and 2) the substrate layer convergence. The Fourier series convergence Table 4-1 shows how the minimum LJ energies of Para, Heli, and Cart molecules (at their optimum heights over an Mg2+ site on the surface) vary when more Fourier vector sets are included in the potential. In all these cases, it is seen that, full convergence to a constant energy is achieved after the inclusion of the (1,2) vector set. 86 Table 4-1 The minimum LJ energies in the UBE case after adding each Fourier vector set to the energy. The data are presented for the Para/Mg2+ system at the height of 3.25 Å, and for the Heli/Mg2+ and Cart/Mg2+ systems at 3.24 Å. The (𝒌,𝒋) vector set is defined in section 4.2 of the text. All the energies are in kcal/mol. (0,0) (0,1) (1,1) (0,2) (1,2) (2,2) (0,3) Heli - 0.30474 - 0.46805 - 0.43587 - 0.43378 - 0.43502 - 0.43499 -0.43500 Para - 0.31105 - 0.47049 - 0.43907 - 0.43704 - 0.43824 - 0.43821 - 0.43822 Cart - 0.31257 - 0.47857 - 0.44548 - 0.44330 - 0.44460 - 0.44457 - 0.44458 In order to see how the energies at heights other than the optimum height change with the number of the vector sets in the potential, in Figure 4-2 the energies for the Heli/Mg2+ case at all heights are presented. The data are only graphed for the vector sets up to (0,3), because going to the next vector sets does not make a visual difference to the curves. In Table 4-2, however, more vector sets (up to (2,4)) were included for the energies shown in Figure 4-2. As seen, at all heights over the surface, a good convergence is achieved at the (2,3) vector set. The maximum difference between the energies of this vector set and the one right before that in the series (i.e., the (1,3) vector set) is about 0.16% at a very low height of 2.00 Å. At larger distances from the surface, this difference becomes even smaller. Similar trends are obtained for the Para/Mg2+ and Cart/Mg2+ cases (see section C.1 of Appendix C). Nonetheless, in all MC simulations presented in Chapter 7, the vector sets up to (4,4) were used in the calculation of the energies. 87 Figure 4-2 The LJ (dispersion-repulsion) energy for the Heli/Mg2+ system in the UBE case after adding each Fourier term into the potential. (𝒌,𝒋) is defined in section 4.2 of the text. To assure that this convergence does not depend on the interacting site above the surface, these energies are discussed for all H2 species in different rotational states over anion (H2/O2-) and saddle (H2/Saddle) sites. A similar trend (to the H2/Mg2+ system) is obtained with a full convergence achieved after the inclusion of the (2,3) vector set. It should also be noted that for systems of H2/Cation and H2/Saddle, adding the Fourier terms to the core potential (the (0,0) term) shifts all the energies as well as the position of the minimum energy to lower values (compared to the (0,0) term, see Figure 4-2), whereas for the H2/Anion, they all shift to higher energies. -0.6-0.30.00.32.0 4.0 6.0 8.0Energy [kcal/mol]Height [Å](0,0)(0,1)(1,1)(0,2)(1,2)(2,2)(0,3)88 Table 4-2 The LJ energies for the Heli/Mg2+ system in the UBE case after adding each vector set into the potential. The (𝒌,𝒋) vector set is defined in section 4.2 of the text. All the energies are in kcal/mol. Height (0,0) (0,1) (1,1) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,3) (0,4) (1,4) (2,4) 2.00 53.06317 -34.45505 6.20091 15.72139 5.88752 6.83042 6.28536 6.98188 6.47035 6.48104 6.50491 6.47410 6.48401 2.05 41.03505 -24.13005 5.00889 11.55791 4.94264 5.53936 5.20011 5.62650 5.31815 5.32401 5.33741 5.32032 5.32563 2.10 31.81621 -16.96789 4.05088 8.56995 4.10569 4.48487 4.27288 4.53501 4.34828 4.35150 4.35906 4.34955 4.35240 2.15 24.72447 -11.97844 3.26997 6.39866 3.37597 3.61786 3.48489 3.64668 3.53310 3.53488 3.53916 3.53384 3.53538 2.20 19.24829 -8.49286 2.62653 4.79998 2.74647 2.90134 2.81764 2.91786 2.84850 2.84948 2.85192 2.84893 2.84976 2.25 15.00371 -6.05435 2.09236 3.60735 2.20757 2.30709 2.25421 2.31652 2.27399 2.27454 2.27592 2.27424 2.27469 2.30 11.70163 -4.34793 1.64666 2.70622 1.74891 1.81307 1.77956 1.81842 1.79224 1.79255 1.79334 1.79239 1.79264 2.35 9.12383 -3.15492 1.27377 2.01721 1.36042 1.40191 1.38061 1.40492 1.38875 1.38892 1.38937 1.38884 1.38897 2.40 7.10488 -2.32262 0.96143 1.48472 1.03269 1.05962 1.04603 1.06128 1.05125 1.05135 1.05161 1.05131 1.05138 2.45 5.51891 -1.74389 0.69987 1.06932 0.75729 0.77481 0.76612 0.77571 0.76947 0.76953 0.76968 0.76950 0.76955 2.50 4.26973 -1.34327 0.48116 0.74277 0.52676 0.53819 0.53261 0.53866 0.53477 0.53480 0.53489 0.53479 0.53481 2.55 3.28354 -1.06753 0.29875 0.48451 0.33455 0.34203 0.33845 0.34226 0.33983 0.33985 0.33990 0.33984 0.33985 2.60 2.50345 -0.87903 0.14713 0.27939 0.17501 0.17992 0.17760 0.18002 0.17849 0.17850 0.17853 0.17850 0.17851 2.65 1.88545 -0.75117 0.02167 0.11609 0.04325 0.04648 0.04498 0.04651 0.04555 0.04555 0.04557 0.04555 0.04556 2.70 1.39534 -0.66513 -0.08155 -0.01398 -0.06494 -0.06281 -0.06378 -0.06281 -0.06341 -0.06341 -0.06340 -0.06341 -0.06341 2.75 1.00646 -0.60765 -0.16588 -0.11741 -0.15315 -0.15174 -0.15237 -0.15175 -0.15214 -0.15214 -0.15213 -0.15214 -0.15213 2.80 0.69793 -0.56940 -0.23420 -0.19935 -0.22446 -0.22353 -0.22394 -0.22355 -0.22379 -0.22379 -0.22379 -0.22379 -0.22379 2.85 0.45334 -0.54385 -0.28894 -0.26382 -0.28152 -0.28090 -0.28117 -0.28091 -0.28107 -0.28107 -0.28107 -0.28107 -0.28107 2.90 0.25976 -0.52648 -0.33220 -0.31407 -0.32656 -0.32614 -0.32632 -0.32615 -0.32625 -0.32625 -0.32625 -0.32625 -0.32625 2.95 0.10695 -0.51414 -0.36578 -0.35266 -0.36149 -0.36122 -0.36133 -0.36122 -0.36129 -0.36129 -0.36128 -0.36129 -0.36129 89 Height (0,0) (0,1) (1,1) (0,2) (1,2) (2,2) (0,3) (1,3) (2,3) (3,3) (0,4) (1,4) (2,4) 3.00 -0.01322 -0.50472 -0.39121 -0.38169 -0.38795 -0.38777 -0.38785 -0.38777 -0.38781 -0.38781 -0.38781 -0.38781 -0.38781 3.05 -0.10720 -0.49681 -0.40980 -0.40289 -0.40733 -0.40721 -0.40726 -0.40720 -0.40723 -0.40723 -0.40723 -0.40723 -0.40723 3.10 -0.18016 -0.48949 -0.42266 -0.41764 -0.42080 -0.42072 -0.42075 -0.42071 -0.42073 -0.42073 -0.42073 -0.42073 -0.42073 3.15 -0.23623 -0.48217 -0.43077 -0.42710 -0.42936 -0.42930 -0.42932 -0.42929 -0.42930 -0.42930 -0.42930 -0.42930 -0.42930 3.20 -0.27871 -0.47452 -0.43492 -0.43225 -0.43386 -0.43382 -0.43383 -0.43381 -0.43381 -0.43381 -0.43381 -0.43381 -0.43381 3.25 -0.31029 -0.46637 -0.43582 -0.43386 -0.43501 -0.43499 -0.43500 -0.43497 -0.43498 -0.43498 -0.43498 -0.43498 -0.43498 3.30 -0.33309 -0.45765 -0.43405 -0.43261 -0.43344 -0.43342 -0.43343 -0.43340 -0.43341 -0.43341 -0.43341 -0.43341 -0.43341 3.35 -0.34886 -0.44837 -0.43010 -0.42905 -0.42964 -0.42963 -0.42963 -0.42961 -0.42961 -0.42961 -0.42961 -0.42961 -0.42961 3.40 -0.35899 -0.43856 -0.42441 -0.42363 -0.42406 -0.42405 -0.42405 -0.42403 -0.42403 -0.42403 -0.42403 -0.42403 -0.42403 3.45 -0.36463 -0.42830 -0.41732 -0.41675 -0.41706 -0.41705 -0.41705 -0.41703 -0.41703 -0.41703 -0.41703 -0.41703 -0.41703 3.50 -0.36668 -0.41766 -0.40914 -0.40872 -0.40894 -0.40894 -0.40894 -0.40892 -0.40892 -0.40892 -0.40892 -0.40892 -0.40892 3.55 -0.36589 -0.40675 -0.40012 -0.39981 -0.39997 -0.39997 -0.39997 -0.39995 -0.39995 -0.39995 -0.39995 -0.39995 -0.39995 3.60 -0.36288 -0.39563 -0.39047 -0.39024 -0.39036 -0.39035 -0.39035 -0.39034 -0.39034 -0.39034 -0.39034 -0.39034 -0.39034 3.65 -0.35813 -0.38439 -0.38037 -0.38020 -0.38028 -0.38028 -0.38028 -0.38027 -0.38027 -0.38027 -0.38027 -0.38027 -0.38027 3.70 -0.35204 -0.37310 -0.36997 -0.36984 -0.36990 -0.36990 -0.36990 -0.36989 -0.36989 -0.36989 -0.36989 -0.36989 -0.36989 3.75 -0.34493 -0.36183 -0.35938 -0.35929 -0.35933 -0.35933 -0.35933 -0.35932 -0.35932 -0.35932 -0.35932 -0.35932 -0.35932 3.80 -0.33708 -0.35063 -0.34872 -0.34865 -0.34868 -0.34868 -0.34868 -0.34866 -0.34866 -0.34866 -0.34866 -0.34866 -0.34866 3.85 -0.32868 -0.33954 -0.33805 -0.33800 -0.33802 -0.33802 -0.33802 -0.33801 -0.33801 -0.33801 -0.33801 -0.33801 -0.33801 3.90 -0.31992 -0.32863 -0.32746 -0.32742 -0.32744 -0.32744 -0.32744 -0.32742 -0.32742 -0.32742 -0.32742 -0.32742 -0.32742 3.95 -0.31093 -0.31790 -0.31699 -0.31696 -0.31697 -0.31697 -0.31697 -0.31696 -0.31696 -0.31696 -0.31696 -0.31696 -0.31696 4.00 -0.30182 -0.30740 -0.30668 -0.30666 -0.30667 -0.30667 -0.30667 -0.30666 -0.30666 -0.30666 -0.30666 -0.30666 -0.30666 90 The substrate layer convergence (the bulk potential) As stated earlier, in Model II, the contribution from the bulk energy to the H2-ionic crystal energy is included into the potential by adding the layers of ions below the top layer (𝑝 < 0 in Eq. (3-23)). In this section, the convergence of the LJ potential will be discussed based on the number of these layers of ions (below the top layer) that must be included into the potential to get a well-converged energy. Figure 4-3 illustrates how the layers below the top layer affect the LJ energy. In this figure, the results are presented only for the Heli/Mg2+ case, but same trends are observed for other rotational states of H2 regardless of its position over the surface, e.g., over the anion and the saddle sites. The softening of the repulsive wall as well as the deepening of the potential minimum indicates that the contributions from the ions below the top layer are mostly of dispersive nature. 91 Figure 4-3 The LJ energy for the Heli/Mg2+ system in the UBE case. The curves show how adding each layer of ions in the MgO crystal below the surface layer affects the potential. In order to see the energies in Figure 4-3 more closely, in Table 4-3 these energies are shown for heights from 2.0 up to 4.0 Å above the surface. As seen, the convergence is achieved after adding the fourth layer into the potential, with a deviation of less than 0.05% between the fourth and fifth layers at 2.00 Å. Table 4-4 shows how adding each layer in the calculation of the LJ potential in the Heli/Mg2+ system changes the minimum energy, its height, and the height at which the energy becomes zero. As seen in the table, there is no change in both the heights of the minimum and zero energies after adding the fourth layer. The shift of 1% (~0.00202 kcal/mol or 0.09 meV) in the LJ minimum energy after adding the fifth layer is considered negligible in the calculations. -0.5-0.10.30.72.0 3.0 4.0 5.0 6.0Energy [kcal/mol]Height [Å]top layerdown to 1stdown to 2nddown to 3rddown to 4thdown to 5thdown to 6th92 Therefore, to get an acceptable convergence in the energy, a total of five layers must be used in the calculation of the H2-MgO LJ interactions; the top layer plus four layers of ions beneath the top layer. Also, the effect of the number of the Fourier vector sets on the layer convergence was examined. The results show that the number of Fourier terms does not affect the number of ionic layers that must be included into the potential to get a well-converged LJ energy. Similar trends are observed for other H2 species over other interacting sites. It should also be added that, the parameterization procedure in section 4.1 was done after the convergence analysis, i.e., it was done for a total of five layers of ions. 93 Table 4-3 The LJ energies in kcal/mol for the Heli/Mg2+ system in the UBE case when the layers below the top layer are added into the potential. Here, “top layer” means that only the surface layer (𝒑 = 𝟎 in Eq. (3-23)) was used in the calculation of the LJ energy. “down to 6th” means that a total of six layers below the top layer have been included in the calculation. Height [Å] top layer down to 1st down to 2nd down to 3rd down to 4th down to 5th down to 6th 2.00 6.75850 6.54479 6.49667 6.48150 6.47533 6.47237 6.47079 2.05 5.59381 5.38762 5.34097 5.32615 5.32010 5.31719 5.31562 2.10 4.61332 4.41449 4.36925 4.35478 4.34884 4.34597 4.34442 2.15 3.78859 3.59690 3.55304 3.53890 3.53307 3.53025 3.52872 2.20 3.09506 2.91034 2.86779 2.85398 2.84826 2.84548 2.84397 2.25 2.51208 2.33410 2.29282 2.27933 2.27371 2.27098 2.26949 2.30 2.02221 1.85074 1.81070 1.79751 1.79200 1.78931 1.78784 2.35 1.61088 1.44572 1.40686 1.39397 1.38856 1.38591 1.38446 2.40 1.26582 1.10675 1.06903 1.05643 1.05112 1.04851 1.04708 2.45 0.97674 0.82353 0.78691 0.77460 0.76938 0.76681 0.76540 2.50 0.73498 0.58742 0.55187 0.53983 0.53471 0.53218 0.53079 2.55 0.53324 0.39112 0.35659 0.34482 0.33980 0.33730 0.33593 2.60 0.36535 0.22846 0.19492 0.18341 0.17848 0.17602 0.17467 2.65 0.22609 0.09423 0.06165 0.05039 0.04554 0.04313 0.04179 2.70 0.11105 -0.01598 -0.04765 -0.05865 -0.06341 -0.06580 -0.06712 2.75 0.01647 -0.10592 -0.13669 -0.14746 -0.15214 -0.15448 -0.15578 2.80 -0.06081 -0.17875 -0.20866 -0.21919 -0.22379 -0.22610 -0.22739 2.85 -0.12351 -0.23717 -0.26625 -0.27655 -0.28107 -0.28334 -0.28461 2.90 -0.17390 -0.28346 -0.31174 -0.32182 -0.32625 -0.32850 -0.32975 2.95 -0.21394 -0.31956 -0.34707 -0.35693 -0.36129 -0.36350 -0.36474 3.00 -0.24528 -0.34712 -0.37388 -0.38354 -0.38782 -0.39000 -0.39122 3.05 -0.26933 -0.36755 -0.39358 -0.40303 -0.40724 -0.40939 -0.41059 3.10 -0.28727 -0.38202 -0.40735 -0.41660 -0.42074 -0.42285 -0.42404 3.15 -0.30013 -0.39154 -0.41619 -0.42525 -0.42932 -0.43140 -0.43257 3.20 -0.30876 -0.39696 -0.42096 -0.42983 -0.43383 -0.43588 -0.43704 3.25 -0.31388 -0.39901 -0.42238 -0.43107 -0.43499 -0.43702 -0.43816 3.30 -0.31611 -0.39830 -0.42106 -0.42956 -0.43342 -0.43542 -0.43655 94 Height [Å] top layer down to 1st down to 2nd down to 3rd down to 4th down to 5th down to 6th 3.35 -0.31598 -0.39534 -0.41750 -0.42584 -0.42963 -0.43160 -0.43271 3.40 -0.31392 -0.39057 -0.41216 -0.42032 -0.42405 -0.42599 -0.42709 3.45 -0.31030 -0.38435 -0.40539 -0.41338 -0.41705 -0.41896 -0.42005 3.50 -0.30545 -0.37700 -0.39749 -0.40533 -0.40894 -0.41082 -0.41190 3.55 -0.29962 -0.36877 -0.38874 -0.39642 -0.39997 -0.40182 -0.40289 3.60 -0.29303 -0.35987 -0.37934 -0.38687 -0.39035 -0.39218 -0.39323 3.65 -0.28586 -0.35050 -0.36948 -0.37685 -0.38028 -0.38209 -0.38312 3.70 -0.27828 -0.34079 -0.35930 -0.36653 -0.36990 -0.37168 -0.37270 3.75 -0.27041 -0.33088 -0.34893 -0.35601 -0.35933 -0.36109 -0.36210 3.80 -0.26236 -0.32086 -0.33847 -0.34541 -0.34868 -0.35041 -0.35141 3.85 -0.25421 -0.31083 -0.32800 -0.33481 -0.33802 -0.33973 -0.34072 3.90 -0.24603 -0.30084 -0.31760 -0.32428 -0.32744 -0.32912 -0.33009 3.95 -0.23790 -0.29096 -0.30731 -0.31386 -0.31697 -0.31863 -0.31959 4.00 -0.22984 -0.28124 -0.29719 -0.30361 -0.30667 -0.30831 -0.30926 Table 4-4 The shift in the minimum LJ energy, its height as well as the height of the zero energy for the Heli/Mg2+ system in the UBE case after adding each layer of ions into the potential. only top layer down to 1st down to 2nd down to 3rd down to 4th down to 5th down to 6th Height at E=0 [Å] 2.76 2.69 2.68 2.67 2.67 2.67 2.67 Height at Emin [Å] 3.32 3.26 3.25 3.25 3.24 3.24 3.24 Emin [kcal/mol] -0.31631 -0.39907 -0.42238 -0.43107 -0.43500 -0.43702 -0.43817 95 4.2.1.2.2 H2 quadrupole-ionic lattice interaction (quadrupole potential) In section 4.2.1.1, it was mentioned that in Model I, only Fourier terms up to the (0,1) vector sets were used in the calculations. In Model II, however, more Fourier terms were used in the calculation of the energy. Figure 4-4 shows the interaction between the quadrupole moment of the Heli molecule with MgO in the Heli/Mg2+ system when different Fourier vector sets, (𝑘, 𝑗), are added into the 𝑉෠ுమି௦௨௥௙ொ potential operator. Table 4-5 presents this data at the height range of 2.0 Å to 4.0 Å. It is seen that the convergence is achieved even after the (0,1) vector set, which agrees with the comments by Kroes and Mowrey53,54 that suggested that only terms with |𝑛| +|𝑚| ≤ 1 are important in the potential. Nonetheless, they included Fourier terms up to the (0,3) vector set in their calculations. In Figure 4-4 and Table 4-5, one more set was included, which is the (2,3) vector set, but the results in Table 4-5 confirm that the shift in the energy compared to the (0,3) vector set is negligible. For example, in Table 4-5, at the height of 2.0 Å, the difference between the energies obtained from the (0,3) and (2,3) vector sets is 0.03%. This difference decreases as the molecule gets further away from the surface. Similar trends are observed for all H2 species above the other interacting sites (see Figure C2-2 and Figure C2-3 in section C.2 of Appendix C). Nonetheless, in this study, all the vector sets up to (4,4) were used in the calculation of the energy. 96 Figure 4-4 The interaction of the quadrupole moment of Heli H2 with the MgO surface in the UBE case for the Heli/Mg2+ system after adding each Fourier term into this potential. The (𝒌, 𝒋) vector set is defined in section 4.2 of the text. 4.2.1.2.3 H2 induced dipole-surface interactions (induction potential) For this type of the interaction potentials, only Fourier terms up to the vector set of (0,1) was used in the calculations. This, in fact, follows the argument by Kroes and Mowrey53,54, where they showed that only terms with |𝑛| + |𝑚| ≤ 1 (which is equivalent to going up to the vector set of (0,1) in our calculations) are important in the energy. -0.60-0.300.002.5 3.0 3.5 4.0Energy [kcal/mol]Height [Å](0,1)(1,2)(0,3)(2,3)97 Table 4-5 The interaction of the quadrupole moment of Heli H2 with the MgO surface in the LBE case for the Heli/Mg2+ system after adding each Fourier term into this potential. The (𝒌, 𝒋) vector set is defined in section 4.2 of the text. Height [Å] (0,1) (1,2) (0,3) (2,3) 2.00 -2.71184 -2.87653 -2.88104 -2.88187 2.05 -2.13104 -2.23014 -2.23262 -2.23305 2.10 -1.72406 -1.78568 -1.78709 -1.78732 2.15 -1.42902 -1.46876 -1.46960 -1.46973 2.20 -1.20724 -1.23384 -1.23435 -1.23443 2.25 -1.03449 -1.05293 -1.05326 -1.05330 2.30 -0.89557 -0.90875 -0.90896 -0.90899 2.35 -0.78085 -0.79049 -0.79064 -0.79065 2.40 -0.68412 -0.69131 -0.69141 -0.69142 2.45 -0.60132 -0.60675 -0.60682 -0.60682 2.50 -0.52969 -0.53382 -0.53387 -0.53388 2.55 -0.46728 -0.47045 -0.47048 -0.47049 2.60 -0.41265 -0.41509 -0.41512 -0.41512 2.65 -0.36471 -0.36659 -0.36661 -0.36661 2.70 -0.32255 -0.32400 -0.32401 -0.32401 2.75 -0.28544 -0.28657 -0.28657 -0.28657 2.80 -0.25276 -0.25363 -0.25363 -0.25363 2.85 -0.22396 -0.22463 -0.22464 -0.22464 2.90 -0.19857 -0.19909 -0.19910 -0.19910 2.95 -0.17618 -0.17658 -0.17659 -0.17659 3.00 -0.15643 -0.15674 -0.15674 -0.15674 3.05 -0.13899 -0.13923 -0.13923 -0.13923 3.10 -0.12358 -0.12377 -0.12377 -0.12377 3.15 -0.10996 -0.11010 -0.11011 -0.11011 3.20 -0.09791 -0.09802 -0.09803 -0.09803 3.25 -0.08725 -0.08733 -0.08733 -0.08733 3.30 -0.07779 -0.07786 -0.07786 -0.07786 3.35 -0.06941 -0.06946 -0.06946 -0.06946 98 Height [Å] (0,1) (1,2) (0,3) (2,3) 3.40 -0.06197 -0.06201 -0.06201 -0.06201 3.45 -0.05536 -0.05539 -0.05539 -0.05539 3.50 -0.04949 -0.04951 -0.04951 -0.04951 3.55 -0.04426 -0.04428 -0.04428 -0.04428 3.60 -0.03960 -0.03961 -0.03961 -0.03961 3.65 -0.03545 -0.03546 -0.03546 -0.03546 3.70 -0.03175 -0.03176 -0.03176 -0.03176 3.75 -0.02844 -0.02845 -0.02845 -0.02845 3.80 -0.02549 -0.02550 -0.02550 -0.02550 3.85 -0.02286 -0.02286 -0.02286 -0.02286 3.90 -0.02050 -0.02050 -0.02050 -0.02050 3.95 -0.01839 -0.01839 -0.01839 -0.01839 4.00 -0.01650 -0.01650 -0.01650 -0.01650 4.2.2 The PT series convergence Based on the potential model presented in Chapter 3 for the H2-surface interactions (Eq. (3-5)), four different types of the interactions between the H2 molecule and the ionic surface are investigated in this section. As discussed in Chapter 3, in Model I, the PT formalism is used only for the electrostatic interactions and is used only up to the first-order PT correction terms. In Model II, however, the PT formalism is used to model all components of the interaction potentials. Also, higher order PT corrections are added into the calculations of the energy if need be. In fact, the procedure here is that, first, the second-order PT energies are calculated for each component of the interaction potential. Then, higher-order terms are included if these second-order PT corrections are not small compared to the first-order PT terms as well as the total energy, both in terms of the percentage and amount of the energy. The calculations show, with 99 the exception of the quadrupole interaction, that for all components of the interaction energies the second-order PT energies are small compared to the first-order corrections as well as the total energy. The only exception is the interaction between the quadrupole moment of H2 and the ionic surface. For the quadrupole electrostatic interaction potential, going to third- (or even higher-) order correction is essential. It should also be noted that for the isotropic H2 induced dipole-ionic surface (henceforth referred to as isotropic induction) interaction, the higher-order corrections to the energy are all zero regardless of the rotational state of the H2 molecule. In order to fully investigate the effect of each PT correction term on the energy, nine molecule/site combinations {Para, Heli, Cart}/{Cation, Saddle, Anion} are studied here. These systems are discussed for the MgO substrate in both energy limits (UBE and LBE), as well as for the NaCl substrate. But before proceeding, two important notions should be explained here. Firstly, based on the interaction results of a single H2 molecule with the surface, it was found that, at very low heights, the perturbation energy series for the interaction of the H2 quadrupole moment and the ionic surface is either divergent or very-slowly convergent. Thus, a lower height limit for the interaction must be specified, below which the energies cannot be trusted in terms of the convergence of the series. This lower limit is considered to be at 2.40 Å for both NaCl and MgO substrates. The implication for the simulations is that if the H2 molecule gets this close to the surface, the movement is rejected. This occurrence is, of course, highly unlikely, because at such a low height there is a very strong repulsion between the molecule and the ionic surface. In Chapter 7, the impact of this restriction on the height of the molecule will be studied further when the simulation results of the H2 layers are presented. Secondly, in the MC simulations of the H2 layers over the ionic crystals, H2-H2 interactions are also included in the calculations. In order to minimize the energy of the layer, these 100 interactions may cause the H2 molecules to sit at heights other than the optimum heights over the surface. Here, the optimum height is defined as the height at which the binding energy of a single H2 molecule to the surface is minimum. Since in the calculation of this energy, the PT terms are truncated, the convergence analysis of this energy is also necessary. In order to understand how this convergence in the H2-surface energy affect the energy of the layer, first the energy analysis must be done for H2-surface interactions at all heights over the surface. The preliminary MC simulation results show that in the layer and at all temperatures, on average, the molecules tend to sit at heights close to their optimum heights. This makes the H2-surface energy analysis easier in terms of the range of heights that is required to be investigated. On the other hand, the calculated H2-surface interaction energies show that at heights equal to the optimum value and higher, the energy is fully convergent, regardless of the type and the x-y position of the H2 molecule over the surface. At lower heights, the convergence is not as perfect, but it is good enough especially at heights close to the optimum. This will be discussed in great detail in the rest of this chapter for the nine systems (molecule/site combinations) mentioned before and over both the MgO and NaCl substrates. Another important notion is that the purpose of this study is to compare the properties of the simulated layers of H2/MgO and H2/NaCl with the experimental data. As the temperature of the layer increases, the average height of the molecules in the layer shifts to greater values than the optimum height. This fact will remove concerns about the impact of the single H2 energy convergence on the energy of the layer, which is due to the fact that the convergence is fully achieved at the optimum height and beyond. Nevertheless, in Chapter 7, where the simulation results are presented, the impact of the H2-surface energy convergence on the final properties of the layer will be discussed based on the average height of the molecules in the layer. 101 In order to simplify the convergence analysis, only a specific range of heights is considered in each of the systems mentioned earlier. This range is considered to be from 0.40 Å below the optimum height up to this height. It is called the Range of Interest (RI), which is used in the rest of this thesis. As stated above, at heights higher than optimum, the energy is fully converged and hence, the data will not be discussed in that range. Also, to simplify the convergence discussion even further, in each system, the examples will be given and discussed only at three heights in the RI range; at the optimum height, at 0.40 below the optimum height, and 0.20 Å below the optimum height. The procedure in the convergence analysis is that, first for each system, the absolute value of the second- to first-order Perturbation Energy Percentage Ratio (PEPR), is investigated for each type of the interaction. The higher-order corrections are then added to the potential if the energy is not “converged enough”. The energy is called well-converged if two conditions are satisfied for that type of the interaction; 1) 2nd/1st PEPR of less than 5%, where 2nd is the second-order PT correction to the energy and 1st is the first-order correction term, and 2) 2nd/Etot-2 of less than 5%, where Etot-2 in this case means that in the calculation of the total energy between the H2 molecule and the surface only up to second-order corrections were used in all components of the interaction potential. If these requirements are not met, then higher order corrections must be included in the calculations. For that, the criteria are the same as above, with the “2nd” term replaced by the “last” correction term and Etot-2 by Etot, which is the sum of the energies calculated up to the last correction term in each component of the potential. It should also be mentioned that, the tolerance of 5% seems to be a reasonable choice because of the large uncertainties in the experimental data for NaCl and the fact that for MgO the energy of the layer is not known from the experiments. It should be added that the uncertainties in the experimental 102 energies reported in the literature22 (the data are only available for NaCl) are more than 20% of the total energy which in terms of energy is much larger than the error of 5% in the energies calculated here. It should also be added that, as will be seen throughout the rest of this chapter, at minimum energies, the errors in our data are very small (~0%). In some cases, going up to fifth-order PT terms may become necessary. These terms were not included in the calculations, but they can be estimated in each system. Here the assumption is based on the ratios between each two consecutive PT terms at each height. This will be discussed in more detail in the next section. It should be stressed that, the results discussed in the following sections are for the case when the 𝑉෠௦௘௣ method is used in the calculations. The Heli/Mg2+ system is presented first, because, as stated earlier, this is the case where the H2 molecule sits closest to the surface, so it is presumably more likely for the energy to deviate from the acceptance criteria. 4.2.2.1 The MgO substrate In Section 4.1, it was explained that two different reference values are used for the H2-MgO binding energy (UBE and LBE). Therefore, the convergence analysis is done for both of these values. As stated earlier, the convergence analysis is done for nine (molecule/ion) systems. For all these systems, the procedure for the convergence analysis is the same. Thus, the details are only given for the first system which is Heli/Mg2+. All the other systems follow the same procedure, but only the final results are presented. In addition, as explained in section 4.2.2, only the energy convergence is important at heights within RI. But, based on the H2-surface energy results, for cases where the optimum height of the molecule occurs at a value lower than 3.10 Å, the full convergence may not be achieved even in the RI. Therefore, these systems will also be discussed in detail. It should also be added that in some systems, going up to third-order PT term 103 is enough to obtain a well-converged energy in the RI. In particular, this happens when the optimum height occurs at heights higher than 3.10 Å. In order to shorten the discussion in this section, the convergence analysis is discussed in detail only for the two most important H2/MgO systems (Heli/Mg2+ and Para/Mg2+) in the UBE case. The discussion for the rest of the systems is similar to these two systems and are presented in section C.3 of Appendix C. Also, the results of the LBE case are presented fully in section C.4 of Appendix C. 4.2.2.1.1 Heli/Mg2+ (the UBE case) Figure 4-5 shows the 2nd/1st PEPR for the dispersion (ED), repulsion (ER), the sum of these energies (EDR, which is the LJ potential), anisotropic induction (Eidnd2), as well as the energy between the quadrupole moment of H2 with the surface (EQels). The RI for this case is from 2.50 to 2.90 Å, where the latter is the optimum height. All the data are shown for heights starting from 2.40 Å up to the height where the PEPR drops to a value close to zero. As seen in subfigure (a) of Figure 4-5, at all heights, the 2nd/1st PEPR values are below 0.04%, so in terms of the percentage this may imply that going up to second-order correction is enough in the calculation of the energy, but this must also be investigated in terms of the quantities of the energies (which will be done later in this section in the discussion related to Table 4-6 and Table 4-7). Figure 4-5b illustrates the 2nd/1st PEPR for the anisotropic induction energies. For the heights shown in this graph, the second-order correction energy falls within 5% of the first correction. Compared to the dispersion and repulsion potentials the ratios here are an order of magnitude larger. Similar to the EDR energy (Figure 4-5a), the magnitude of the energies must also be 104 examined to decide if going up to second-order PT corrections is enough for the calculation of this energy (which will be done later in this section). Figure 4-5c presents the 2nd/1st PEPR ratio for EQels. Compared to the LJ and induction potentials, the ratios as well as the energies are very large even at heights around the optimum value. This indicates that, going up to second-order correction is not enough in the calculation of this energy, and hence, higher-order corrections are required to get a well-converged quantity. To further investigate if going up to second-order energy corrections is enough for the calculation of the energy, in Table 4-6, the second-order EDR, Eind2, as well as EQels energies as well as their corresponding 2nd/1st PEPR and 2nd E/Etot-2 ratios are presented at three different heights, i.e., at the optimum height, at 0.40 Å and 0.20 Å below this height. Once again, Etot-2 in Table 4-6 is the total energy only up to the second-order PT corrections in all types of the potentials. The results in this table show that, the second-order energies for EDR and Eind2 are very small compared to Etot-2 energies and hence, the energy can be truncated after the second order. For EQels, however, this energy is large compared to Etot-2 and going up to third- or even higher-order correction terms is necessary. 105 (a) (b) (c) Figure 4-5 The 2nd/1st PEPR for a system of Heli/Mg2+ in the UBE case for a. dispersion, repulsion and the sum of these interactions, b. anisotropic H2 induced dipole-surface (induction), and c. H2 quadrupole-surface interaction energy. Table 4-7 shows the amounts of different types of the energy in the Heli/Mg2+ system at different heights over the surface. It should be noted that in this table, the EDR and Eind2 energies were calculated up to second order, because as was seen in Table 4-7, going up to second order is enough for the calculation of these energies. For EQels, however, up to fourth-order energies were 0.000.010.020.030.042.4 2.8 3.2 3.62nd /1stPEPR (%)Height [Å]VRVDVDR0.01.02.03.04.05.02.4 2.8 3.2 3.62nd /1stPEPR (%)Height [Å]Anisotropic Induction0204060802.4 2.8 3.2 3.62nd /1stPEPR (%)Height [Å]H₂ quadrupole-surface106 included in the calculations. This table shows how the amount of each correction term in each component of the energy changes when higher-order correction terms are used. It can be seen that, at very low heights (< 2.30 Å), the EQels correction terms are diverging instead of converging. At higher heights, the series is converging, and does so even faster as the molecule gets further away from the surface. Table 4-6 The second order energies and the corresponding ratios for dispersion-repulsion (DR), anisotropic induction (ind2), and the quadrupole electrostatic interaction (Qels) for the Heli/Mg2+ system and in the UBE case. The optimum height occurs at 2.90 Å. Etot-2 in this table is the sum of all components of the energies up to second order PT corrections and is different from the Etot shown in Table 4-7. Height DR ind2 Qels 0.40 Å below optimum (2.50 Å) Energy (2nd) [kcal/mol] -0.00009 -0.00071 -0.22788 2nd/1st [%] 0.02 3.29 58.80 2nd/Etot-2 [%] 0.02 0.15 46.52 0.20 Å below optimum (2.70 Å) Energy (2nd) [kcal/mol] 0.00000 -0.00013 -0.09724 2nd/1st [%] 0.01 1.41 38.41 2nd/Etot-2 [%] 0.00 0.02 16.50 optimum (2.90 Å) Energy (2nd) [kcal/mol] 0.00000 -0.00002 -0.04158 2nd/1st [%] 0.00 0.60 25.12 2nd/Etot-2 [%] 0.00 0.00 6.83 107 Table 4-7 The energy results for the Heli/Mg2+ system in the UBE case. The numbers in brackets (superscripts) indicate the order of the correction term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic H2 induced dipole-surface (induction), and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 0.15639 10.18177 -0.01348 -3.69926 -0.00099 -1.13657 -1.95980 3.01077 -2.79626 -3.56778 0.00000 0.17969 -0.04898 2.05 0.31293 8.74944 -0.00967 -3.42457 -0.00086 -1.01937 -1.57646 2.17212 -1.80934 -2.88746 0.00000 0.14542 -0.03208 2.10 0.32132 7.52742 -0.00701 -3.17541 -0.00075 -0.91460 -1.26906 1.56885 -1.17251 -2.33687 0.00000 0.11769 -0.02101 2.15 0.25357 6.48392 -0.00514 -2.94874 -0.00066 -0.82087 -1.02228 1.13426 -0.76083 -1.89126 0.00000 0.09525 -0.01376 2.20 0.15127 5.59164 -0.00380 -2.74198 -0.00059 -0.73696 -0.82396 0.82076 -0.49427 -1.53063 0.00000 0.07709 -0.00902 2.25 0.03813 4.82760 -0.00284 -2.55296 -0.00052 -0.66179 -0.66445 0.59436 -0.32142 -1.23876 0.00000 0.06239 -0.00591 2.30 -0.07292 4.17239 -0.00214 -2.37978 -0.00047 -0.59443 -0.53606 0.43070 -0.20921 -1.00255 0.00000 0.05049 -0.00387 2.35 -0.17514 3.60979 -0.00163 -2.22083 -0.00042 -0.53402 -0.43265 0.31229 -0.13628 -0.81138 0.00000 0.04086 -0.00253 2.40 -0.26555 3.12607 -0.00124 -2.07470 -0.00038 -0.47984 -0.34930 0.22655 -0.08883 -0.65666 0.00000 0.03307 -0.00166 2.45 -0.34321 2.70969 -0.00096 -1.94014 -0.00034 -0.43121 -0.28210 0.16442 -0.05794 -0.53145 0.00000 0.02677 -0.00109 2.50 -0.40832 2.35088 -0.00074 -1.81607 -0.00030 -0.38757 -0.22788 0.11938 -0.03781 -0.43011 0.00000 0.02166 -0.00071 2.55 -0.46172 2.04138 -0.00057 -1.70153 -0.00027 -0.34838 -0.18413 0.08670 -0.02468 -0.34809 0.00000 0.01753 -0.00047 2.60 -0.50448 1.77415 -0.00044 -1.59564 -0.00025 -0.31318 -0.14881 0.06299 -0.01612 -0.28172 0.00000 0.01419 -0.00031 2.65 -0.53778 1.54322 -0.00035 -1.49766 -0.00022 -0.28157 -0.12028 0.04578 -0.01053 -0.22800 0.00000 0.01148 -0.00020 2.70 -0.56279 1.34349 -0.00027 -1.40690 -0.00020 -0.25317 -0.09724 0.03327 -0.00688 -0.18452 0.00000 0.00929 -0.00013 2.75 -0.58061 1.17060 -0.00021 -1.32273 -0.00018 -0.22765 -0.07862 0.02419 -0.00450 -0.14934 0.00000 0.00752 -0.00009 2.80 -0.59225 1.02083 -0.00017 -1.24462 -0.00017 -0.20471 -0.06358 0.01759 -0.00294 -0.12086 0.00000 0.00609 -0.00006 2.85 -0.59863 0.89098 -0.00013 -1.17205 -0.00015 -0.18409 -0.05141 0.01279 -0.00192 -0.09782 0.00000 0.00493 -0.00004 108 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.90 -0.60055 0.77833 -0.00010 -1.10458 -0.00014 -0.16556 -0.04158 0.00931 -0.00126 -0.07916 0.00000 0.00399 -0.00002 2.95 -0.59873 0.68050 -0.00008 -1.04179 -0.00012 -0.14890 -0.03364 0.00677 -0.00082 -0.06407 0.00000 0.00323 -0.00002 3.00 -0.59381 0.59550 -0.00006 -0.98331 -0.00011 -0.13392 -0.02721 0.00492 -0.00054 -0.05185 0.00000 0.00261 -0.00001 3.05 -0.58633 0.52158 -0.00005 -0.92881 -0.00010 -0.12045 -0.02201 0.00358 -0.00035 -0.04196 0.00000 0.00211 -0.00001 3.10 -0.57676 0.45724 -0.00004 -0.87797 -0.00009 -0.10834 -0.01781 0.00261 -0.00023 -0.03396 0.00000 0.00171 0.00000 3.15 -0.56553 0.40120 -0.00003 -0.83051 -0.00008 -0.09745 -0.01441 0.00190 -0.00015 -0.02749 0.00000 0.00138 0.00000 3.20 -0.55298 0.35235 -0.00002 -0.78617 -0.00008 -0.08765 -0.01166 0.00138 -0.00010 -0.02225 0.00000 0.00112 0.00000 3.25 -0.53942 0.30974 -0.00002 -0.74471 -0.00007 -0.07884 -0.00943 0.00101 -0.00006 -0.01800 0.00000 0.00091 0.00000 3.30 -0.52512 0.27253 -0.00002 -0.70593 -0.00006 -0.07092 -0.00763 0.00073 -0.00004 -0.01457 0.00000 0.00073 0.00000 3.35 -0.51029 0.24001 -0.00001 -0.66963 -0.00006 -0.06380 -0.00617 0.00053 -0.00003 -0.01179 0.00000 0.00059 0.00000 3.40 -0.49513 0.21157 -0.00001 -0.63561 -0.00005 -0.05739 -0.00500 0.00039 -0.00002 -0.00954 0.00000 0.00048 0.00000 3.45 -0.47978 0.18668 -0.00001 -0.60371 -0.00005 -0.05162 -0.00404 0.00028 -0.00001 -0.00772 0.00000 0.00039 0.00000 3.50 -0.46438 0.16487 -0.00001 -0.57379 -0.00004 -0.04644 -0.00327 0.00021 -0.00001 -0.00625 0.00000 0.00031 0.00000 3.55 -0.44905 0.14575 -0.00001 -0.54570 -0.00004 -0.04177 -0.00265 0.00015 -0.00001 -0.00506 0.00000 0.00025 0.00000 3.60 -0.43386 0.12897 0.00000 -0.51931 -0.00004 -0.03758 -0.00214 0.00011 0.00000 -0.00409 0.00000 0.00021 0.00000 3.65 -0.41889 0.11423 0.00000 -0.49450 -0.00003 -0.03380 -0.00173 0.00008 0.00000 -0.00331 0.00000 0.00017 0.00000 3.70 -0.40420 0.10127 0.00000 -0.47116 -0.00003 -0.03041 -0.00140 0.00006 0.00000 -0.00268 0.00000 0.00014 0.00000 3.80 -0.37585 0.07983 0.00000 -0.42849 -0.00003 -0.00176 0.00000 0.00009 0.00000 -0.00176 0.00000 0.00009 0.00000 3.90 -0.34903 0.06317 0.00000 -0.39059 -0.00002 -0.00115 0.00000 0.00006 0.00000 -0.00115 0.00000 0.00006 0.00000 4.00 -0.32389 0.05017 0.00000 -0.35683 -0.00002 -0.00075 0.00000 0.00004 0.00000 -0.00075 0.00000 0.00004 0.00000 109 In order to investigate how including the higher-order PT terms in EQels affect the convergence of the total energy, in Table 4-8, the ratios between these correction terms are presented. In this table, 1st, 2nd, 3rd, and 4th refer to the first-, second-, third-, and fourth-order correction terms in EQels (the corresponding energies are shown in Table 4-7). For example, 2nd/1st is the ratio between the second- and the first-order EQels correction terms. As seen, at each height shown in Table 4-8, the PEPR ratios decrease as the order of the PT correction term increases. For example, at 2.50 Å, the 2nd/1st PEPR is 58.80% while the 3rd/2nd and 4th/3rd are, in turn, 52.39 and 31.67%. Also, at all heights, the decrease in the ratios is more dramatic from 3rd/2nd PEPR to 4th/3rd than from 2nd/1st to 3rd/2nd. From the results in Table 4-8, only for heights lower than 2.55 Å, the 4th/Etot ratio is larger than the acceptable amount in our criteria for a well-converged energy and hence, adding fifth-order corrections (EQels(5) or 5th in Table 4-8) in EQels may be necessary for these heights. These fifth-order correction equations have not been developed for EQels, and hence, these terms have not been included in the calculations. However, as stated earlier, these fifth-order corrections to the energy can be estimated using an assumption, which is based on the (n+1)th/nth PEPR ratios between the consecutive correction terms in EQels (n is the order of the correction energy in the PT series presented in Table 4-8). As stated above, at all heights, these PEPR quantities decrease as the order of the terms increases. Based on this, it is assumed that the 5th/4th PEPR is the same as 4th/3rd. In fact, this is the worst-case scenario for the 5th/4th PEPR quantity, because this ratio is expected to be smaller than the 4th/3rd PEPR quantity. Using this assumption, the fifth-order terms can be calculated by multiplying the 5th/4th PEPR by the fourth-order correction term in EQels, i.e., EQels(5) = 5th/4th × EQels(4), where 5th/4th =4th/3rd. The calculated data for EQels(5) in Table 4-8 show that even at the lowest height in the RI, i.e., at 2.50 Å, the 5th/Etot quantity is 2.93%. At higher heights in the RI, this quantity is even smaller, 110 suggesting that not including the fifth-order EQels corrections in the calculation of the energy will cause an error that is within the accepted level (5%) for a well-converged energy. Therefore, in the Heli/Mg2+ system, going up to fourth-order correction terms is enough for the calculation of EQels. In addition, to see the effect of including the up-to-fourth-order terms in EQels on the convergence of the EDR and Eind2 energies, in Table 4-8, the EDR(2)/Etot and Eind2(2)/Etot ratios are also presented. It is seen that these ratios are still small and going up to second-order PT terms is enough for the calculation of EDR and Eind2 even when up-to-fourth-order EQels is used in the calculation of the energy. 111 Table 4-8 The energies and PEPR quantities for the Heli/Mg2+ system in the UBE case after including the fourth-order EQels PT term into the total energy. The RI for this system is from 2.50 up to 2.90 Å. EQels(tot) is the sum of all EQels PT terms up to the fourth-order energy (shown in Table 4-7). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels correction terms in its PT series (presented in Table 4-7). EQels(5) is estimated using the assumption explained in the text. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %. Height Etot EQels(tot) EQels(5) 2nd/1st 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.00 0.15639 -2.88187 2.59704 172.43 153.63 92.88 97.03 90.12 1788.04 1660.65 4.59 31.32 2.05 0.31293 -2.23305 1.50714 154.65 137.78 83.30 81.03 67.49 578.20 481.63 1.52 10.25 2.10 0.32132 -1.78732 0.87630 138.76 123.62 74.74 65.60 49.03 364.90 272.72 0.99 6.54 2.15 0.25357 -1.46973 0.51035 124.54 110.95 67.08 51.77 34.72 300.05 201.27 0.83 5.43 2.20 0.15127 -1.23443 0.29765 111.80 99.61 60.22 40.04 24.11 326.74 196.76 0.93 5.96 2.25 0.03813 -1.05330 0.17382 100.40 89.45 54.08 30.52 16.50 843.00 455.88 2.44 15.49 2.30 -0.07292 -0.90899 0.10162 90.18 80.35 48.57 23.02 11.18 286.90 139.36 0.84 5.30 2.35 -0.17514 -0.79065 0.05947 81.02 72.18 43.64 17.24 7.52 77.81 33.95 0.23 1.45 2.40 -0.26555 -0.69142 0.03483 72.80 64.86 39.21 12.85 5.04 33.45 13.12 0.10 0.62 2.45 -0.34321 -0.60682 0.02041 65.42 58.29 35.24 9.55 3.36 16.88 5.95 0.05 0.32 2.50 -0.40832 -0.53388 0.01197 58.80 52.39 31.67 7.08 2.24 9.26 2.93 0.02 0.17 2.55 -0.46172 -0.47049 0.00703 52.85 47.09 28.47 5.25 1.49 5.35 1.52 0.01 0.10 2.60 -0.50448 -0.41512 0.00413 47.51 42.33 25.59 3.88 0.99 3.20 0.82 0.01 0.06 2.65 -0.53778 -0.36661 0.00242 42.72 38.06 23.01 2.87 0.66 1.96 0.45 0.00 0.04 2.70 -0.56279 -0.32401 0.00142 38.41 34.22 20.69 2.12 0.44 1.22 0.25 0.00 0.02 2.75 -0.58061 -0.28657 0.00084 34.54 30.77 18.60 1.57 0.29 0.78 0.14 0.00 0.01 2.80 -0.59225 -0.25363 0.00049 31.06 27.67 16.73 1.16 0.19 0.50 0.08 0.00 0.01 2.85 -0.59863 -0.22464 0.00029 27.93 24.88 15.04 0.86 0.13 0.32 0.05 0.00 0.01 2.90 -0.60055 -0.19910 0.00017 25.12 22.38 13.53 0.63 0.09 0.21 0.03 0.00 0.00 112 Height Etot EQels(tot) EQels(5) 2nd/1st 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.95 -0.59873 -0.17659 0.00010 22.59 20.13 12.17 0.47 0.06 0.14 0.02 0.00 0.00 3.00 -0.59381 -0.15674 0.00006 20.32 18.10 10.94 0.34 0.04 0.09 0.01 0.00 0.00 3.05 -0.58633 -0.13923 0.00003 18.27 16.28 9.84 0.25 0.02 0.06 0.01 0.00 0.00 3.10 -0.57676 -0.12377 0.00002 16.44 14.64 8.85 0.19 0.02 0.04 0.00 0.00 0.00 3.15 -0.56553 -0.11011 0.00001 14.78 13.17 7.96 0.14 0.01 0.03 0.00 0.00 0.00 3.20 -0.55298 -0.09803 0.00001 13.30 11.85 7.16 0.10 0.01 0.02 0.00 0.00 0.00 3.25 -0.53942 -0.08733 0.00000 11.96 10.66 6.44 0.07 0.00 0.01 0.00 0.00 0.00 3.30 -0.52512 -0.07786 0.00000 10.76 9.59 5.80 0.05 0.00 0.01 0.00 0.00 0.00 3.35 -0.51029 -0.06946 0.00000 9.68 8.62 5.22 0.04 0.00 0.01 0.00 0.00 0.00 3.40 -0.49513 -0.06201 0.00000 8.71 7.76 4.70 0.03 0.00 0.00 0.00 0.00 0.00 3.45 -0.47978 -0.05539 0.00000 7.83 6.98 4.22 0.02 0.00 0.00 0.00 0.00 0.00 3.50 -0.46438 -0.04951 0.00000 7.05 6.28 3.80 0.02 0.00 0.00 0.00 0.00 0.00 3.55 -0.44905 -0.04428 0.00000 6.34 5.65 3.41 0.01 0.00 0.00 0.00 0.00 0.00 3.60 -0.43386 -0.03961 0.00000 5.70 5.08 3.03 0.01 0.00 0.00 0.00 0.00 0.00 3.65 -0.41889 -0.03546 0.00000 5.13 4.57 2.78 0.01 0.00 0.00 0.00 0.00 0.00 3.70 -0.40420 -0.03176 0.00000 4.61 4.11 2.43 0.00 0.00 0.00 0.00 0.00 0.00 3.75 -0.38984 -0.02845 0.00000 4.15 3.70 2.14 0.00 0.00 0.00 0.00 0.00 0.00 3.80 -0.37585 -0.02550 0.00000 3.73 3.33 1.96 0.00 0.00 0.00 0.00 0.00 0.00 3.85 -0.36223 -0.02286 0.00000 3.36 3.00 1.79 0.00 0.00 0.00 0.00 0.00 0.00 3.90 -0.34903 -0.02050 0.00000 3.02 2.69 1.85 0.00 0.00 0.00 0.00 0.00 0.00 3.95 -0.33625 -0.01839 0.00000 2.72 2.42 1.69 0.00 0.00 0.00 0.00 0.00 0.00 4.00 -0.32389 -0.01650 0.00000 2.45 2.18 1.16 0.00 0.00 0.00 0.00 0.00 0.00 113 4.2.2.1.2 Para/Mg2+ (the UBE case) The energies for a Para/Mg2+ system has also been examined. The procedure is the same as the one used for the Heli/Mg2+ system. The RI for this system is between 2.60 and 3.0 Å. The quantities for different types of the energy for this system are presented in Table 4-9. As seen in the table, the first-order isotropic and anisotropic induction energies, i.e., Eind0(1) and Eind2(1), respectively, are both zero at all heights over the surface. Eind2(2) is nonzero, but the results show that it is small compared to the total energy. The same is true for EDR and hence, both of these energies have been truncated after the second order. For EQels, however, going up to second-order PT energies does not give a well-converged energy and hence, like Heli/Mg2+ system, the EQels PT terms were included up to fourth order. The results are presented in Table 4-10. The energies in this table show that, at heights lower than 2.65 Å, the fifth-order energy contributes more than 5% to the total energy and hence, going up to fifth-order EQels PT terms is necessary to get a well-converged energy based on the criteria explained in section 4.2.2. It should be added that, the total energy at 2.65 Å is about +0.15030 kcal/mol higher than the energy at the optimum height (3.0 Å). This amount is about 25% of the minimum energy. This quantity is, in fact, the barrier against the vertical motion of an H2 molecule (in this case Para H2) when this molecule sits right above the Mg2+ cation. It is totally different from the energy barrier against the lateral motion of an H2 molecule over the surface (which will be discussed fully in Chapter 7. These two barriers are included in the MC simulations and hence, the effect of the energy convergence on the simulated layers can be investigated by examining the heights of the molecules in the simulated layers. These heights can be calculated in the simulations by using the Height Distribution Functions (HDF). This function gives information about the heights of the molecules in the adsorbed H2 layer. Based on this average height, the impact of ignoring the 114 fifth-order energy at heights lower than 2.65 Å can be explored. This will be discussed in more detail in Chapter 7. For the rest of the systems over MgO in the UBE case, the results are discussed in detail in section C.3 of Appendix C. In addition, the results of the LBE case of MgO are discussed in section C.4 of Appendix C. It should be mentioned that, in all these cases, the general conclusion is that going up to second-order correction term is enough in the calculation of all components of the energy, except EQels. For this energy, and in most cases, going up to fourth order is necessary in order to get a well-converged energy. The effect of not including the fifth-order corrections is also discussed if need be. 115 Table 4-9 The energy results for Para/Mg2+ system in the UBE case. The numbers in brackets (superscripts) indicate the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 21.32309 10.27591 -0.04366 -3.72478 -0.00320 0.00000 -6.35122 10.14026 14.73268 -3.56778 0.00000 0.00000 -0.15874 2.05 14.11360 8.82920 -0.03131 -3.44837 -0.00278 0.00000 -5.10891 7.31568 9.53286 -2.88746 0.00000 0.00000 -0.10398 2.10 9.33124 7.59536 -0.02270 -3.19769 -0.00244 0.00000 -4.11270 5.28389 6.17762 -2.33687 0.00000 0.00000 -0.06810 2.15 6.14557 6.54209 -0.01665 -2.96965 -0.00215 0.00000 -3.31293 3.82016 4.00860 -1.89126 0.00000 0.00000 -0.04461 2.20 4.01389 5.64171 -0.01233 -2.76166 -0.00190 0.00000 -2.67024 2.76431 2.60415 -1.53063 0.00000 0.00000 -0.02922 2.25 2.58046 4.87089 -0.00922 -2.57152 -0.00169 0.00000 -2.15331 2.00181 1.69349 -1.23876 0.00000 0.00000 -0.01914 2.30 1.61124 4.20999 -0.00695 -2.39732 -0.00151 0.00000 -1.73723 1.45061 1.10226 -1.00255 0.00000 0.00000 -0.01253 2.35 0.95194 3.64255 -0.00528 -2.23743 -0.00135 0.00000 -1.40210 1.05179 0.71800 -0.81138 0.00000 0.00000 -0.00821 2.40 0.50046 3.15471 -0.00403 -2.09042 -0.00122 0.00000 -1.13200 0.76301 0.46801 -0.65666 0.00000 0.00000 -0.00538 2.45 0.18907 2.73479 -0.00310 -1.95505 -0.00109 0.00000 -0.91420 0.55377 0.30525 -0.53145 0.00000 0.00000 -0.00352 2.50 -0.02726 2.37293 -0.00239 -1.83022 -0.00098 0.00000 -0.73851 0.40206 0.19919 -0.43011 0.00000 0.00000 -0.00231 2.55 -0.17861 2.06079 -0.00185 -1.71496 -0.00089 0.00000 -0.59671 0.29202 0.13005 -0.34809 0.00000 0.00000 -0.00151 2.60 -0.28510 1.79126 -0.00144 -1.60841 -0.00080 0.00000 -0.48224 0.21216 0.08494 -0.28172 0.00000 0.00000 -0.00099 2.65 -0.36030 1.55832 -0.00112 -1.50980 -0.00072 0.00000 -0.38980 0.15418 0.05549 -0.22800 0.00000 0.00000 -0.00065 2.70 -0.41336 1.35684 -0.00088 -1.41844 -0.00066 0.00000 -0.31512 0.11207 0.03627 -0.18452 0.00000 0.00000 -0.00042 2.75 -0.45054 1.18241 -0.00069 -1.33372 -0.00059 0.00000 -0.25479 0.08148 0.02371 -0.14934 0.00000 0.00000 -0.00028 2.80 -0.47612 1.03128 -0.00054 -1.25508 -0.00054 0.00000 -0.20603 0.05925 0.01550 -0.12086 0.00000 0.00000 -0.00018 2.85 -0.49309 0.90025 -0.00042 -1.18201 -0.00049 0.00000 -0.16662 0.04309 0.01014 -0.09782 0.00000 0.00000 -0.00012 116 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.90 -0.50356 0.78654 -0.00033 -1.11406 -0.00044 0.00000 -0.13476 0.03134 0.00663 -0.07916 0.00000 0.00000 -0.00008 2.95 -0.50903 0.68779 -0.00026 -1.05082 -0.00040 0.00000 -0.10900 0.02280 0.00434 -0.06407 0.00000 0.00000 -0.00005 3.00 -0.51060 0.60197 -0.00021 -0.99192 -0.00036 0.00000 -0.08817 0.01659 0.00284 -0.05185 0.00000 0.00000 -0.00003 3.05 -0.50911 0.52732 -0.00016 -0.93701 -0.00033 0.00000 -0.07133 0.01207 0.00186 -0.04196 0.00000 0.00000 -0.00002 3.10 -0.50516 0.46235 -0.00013 -0.88579 -0.00030 0.00000 -0.05771 0.00878 0.00122 -0.03396 0.00000 0.00000 -0.00001 3.15 -0.49926 0.40574 -0.00010 -0.83796 -0.00027 0.00000 -0.04669 0.00639 0.00080 -0.02749 0.00000 0.00000 -0.00001 3.20 -0.49179 0.35639 -0.00008 -0.79328 -0.00025 0.00000 -0.03777 0.00465 0.00052 -0.02225 0.00000 0.00000 -0.00001 3.25 -0.48306 0.31333 -0.00006 -0.75150 -0.00023 0.00000 -0.03056 0.00338 0.00034 -0.01800 0.00000 0.00000 0.00000 3.30 -0.47335 0.27573 -0.00005 -0.71241 -0.00021 0.00000 -0.02473 0.00246 0.00022 -0.01457 0.00000 0.00000 0.00000 3.35 -0.46286 0.24286 -0.00004 -0.67581 -0.00019 0.00000 -0.02001 0.00179 0.00015 -0.01179 0.00000 0.00000 0.00000 3.40 -0.45179 0.21411 -0.00003 -0.64151 -0.00017 0.00000 -0.01619 0.00131 0.00010 -0.00954 0.00000 0.00000 0.00000 3.45 -0.44028 0.18895 -0.00003 -0.60935 -0.00016 0.00000 -0.01310 0.00095 0.00006 -0.00772 0.00000 0.00000 0.00000 3.50 -0.42846 0.16690 -0.00002 -0.57918 -0.00014 0.00000 -0.01060 0.00069 0.00004 -0.00625 0.00000 0.00000 0.00000 3.55 -0.41646 0.14756 -0.00002 -0.55085 -0.00013 0.00000 -0.00858 0.00050 0.00003 -0.00506 0.00000 0.00000 0.00000 3.60 -0.40435 0.13058 -0.00001 -0.52423 -0.00012 0.00000 -0.00694 0.00037 0.00002 -0.00409 0.00000 0.00000 0.00000 3.65 -0.39224 0.11567 -0.00001 -0.49921 -0.00011 0.00000 -0.00562 0.00027 0.00001 -0.00331 0.00000 0.00000 0.00000 3.70 -0.38018 0.10256 -0.00001 -0.47566 -0.00010 0.00000 -0.00455 0.00019 0.00001 -0.00268 0.00000 0.00000 0.00000 3.80 -0.35643 0.08086 -0.00001 -0.43262 -0.00008 0.00000 -0.00298 0.00010 0.00000 -0.00176 0.00000 0.00000 0.00000 3.90 -0.33347 0.06400 0.00000 -0.39438 -0.00007 0.00000 -0.00195 0.00005 0.00000 -0.00115 0.00000 0.00000 0.00000 4.00 -0.31151 0.05084 0.00000 -0.36031 -0.00006 0.00000 -0.00128 0.00003 0.00000 -0.00075 0.00000 0.00000 0.00000 117 Table 4-10 The energies and PEPR quantities for a system of Para/Mg2+ in the UBE case after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.60 up to 3.00 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table 4-9). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels PT correction terms in the series presented in Table 4-9. EQels(5) was estimated using the assumption explained in the text. 5th in the ratios refers to this estimated energy. Heights are in Å, the energies are in kcal/mol and the ratios are in %. Height Etot EQels(tot) EQels(5) 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.00 21.32309 18.52172 21.40498 159.66 145.29 79.54 115.57 69.09 100.38 0.11 0.74 2.05 14.11360 11.73964 12.42200 143.19 130.31 81.20 105.81 67.54 88.01 0.11 0.74 2.10 9.33124 7.34880 7.22252 128.48 116.91 84.06 98.28 66.20 77.40 0.11 0.73 2.15 6.14557 4.51583 4.20633 115.31 104.93 88.77 93.15 65.23 68.44 0.11 0.73 2.20 4.01389 2.69823 2.45327 103.52 94.21 96.51 90.92 64.88 61.12 0.11 0.73 2.25 2.58046 1.54199 1.43265 92.96 84.60 109.82 92.91 65.63 55.52 0.12 0.74 2.30 1.61124 0.81563 0.83756 83.50 75.99 135.14 102.69 68.41 51.98 0.12 0.78 2.35 0.95194 0.36769 0.49013 75.02 68.26 195.27 133.30 75.42 51.49 0.14 0.86 2.40 0.50046 0.09903 0.28707 67.40 61.34 472.61 289.89 93.52 57.36 0.16 1.07 2.45 0.18907 -0.05519 0.16826 60.57 55.12 553.10 304.88 161.45 88.99 0.27 1.86 2.50 -0.02726 -0.13725 0.09869 54.44 49.54 145.13 71.90 730.65 361.99 1.13 8.46 2.55 -0.17861 -0.17465 0.05791 48.94 44.53 74.46 33.16 72.81 32.43 0.10 0.85 2.60 -0.28510 -0.18515 0.03400 43.99 40.03 45.88 18.37 29.79 11.93 0.03 0.35 2.65 -0.36030 -0.18013 0.01997 39.55 35.99 30.81 11.09 15.40 5.54 0.01 0.18 2.70 -0.41336 -0.16679 0.01174 35.56 32.36 21.75 7.04 8.77 2.84 0.00 0.10 2.75 -0.45054 -0.14960 0.00690 31.98 29.10 15.85 4.61 5.26 1.53 0.00 0.06 2.80 -0.47612 -0.13128 0.00406 28.76 26.17 11.81 3.09 3.26 0.85 0.00 0.04 2.85 -0.49309 -0.11339 0.00239 25.86 23.53 8.94 2.10 2.06 0.48 0.00 0.02 2.90 -0.50356 -0.09679 0.00140 23.26 21.16 6.85 1.45 1.32 0.28 0.00 0.02 118 Height Etot EQels(tot) EQels(5) 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.95 -0.50903 -0.08186 0.00083 20.92 19.03 5.30 1.01 0.85 0.16 0.00 0.01 3.00 -0.51060 -0.06875 0.00049 18.81 17.12 4.13 0.71 0.56 0.10 0.00 0.01 3.05 -0.50911 -0.05740 0.00029 16.92 15.40 3.24 0.50 0.36 0.06 0.01 0.00 3.10 -0.50516 -0.04771 0.00017 15.22 13.85 2.55 0.35 0.24 0.03 0.01 0.00 3.15 -0.49926 -0.03950 0.00010 13.69 12.46 2.02 0.25 0.16 0.02 0.01 0.00 3.20 -0.49179 -0.03260 0.00006 12.31 11.20 1.60 0.18 0.11 0.01 0.01 0.00 3.25 -0.48306 -0.02684 0.00003 11.08 10.08 1.27 0.13 0.07 0.01 0.01 0.00 3.30 -0.47335 -0.02204 0.00002 9.96 9.06 1.01 0.09 0.05 0.00 0.01 0.00 3.35 -0.46286 -0.01807 0.00001 8.96 8.15 0.81 0.07 0.03 0.00 0.01 0.00 3.40 -0.45179 -0.01479 0.00001 8.06 7.33 0.65 0.05 0.02 0.00 0.01 0.00 3.45 -0.44028 -0.01209 0.00000 7.25 6.60 0.52 0.03 0.01 0.00 0.01 0.00 3.50 -0.42846 -0.00987 0.00000 6.52 5.94 0.42 0.02 0.01 0.00 0.01 0.00 3.55 -0.41646 -0.00805 0.00000 5.87 5.34 0.33 0.02 0.01 0.00 0.01 0.00 3.60 -0.40435 -0.00656 0.00000 5.28 4.80 0.27 0.01 0.00 0.00 0.01 0.00 3.65 -0.39224 -0.00534 0.00000 4.75 4.31 0.22 0.01 0.00 0.00 0.01 0.00 3.70 -0.38018 -0.00434 0.00000 4.27 3.91 0.17 0.01 0.00 0.00 0.01 0.00 3.75 -0.36823 -0.00353 0.00000 3.84 3.47 0.14 0.00 0.00 0.00 0.01 0.00 3.80 -0.35643 -0.00287 0.00000 3.46 3.11 0.11 0.00 0.00 0.00 0.01 0.00 3.85 -0.34483 -0.00233 0.00000 3.11 2.80 0.09 0.00 0.00 0.00 0.01 0.00 3.90 -0.33347 -0.00189 0.00000 2.80 2.56 0.07 0.00 0.00 0.00 0.01 0.00 4.00 -0.31151 -0.00125 0.00000 2.26 2.08 0.05 0.00 0.00 0.00 0.01 0.00 119 4.2.2.2 The NaCl substrate For this substrate, only one value was considered to be the binding energy of the molecule and so no UBE and LBE cases are considered here. The procedure and the criteria for the convergence of the energy is exactly the same as the one explained in section 4.2.2. Similar to the MgO case, nine systems of molecule/ion (i.e., Heli/Na+, Para/Na+, Cart/Na+, Heli/Saddle, Para/Saddle, Cart/Saddle, Heli/Cl-, Para/Cl-, and Cart/Cl-) are considered for the convergence analysis. Among these systems, some has optimum heights far from the surface (>3.20 Å). For these systems, the convergence of the energy in the RI range is good enough even up to second order for “ind” and “DR” energies and up to fourth order for “Qels” and hence, the convergence analysis is only presented and discussed for the cases with an optimum height below 3.20 Å. Among these systems, the Heli/Na+ and Para/Na+ systems are discussed in the following paragraphs, and then the rest of the systems are discussed in section C.5 of Appendix C. 4.2.2.2.1 Heli/Na+ Table 4-11 presents the data for this system. The RI for this system is from 2.30 up to 2.70 Å. In order to investigate the convergence of the energy, the procedure in section 4.2.2 is used. It is found that except for EQels, in all components of the potentials, going up to second-order PT correction is enough to obtain a well-converged energy (like Heli/Mg2+). The results for the EQels quantities in the PT series are shown in Table 4-11. To analyze the convergence of this series, the data in Table 4-12 are used. As seen, at the height of 2.35 Å, the 5th/Etot ratio is around 4%, which is in the acceptable range (< 5%) for a well-converged energy. Also, it is seen that below 2.35 Å, going up to fourth order is not enough to obtain a well-converged energy and fifth-order corrections must be included in the calculation of the energy. To “push” the molecule into the 120 surface from the equilibrium height of 2.70 Å to 2.35 Å requires 0.40303 kcal/mol, which is about 51% of the binding energy. This is a large amount compared to the barriers discussed for the MgO substrate. It should be emphasized that, both this barrier and the barrier against the lateral motion of the molecule are taken into account in the MC simulations and their effects are studied by the HDF of the H2 molecules in the simulated adsorbed film. This will be discussed in great detail in Chapter 7. 4.2.2.2.2 Para/Na+ The results for this system are shown in Table 4-13. The RI for this system is from 2.35 up to 2.75 Å. The analysis of the data in this range indicates that except for EQels, going up to second-order energy is enough to get a well-converged energy. Table 4-14 illustrates the final data for the convergence analysis of EQels. The 5th/Etot ratios in this table show that at heights below 2.55 Å, going up to fifth order is necessary to get a well-converged energy. At heights of 2.35 as well as 2.40 Å, the 5th/Etot ratio is very large and including even up to sixth-order PT terms in EQels may not result in a well-converged energy. However, the energy barrier against the vertical motion of the molecule between these heights and the optimum height of 2.75 Å is about 0.72351 and 0.48528 kcal/mol, respectively. These energies are in turn about 108% and 73% of the minimum energy. This large energy means that, on average, the molecule tends to sit at heights close to the optimum height, which will have a large impact on the HDF of the simulated H2 layer. In Chapter 7, the effect of the convergence of the Para-surface energy on the energy of the layer will be discussed in detail based on the HDF data. 121 For the rest of the systems on NaCl, the results are discussed in detail in section C.5 of Appendix C. In all these cases, the general conclusion is that going up to the second-order correction term is sufficient for all components of the energy, except EQels. For this energy, going up to fourth order is necessary in order to get a well converged energy. Also, including fifth-order corrections is discussed if need be. 122 Table 4-11 The energy results for the Heli/Na+ system. The numbers in brackets (superscripts) indicate the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 2.92192 9.68081 -0.05143 -3.18822 -0.00076 -0.74163 -0.83443 0.83645 -0.50691 -2.38005 0.00000 0.11987 -0.02180 2.05 2.08438 8.10090 -0.03147 -2.95173 -0.00062 -0.67939 -0.70025 0.64304 -0.35699 -2.03312 0.00000 0.10240 -0.01591 2.10 1.39173 6.77576 -0.01973 -2.74036 -0.00051 -0.62289 -0.58862 0.49558 -0.25225 -1.73677 0.00000 0.08747 -0.01161 2.15 0.83567 5.67528 -0.01261 -2.55046 -0.00042 -0.57151 -0.49553 0.38279 -0.17877 -1.48361 0.00000 0.07472 -0.00847 2.20 0.39904 4.76631 -0.00818 -2.37901 -0.00035 -0.52474 -0.41773 0.29628 -0.12704 -1.26735 0.00000 0.06383 -0.00618 2.25 0.06222 4.01701 -0.00538 -2.22354 -0.00030 -0.48208 -0.35258 0.22975 -0.09051 -1.08261 0.00000 0.05452 -0.00451 2.30 -0.19343 3.39927 -0.00358 -2.08197 -0.00026 -0.44315 -0.29793 0.17846 -0.06462 -0.92481 0.00000 0.04658 -0.00329 2.35 -0.38435 2.88902 -0.00240 -1.95259 -0.00023 -0.40756 -0.25200 0.13883 -0.04623 -0.79000 0.00000 0.03979 -0.00240 2.40 -0.52431 2.46630 -0.00163 -1.83393 -0.00020 -0.37501 -0.21336 0.10815 -0.03314 -0.67485 0.00000 0.03399 -0.00175 2.45 -0.62461 2.11473 -0.00112 -1.72477 -0.00017 -0.34520 -0.18079 0.08435 -0.02379 -0.57648 0.00000 0.02903 -0.00128 2.50 -0.69423 1.82109 -0.00078 -1.62407 -0.00016 -0.31788 -0.15330 0.06587 -0.01711 -0.49245 0.00000 0.02480 -0.00093 2.55 -0.74032 1.57466 -0.00055 -1.53093 -0.00014 -0.29283 -0.13009 0.05149 -0.01232 -0.42067 0.00000 0.02119 -0.00068 2.60 -0.76849 1.36682 -0.00039 -1.44460 -0.00013 -0.26983 -0.11046 0.04029 -0.00888 -0.35935 0.00000 0.01810 -0.00050 2.65 -0.78306 1.19066 -0.00028 -1.36440 -0.00011 -0.24871 -0.09384 0.03155 -0.00641 -0.30697 0.00000 0.01546 -0.00036 2.70 -0.78738 1.04066 -0.00020 -1.28976 -0.00010 -0.22930 -0.07977 0.02472 -0.00463 -0.26222 0.00000 0.01321 -0.00026 2.75 -0.78404 0.91232 -0.00015 -1.22019 -0.00010 -0.21146 -0.06784 0.01939 -0.00335 -0.22400 0.00000 0.01128 -0.00019 2.80 -0.77502 0.80203 -0.00011 -1.15523 -0.00009 -0.19504 -0.05771 0.01521 -0.00242 -0.19135 0.00000 0.00964 -0.00014 2.85 -0.76184 0.70685 -0.00008 -1.09450 -0.00008 -0.17994 -0.04912 0.01195 -0.00176 -0.16346 0.00000 0.00823 -0.00010 2.90 -0.74566 0.62440 -0.00006 -1.03765 -0.00007 -0.16603 -0.04182 0.00938 -0.00127 -0.13963 0.00000 0.00703 -0.00008 123 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.95 -0.72737 0.55271 -0.00005 -0.98437 -0.00007 -0.15322 -0.03562 0.00738 -0.00092 -0.11928 0.00000 0.00601 -0.00005 3.00 -0.70765 0.49018 -0.00004 -0.93439 -0.00006 -0.14142 -0.03034 0.00580 -0.00067 -0.10189 0.00000 0.00513 -0.00004 3.05 -0.68701 0.43546 -0.00003 -0.88746 -0.00006 -0.13054 -0.02585 0.00456 -0.00049 -0.08704 0.00000 0.00438 -0.00003 3.10 -0.66584 0.38746 -0.00002 -0.84335 -0.00005 -0.12052 -0.02204 0.00359 -0.00035 -0.07435 0.00000 0.00374 -0.00002 3.15 -0.64445 0.34523 -0.00002 -0.80185 -0.00005 -0.11127 -0.01879 0.00283 -0.00026 -0.06351 0.00000 0.00320 -0.00002 3.20 -0.62305 0.30801 -0.00001 -0.76279 -0.00005 -0.10275 -0.01602 0.00222 -0.00019 -0.05426 0.00000 0.00273 -0.00001 3.25 -0.60183 0.27514 -0.00001 -0.72600 -0.00004 -0.09489 -0.01366 0.00175 -0.00014 -0.04635 0.00000 0.00233 -0.00001 3.30 -0.58090 0.24604 -0.00001 -0.69132 -0.00004 -0.08763 -0.01165 0.00138 -0.00010 -0.03959 0.00000 0.00199 -0.00001 3.35 -0.56036 0.22025 -0.00001 -0.65861 -0.00004 -0.08094 -0.00994 0.00109 -0.00007 -0.03382 0.00000 0.00170 0.00000 3.40 -0.54028 0.19735 -0.00001 -0.62774 -0.00004 -0.07476 -0.00848 0.00086 -0.00005 -0.02889 0.00000 0.00146 0.00000 3.45 -0.52071 0.17699 0.00000 -0.59859 -0.00003 -0.06906 -0.00724 0.00068 -0.00004 -0.02468 0.00000 0.00124 0.00000 3.50 -0.50168 0.15887 0.00000 -0.57105 -0.00003 -0.06379 -0.00617 0.00053 -0.00003 -0.02108 0.00000 0.00106 0.00000 3.55 -0.48322 0.14272 0.00000 -0.54502 -0.00003 -0.05893 -0.00527 0.00042 -0.00002 -0.01801 0.00000 0.00091 0.00000 3.60 -0.46535 0.12831 0.00000 -0.52040 -0.00003 -0.05445 -0.00450 0.00033 -0.00001 -0.01538 0.00000 0.00077 0.00000 3.65 -0.44806 0.11543 0.00000 -0.49711 -0.00002 -0.05030 -0.00384 0.00026 -0.00001 -0.01314 0.00000 0.00066 0.00000 3.70 -0.43136 0.10393 0.00000 -0.47506 -0.00002 -0.04648 -0.00328 0.00021 -0.00001 -0.01123 0.00000 0.00057 0.00000 3.75 -0.41525 0.09364 0.00000 -0.45418 -0.00002 -0.04294 -0.00280 0.00016 -0.00001 -0.00959 0.00000 0.00048 0.00000 3.80 -0.39972 0.08442 0.00000 -0.43440 -0.00002 -0.03968 -0.00239 0.00013 0.00000 -0.00819 0.00000 0.00041 0.00000 3.85 -0.38476 0.07616 0.00000 -0.41565 -0.00002 -0.03666 -0.00204 0.00010 0.00000 -0.00700 0.00000 0.00035 0.00000 3.90 -0.37036 0.06875 0.00000 -0.39788 -0.00002 -0.03388 -0.00174 0.00008 0.00000 -0.00598 0.00000 0.00030 0.00000 4.00 -0.34318 0.05613 0.00000 -0.36501 -0.00002 -0.02893 -0.00127 0.00005 0.00000 -0.00436 0.00000 0.00022 0.00000 124 Table 4-12 The energies and PEPR quantities for the system of Heli/Na+ after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.30 up to 2.70 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table 4-11). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels correction terms in the PT series presented in Table 4-11. EQels(5) was estimated using the assumption explained in in section 4.2.2. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %. Height Etot EQels(tot) EQels(5) 2nd/1st 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.00 2.92192 -1.24651 0.30720 112.51 100.24 60.60 40.67 24.64 17.35 10.51 1.44 0.75 2.05 2.08438 -1.09359 0.19819 103.07 91.83 55.52 32.64 18.12 17.13 9.51 1.18 0.76 2.10 1.39173 -0.96818 0.12839 94.50 84.19 50.90 26.05 13.26 18.12 9.23 1.05 0.83 2.15 0.83567 -0.86302 0.08349 86.70 77.25 46.70 20.71 9.67 21.39 9.99 1.05 1.01 2.20 0.39904 -0.77323 0.05447 79.61 70.93 42.88 16.43 7.05 31.84 13.65 1.33 1.55 2.25 0.06222 -0.69543 0.03565 73.14 65.16 39.39 13.01 5.13 145.47 57.31 5.20 7.25 2.30 -0.19343 -0.62724 0.02340 67.23 59.90 36.21 10.30 3.73 33.41 12.10 1.02 1.70 2.35 -0.38435 -0.56698 0.01540 61.83 55.09 33.30 8.15 2.72 12.03 4.01 0.31 0.62 2.40 -0.52431 -0.51336 0.01016 56.89 50.69 30.64 6.46 1.98 6.32 1.94 0.14 0.33 2.45 -0.62461 -0.46543 0.00671 52.37 46.66 28.21 5.11 1.44 3.81 1.07 0.07 0.20 2.50 -0.69423 -0.42243 0.00444 48.23 42.97 25.98 4.05 1.05 2.46 0.64 0.04 0.13 2.55 -0.74032 -0.38375 0.00295 44.43 39.58 23.93 3.21 0.77 1.66 0.40 0.02 0.09 2.60 -0.76849 -0.34889 0.00196 40.94 36.47 22.05 2.55 0.56 1.16 0.25 0.01 0.06 2.65 -0.78306 -0.31742 0.00130 37.73 33.62 20.32 2.02 0.41 0.82 0.17 0.00 0.05 2.70 -0.78738 -0.28898 0.00087 34.79 30.99 18.74 1.60 0.30 0.59 0.11 0.00 0.03 2.75 -0.78404 -0.26325 0.00058 32.08 28.58 17.28 1.27 0.22 0.43 0.07 0.00 0.02 2.80 -0.77502 -0.23996 0.00039 29.59 26.36 15.94 1.01 0.16 0.31 0.05 0.00 0.02 2.85 -0.76184 -0.21886 0.00026 27.30 24.32 14.70 0.80 0.12 0.23 0.03 0.00 0.01 2.90 -0.74566 -0.19973 0.00017 25.19 22.44 13.57 0.64 0.09 0.17 0.02 0.00 0.01 125 Height Etot EQels(tot) EQels(5) 2nd/1st 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.95 -0.72737 -0.18238 0.00012 23.24 20.71 12.52 0.51 0.06 0.13 0.02 0.00 0.01 3.00 -0.70765 -0.16663 0.00008 21.45 19.11 11.56 0.40 0.05 0.09 0.01 0.00 0.01 3.05 -0.68701 -0.15232 0.00005 19.80 17.64 10.67 0.32 0.03 0.07 0.01 0.00 0.00 3.10 -0.66584 -0.13932 0.00003 18.28 16.29 9.85 0.25 0.02 0.05 0.01 0.00 0.00 3.15 -0.64445 -0.12749 0.00002 16.88 15.04 9.09 0.20 0.02 0.04 0.00 0.00 0.00 3.20 -0.62305 -0.11673 0.00002 15.59 13.89 8.40 0.16 0.01 0.03 0.00 0.00 0.00 3.25 -0.60183 -0.10693 0.00001 14.40 12.83 7.75 0.13 0.01 0.02 0.00 0.00 0.00 3.30 -0.58090 -0.09800 0.00001 13.29 11.84 7.16 0.10 0.01 0.02 0.00 0.00 0.00 3.35 -0.56036 -0.08986 0.00000 12.28 10.94 6.61 0.08 0.01 0.01 0.00 0.00 0.00 3.40 -0.54028 -0.08244 0.00000 11.34 10.10 6.10 0.06 0.00 0.01 0.00 0.00 0.00 3.45 -0.52071 -0.07566 0.00000 10.48 9.33 5.64 0.05 0.00 0.01 0.00 0.00 0.00 3.50 -0.50168 -0.06946 0.00000 9.68 8.62 5.22 0.04 0.00 0.01 0.00 0.00 0.00 3.55 -0.48322 -0.06380 0.00000 8.94 7.97 4.81 0.03 0.00 0.00 0.00 0.00 0.00 3.60 -0.46535 -0.05863 0.00000 8.26 7.36 4.44 0.03 0.00 0.00 0.00 0.00 0.00 3.65 -0.44806 -0.05389 0.00000 7.63 6.80 4.10 0.02 0.00 0.00 0.00 0.00 0.00 3.70 -0.43136 -0.04956 0.00000 7.05 6.28 3.79 0.02 0.00 0.00 0.00 0.00 0.00 3.75 -0.41525 -0.04558 0.00000 6.51 5.80 3.51 0.01 0.00 0.00 0.00 0.00 0.00 3.80 -0.39972 -0.04194 0.00000 6.02 5.36 3.28 0.01 0.00 0.00 0.00 0.00 0.00 126 Table 4-13 The energy results for the system of Para/Na+. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 6.82563 9.82347 -0.15001 -3.21060 -0.00246 0.00000 -2.70416 2.81716 2.67075 -2.38005 0.00000 0.00000 -0.07064 2.05 4.86999 8.22128 -0.09429 -2.97187 -0.00200 0.00000 -2.26933 2.16576 1.88089 -2.03312 0.00000 0.00000 -0.05155 2.10 3.39004 6.87627 -0.06038 -2.75860 -0.00164 0.00000 -1.90757 1.66911 1.32901 -1.73677 0.00000 0.00000 -0.03762 2.15 2.27894 5.75866 -0.03920 -2.56709 -0.00136 0.00000 -1.60589 1.28925 0.94189 -1.48361 0.00000 0.00000 -0.02745 2.20 1.45057 4.83525 -0.02574 -2.39427 -0.00115 0.00000 -1.35376 0.99788 0.66935 -1.26735 0.00000 0.00000 -0.02003 2.25 0.83697 4.07394 -0.01706 -2.23762 -0.00098 0.00000 -1.14263 0.77379 0.47685 -1.08261 0.00000 0.00000 -0.01462 2.30 0.38559 3.44630 -0.01141 -2.09504 -0.00084 0.00000 -0.96551 0.60104 0.34047 -0.92481 0.00000 0.00000 -0.01067 2.35 0.05610 2.92794 -0.00771 -1.96477 -0.00073 0.00000 -0.81668 0.46757 0.24360 -0.79000 0.00000 0.00000 -0.00778 2.40 -0.18213 2.49859 -0.00525 -1.84534 -0.00064 0.00000 -0.69143 0.36424 0.17461 -0.67485 0.00000 0.00000 -0.00568 2.45 -0.35225 2.14163 -0.00362 -1.73550 -0.00057 0.00000 -0.58588 0.28410 0.12537 -0.57648 0.00000 0.00000 -0.00414 2.50 -0.47169 1.84358 -0.00252 -1.63420 -0.00051 0.00000 -0.49682 0.22185 0.09015 -0.49245 0.00000 0.00000 -0.00302 2.55 -0.55354 1.59356 -0.00177 -1.54053 -0.00045 0.00000 -0.42159 0.17342 0.06491 -0.42067 0.00000 0.00000 -0.00221 2.60 -0.60762 1.38277 -0.00126 -1.45372 -0.00041 0.00000 -0.35797 0.13568 0.04680 -0.35935 0.00000 0.00000 -0.00161 2.65 -0.64123 1.20421 -0.00091 -1.37308 -0.00037 0.00000 -0.30412 0.10625 0.03378 -0.30697 0.00000 0.00000 -0.00118 2.70 -0.65980 1.05222 -0.00066 -1.29805 -0.00034 0.00000 -0.25851 0.08327 0.02441 -0.26222 0.00000 0.00000 -0.00086 2.75 -0.66741 0.92223 -0.00048 -1.22811 -0.00031 0.00000 -0.21984 0.06530 0.01765 -0.22400 0.00000 0.00000 -0.00063 2.80 -0.66706 0.81058 -0.00036 -1.16282 -0.00028 0.00000 -0.18703 0.05124 0.01278 -0.19135 0.00000 0.00000 -0.00046 2.85 -0.66099 0.71427 -0.00027 -1.10178 -0.00026 0.00000 -0.15918 0.04023 0.00925 -0.16346 0.00000 0.00000 -0.00033 2.90 -0.65087 0.63085 -0.00021 -1.04464 -0.00024 0.00000 -0.13553 0.03161 0.00671 -0.13963 0.00000 0.00000 -0.00024 127 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.95 -0.63790 0.55836 -0.00016 -0.99109 -0.00022 0.00000 -0.11542 0.02484 0.00487 -0.11928 0.00000 0.00000 -0.00018 3.00 -0.62300 0.49514 -0.00012 -0.94085 -0.00021 0.00000 -0.09833 0.01953 0.00353 -0.10189 0.00000 0.00000 -0.00013 3.05 -0.60684 0.43984 -0.00009 -0.89367 -0.00019 0.00000 -0.08378 0.01536 0.00256 -0.08704 0.00000 0.00000 -0.00009 3.10 -0.58990 0.39134 -0.00007 -0.84933 -0.00018 0.00000 -0.07141 0.01209 0.00186 -0.07435 0.00000 0.00000 -0.00007 3.15 -0.57254 0.34868 -0.00006 -0.80762 -0.00016 0.00000 -0.06088 0.00952 0.00135 -0.06351 0.00000 0.00000 -0.00005 3.20 -0.55502 0.31109 -0.00005 -0.76835 -0.00015 0.00000 -0.05191 0.00749 0.00098 -0.05426 0.00000 0.00000 -0.00004 3.25 -0.53753 0.27788 -0.00004 -0.73136 -0.00014 0.00000 -0.04427 0.00590 0.00072 -0.04635 0.00000 0.00000 -0.00003 3.30 -0.52022 0.24850 -0.00003 -0.69649 -0.00013 0.00000 -0.03776 0.00465 0.00052 -0.03959 0.00000 0.00000 -0.00002 3.35 -0.50318 0.22246 -0.00002 -0.66359 -0.00012 0.00000 -0.03221 0.00366 0.00038 -0.03382 0.00000 0.00000 -0.00001 3.40 -0.48647 0.19934 -0.00002 -0.63255 -0.00011 0.00000 -0.02748 0.00289 0.00028 -0.02889 0.00000 0.00000 -0.00001 3.45 -0.47014 0.17878 -0.00002 -0.60323 -0.00011 0.00000 -0.02345 0.00227 0.00020 -0.02468 0.00000 0.00000 -0.00001 3.50 -0.45424 0.16049 -0.00001 -0.57553 -0.00010 0.00000 -0.02001 0.00179 0.00015 -0.02108 0.00000 0.00000 -0.00001 3.55 -0.43877 0.14418 -0.00001 -0.54934 -0.00009 0.00000 -0.01708 0.00141 0.00011 -0.01801 0.00000 0.00000 0.00000 3.60 -0.42375 0.12963 -0.00001 -0.52457 -0.00009 0.00000 -0.01457 0.00111 0.00008 -0.01538 0.00000 0.00000 0.00000 3.65 -0.40919 0.11664 -0.00001 -0.50113 -0.00008 0.00000 -0.01244 0.00088 0.00006 -0.01314 0.00000 0.00000 0.00000 3.70 -0.39508 0.10502 -0.00001 -0.47895 -0.00007 0.00000 -0.01062 0.00069 0.00004 -0.01123 0.00000 0.00000 0.00000 3.75 -0.38143 0.09463 0.00000 -0.45793 -0.00007 0.00000 -0.00907 0.00055 0.00003 -0.00959 0.00000 0.00000 0.00000 3.80 -0.36822 0.08532 0.00000 -0.43802 -0.00006 0.00000 -0.00774 0.00043 0.00002 -0.00819 0.00000 0.00000 0.00000 3.85 -0.35546 0.07698 0.00000 -0.41915 -0.00006 0.00000 -0.00661 0.00034 0.00002 -0.00700 0.00000 0.00000 0.00000 3.90 -0.34313 0.06950 0.00000 -0.40125 -0.00006 0.00000 -0.00564 0.00027 0.00001 -0.00598 0.00000 0.00000 0.00000 4.00 -0.31974 0.05675 0.00000 -0.36816 -0.00005 0.00000 -0.00411 0.00017 0.00001 -0.00436 0.00000 0.00000 0.00000 128 Table 4-14 The energies and PEPR quantities for the system of Para/Na+ after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.35 up to 2.75 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table 4-13). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels correction terms in its PT series presented in Table 4-13. EQels(5) was estimated using the assumption explained in in section 4.2.2.1. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %. Height Etot EQels(tot) EQels(5) 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.00 6.82563 2.78375 2.53194 104.18 94.80 95.94 90.95 39.13 37.09 1.76 1.03 2.05 4.86999 1.77731 1.63349 95.44 86.85 105.83 91.91 38.62 33.54 1.48 1.06 2.10 3.39004 1.09055 1.05821 87.50 79.62 121.87 97.04 39.20 31.22 1.29 1.11 2.15 2.27894 0.62524 0.68811 80.28 73.06 150.64 110.06 41.33 30.19 1.18 1.20 2.20 1.45057 0.31346 0.44898 73.71 67.08 213.53 143.23 46.14 30.95 1.14 1.38 2.25 0.83697 0.10801 0.29386 67.72 61.63 441.50 272.08 56.97 35.11 1.21 1.75 2.30 0.38559 -0.02400 0.19287 62.25 56.65 1418.47 803.54 88.30 50.02 1.61 2.77 2.35 0.05610 -0.10552 0.12691 57.25 52.10 230.86 120.27 434.22 226.23 6.71 13.87 2.40 -0.18213 -0.15258 0.08370 52.68 47.94 114.43 54.86 95.87 45.96 1.24 3.12 2.45 -0.35225 -0.17641 0.05532 48.49 44.13 71.07 31.36 35.59 15.71 0.38 1.18 2.50 -0.47169 -0.18482 0.03663 44.65 40.64 48.78 19.82 19.11 7.77 0.17 0.64 2.55 -0.55354 -0.18325 0.02430 41.13 37.43 35.42 13.26 11.73 4.39 0.08 0.40 2.60 -0.60762 -0.17548 0.01614 37.90 34.49 26.67 9.20 7.70 2.66 0.04 0.27 2.65 -0.64123 -0.16409 0.01074 34.94 31.79 20.59 6.55 5.27 1.67 0.02 0.18 2.70 -0.65980 -0.15083 0.00715 32.21 29.31 16.18 4.74 3.70 1.08 0.01 0.13 2.75 -0.66741 -0.13689 0.00477 29.70 27.03 12.89 3.49 2.64 0.71 0.00 0.09 2.80 -0.66706 -0.12301 0.00319 27.40 24.93 10.39 2.59 1.92 0.48 0.00 0.07 2.85 -0.66099 -0.10969 0.00213 25.28 23.00 8.44 1.94 1.40 0.32 0.00 0.05 2.90 -0.65087 -0.09721 0.00142 23.32 21.22 6.90 1.46 1.03 0.22 0.00 0.04 129 Height Etot EQels(tot) EQels(5) 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.95 -0.63790 -0.08571 0.00095 21.52 19.59 5.68 1.11 0.76 0.15 0.00 0.03 3.00 -0.62300 -0.07526 0.00064 19.87 18.08 4.69 0.85 0.57 0.10 0.00 0.02 3.05 -0.60684 -0.06586 0.00043 18.34 16.69 3.89 0.65 0.42 0.07 0.00 0.02 3.10 -0.58990 -0.05746 0.00029 16.93 15.41 3.24 0.50 0.32 0.05 0.00 0.01 3.15 -0.57254 -0.05001 0.00019 15.63 14.22 2.71 0.39 0.24 0.03 0.00 0.01 3.20 -0.55502 -0.04343 0.00013 14.43 13.14 2.27 0.30 0.18 0.02 0.01 0.01 3.25 -0.53753 -0.03765 0.00009 13.33 12.13 1.90 0.23 0.13 0.02 0.01 0.00 3.30 -0.52022 -0.03259 0.00006 12.31 11.20 1.60 0.18 0.10 0.01 0.01 0.00 3.35 -0.50318 -0.02817 0.00004 11.37 10.35 1.35 0.14 0.08 0.01 0.01 0.00 3.40 -0.48647 -0.02432 0.00003 10.50 9.56 1.13 0.11 0.06 0.01 0.01 0.00 3.45 -0.47014 -0.02097 0.00002 9.70 8.83 0.96 0.08 0.04 0.00 0.01 0.00 3.50 -0.45424 -0.01807 0.00001 8.96 8.15 0.81 0.07 0.03 0.00 0.01 0.00 3.55 -0.43877 -0.01556 0.00001 8.28 7.53 0.68 0.05 0.02 0.00 0.01 0.00 3.60 -0.42375 -0.01338 0.00001 7.65 6.96 0.58 0.04 0.02 0.00 0.01 0.00 3.65 -0.40919 -0.01151 0.00000 7.07 6.43 0.49 0.03 0.01 0.00 0.01 0.00 3.70 -0.39508 -0.00989 0.00000 6.53 5.94 0.42 0.02 0.01 0.00 0.01 0.00 3.75 -0.38143 -0.00849 0.00000 6.03 5.49 0.35 0.02 0.01 0.00 0.01 0.00 3.80 -0.36822 -0.00729 0.00000 5.57 5.08 0.30 0.02 0.01 0.00 0.01 0.00 3.85 -0.35546 -0.00625 0.00000 5.15 4.70 0.26 0.01 0.00 0.00 0.01 0.00 3.90 -0.34313 -0.00536 0.00000 4.76 4.32 0.22 0.01 0.00 0.00 0.01 0.00 3.95 -0.33123 -0.00460 0.00000 4.40 4.01 0.18 0.01 0.00 0.00 0.01 0.00 4.00 -0.31974 -0.00394 0.00000 4.06 3.71 0.16 0.01 0.00 0.00 0.01 0.00 130 4.3 Single H2-ionic surface energies (the minimum energies) In this section, the minimum energies for the interaction of a single H2 molecule with the ionic surface in models I and II are presented. These energies are only discussed for cases where the molecule sits either over the cation, or the anion, or the saddle site. Also, in this section, the differences between the binding energies of different H2 molecules over their most attractive sites in the surface is discussed in detail. It should be mentioned that, all these energies between a single H2 molecule and the surface will be used in Chapter 6 and Chapter 7 to explain the simulation results for the adsorbed H2 layers over the MgO and NaCl surfaces. 4.3.1 Model I As explained earlier, in this model, the dispersion and repulsion interactions are treated isotropically. Consequently, these energies are the same for all H2 species over the surface, even though they depend on the position of the molecule over the surface. Similarly, the bulk potential, which is in fact the dispersion potential between the H2 molecule and the layers of ions below the top layer, is also treated isotropically and hence, it is independent of the type of the H2 species. On the other hand, in Model I, the induction and quadrupole electrostatic interactions are modelled anisotropically, which means that, these energies depend on the type of the H2 species used in the interaction potential. Thus, the differences between the total energies of these H2 species with the surface over each one of the aforementioned interacting sites is due to the differences only in the electrostatic interactions. It should be emphasized again that, to calculate these energies in Model I, first the energy of a Heli/Mg2+ system is fitted to the experimental data to obtain the repulsion parameters for the H2 molecule. Then, these parameters are used in the 131 calculation of all H2 species regardless of their position over the surface as well as their rotational state. 4.3.1.1 The MgO substrate As explained in section 4.1.1, for this substrate, two different reference energies28,67 are used as the binding energy of the H2 molecule to the surface. It was also explained that, the only difference between these two binding energies is in the repulsion interactions. In this section, the minimum energies of the interaction between the single H2 molecule with the MgO surface for both of these energy cases are discussed. 4.3.1.1.1 The UBE case Table 4-15 shows the minimum binding energies of different H2 species to the surface when the molecule sits over cation, anion, and the saddle site. Since, the bulk, dispersion and repulsion energies are treated isotropically, they do not depend on the type of the H2 molecule. However, they depend on the position of the molecule over the surface. Therefore, at the same position (height), these energies are the same for all H2 species. This is confirmed in Table 4-15 by the energies over the saddle sites, because on this site all H2 molecules experience their minimum energies at the same height. This occurs because the electric field of the surface is zero due to the two-fold symmetry of the surface, and thus, the interaction between the induced dipole moment of the H2 molecule with the surface (induction interaction) is zero regardless of the type of the H2 species. Also, over the saddle site, the interaction of the quadrupole moment with the electric field gradient results in a zero quantity and thus, EQels vanishes. The zero EQels and Eind as well as the same dispersion, repulsion, and bulk energies over the saddle site result in the same total 132 energies for all H2 species over this site. Besides, up to first-order PT correction the total induction energy always has a negative value (if not zero)94, which is also seen in our data in Table 4-15 (Eind energies). EQels, however, depends on the type of H2 molecule as well as its position over the surface and hence (if not zero), its sign can be positive or negative depending on these two factors. Moreover, for both the Para and Heli molecules, the strongest binding energy occurs when the molecule sits over the cation (the Heli/Mg2+ and Para/Mg2+ systems). For these species, the second most attractive site is the saddle site. For Cart H2, however, the molecule is most strongly bound to the surface when it sits over an anion. Similar to Heli and Para H2, the second most attractive site for Cart is the saddle site. Also, among these three H2 species, the lowest optimum height happens in the Heli/Mg2+ system, where the binding energy to the surface is the strongest. These results are in great agreement with the findings of the previous theoretical and experimental studies10,16,20,22,23,26,29,37,53,56. In addition to this information, the differences between the strongest binding energies for the H2 molecules can be obtained from Table 4-15. For example, the strongest binding energy, which occurs in the system of Heli/Mg2+, is almost 1.2 times the second strongest (Cart/O2-) binding energy. Also, the binding energy of the Heli/Mg2+ system is 1.3 times Para/Mg2+ (the third strongest binding energy over MgO). 133 Table 4-15 The Model I H2-MgO minimum interaction energies for nine different systems over this substrate in the UBE case. Etot, ER, ED, EQels, Eind, and Ebulk are the total, repulsion, dispersion, H2 quadrupole-surface, H2 induced dipole-surface (induction), and bulk energies, respectively. [kcal/mol] Height [Å] Etot ER ED EQels Eind Ebulk Above Mg2+ 2.68 -0.60039 1.50217 -1.43875 -0.36930 -0.19080 -0.10370 Heli Above Saddle 3.49 -0.35436 0.28937 -0.57890 0.00000 0.00000 -0.06484 Above O2- 3.77 -0.28173 0.17385 -0.43449 0.03680 -0.00189 -0.05600 Above Mg2+ 3.24 -0.40891 0.43260 -0.74834 0.00000 -0.01879 -0.07438 Para Above Saddle 3.49 -0.35436 0.28937 -0.57890 0.00000 0.00000 -0.06484 Above O2- 3.63 -0.32437 0.24638 -0.50694 0.00000 -0.00361 -0.06020 Above Mg2+ 3.72 -0.27519 0.14690 -0.44374 0.08182 -0.00271 -0.05745 Cart Above Saddle 3.49 -0.35436 0.28937 -0.57890 0.00000 0.00000 -0.06484 Above O2- 3.30 -0.46646 0.56140 -0.74094 -0.19895 -0.01605 -0.07193 Table 4-16 shows the Minimum Energy Barrier (MiEB) against the lateral motion of the H2 molecule, when it moves from above its most attractive site to over its second most attractive site on the surface as follows: MiEB = 𝐸୫୧୬ (୭୴ୣ୰ ୲୦ୣ ୫୭ୱ୲ ୟ୲୲୰ୟୡ୲୧୴ୣ ୱ୧୲ୣ) − 𝐸୫୧୬ (୭୴ୣ୰ ୲୦ୣ ୱୣୡ୭୬ୢ ୫୭ୱ୲ ୟ୲୲୰ୟୡ୲୧୴ୣ ୱ୧୲ୣ) (4-1) It should be mentioned that the interacting sites that are considered here are only the cation, anion, and saddle sites (the local minima as well as the global minimum in the potential energy surface of the molecule). Based on that, for the Heli and Para molecules, the MiEB is defined as the difference between the minimum binding energy of the molecule over cation and the minimum binding energy of the molecule over the saddle site. For the Cart molecule, it is the 134 difference between the minimum binding energy over anion and the minimum binding energy of the molecule on the saddle site. As seen in Table 4-16, the largest MiEB is against the lateral motion of a Heli molecule over the surface. The second and third largest MiEBs are for the Cart and Para molecules, respectively. The MiEB for the Heli molecule is in turn about 1.3 and 4.5 times that for the Cart and Para molecules, respectively. This means that the Para molecules have more freedom than any other H2 species over the surface. In Table 4-16, the Maximum Energy Barrier (MaEB) against the lateral motion of H2 species is also presented. This barrier is defined as the difference between the binding energy of the molecule when it sits over its most attractive site and the binding energy of the molecule over its third most attractive site on the surface as follows: MaEB = 𝐸୫୧୬ (୭୬ ୲୦ୣ ୫୭ୱ୲ ୟ୲୲୰ୟୡ୲୧୴ୣ ୱ୧୲ୣ) − 𝐸୫୧୬ (୭୬ ୲୦ୣ ୲୦୧୰ୢ ୫୭ୱ୲ ୟ୲୲୰ୟୡ୲୧୴ୣ ୱ୧୲ୣ) (4-2) For the Heli and Para species, these sites are the cation and anion sites, respectively. For Cart H2, however, the situation is reversed, where anions and cations are in turn the most and second most attractive sites for this species over the surface. As expected, the largest MaEB occurs for a Heli molecule followed by the Cart molecule. The MaEB for the Heli molecule is 3.8 and 2.8 times those for the Para and Cart molecules, respectively. 135 Table 4-16 The Model I MiEB and MaEB data for the H2 species over MgO in the UBE case. The MiEB and MaEB terms were defined in the text. H2 species Barrier Energy [kcal/mol] MiEB MaEB Heli 0.24603 0.31866 Para 0.05455 0.08454 Cart 0.19127 0.11210 4.3.1.1.2 The LBE case Table 4-17 shows the Model I interaction energies between a single H2 molecule (in three different rotational states) with the MgO surface in the LBE case. Once again, it is emphasized that the only difference with the UBE case is the repulsion potential. This means that, compared to the UBE case, in each system (for example Heli/Mg2+) and at the same height, all the energies stay the same except for ER. This energy is smaller in the LBE case. In Table 4-17 and similar to the UBE case, the strongest binding energy still occurs in the Heli/Mg2+ system, but in the LBE case this energy is about 11% stronger than the UBE case. For Para/Mg2+ and Cart/O2-, the binding energies are in turn 3.8% and 4.4% stronger in the LBE case. Moreover, in the LBE case (Table 4-17), the strongest binding energy (Heli/Mg2+) is 1.3 times the second strongest binding energy (Cart/O2-) in this case. In the UBE case (Table 4-15), however, this ratio was about 1.2. Besides, in the LBE case (Table 4-17), the strongest binding energy (Heli/Mg2+) is 1.4 times the third strongest binding energy (Para/Mg2+), comparted to 1.2 in the UBE case. 136 Table 4-18 shows the MiEB and MaEB results for the interaction of the H2 molecule with the MgO substrate in the LBE case. Compared to the UBE case, for all H2 species, both MiEB and MaEB are larger in the LBE case. For example, the MiEB for Heli, Para and Cart species are in turn 1.2, 1.1, and 1.1 times the corresponding quantities in the UBE case. Therefore, compared to the UBE case, the largest shift in MiEB happens for the Heli species. The same is true about MaEB, with the largest shift for the Heli molecules (about 18%). Table 4-17 The Model I H2-MgO minimum interaction energies for nine different systems over this substrate in the LBE case. Etot, ER, ED, EQels, Eind, and Ebulk are the total, repulsion, dispersion, H2 quadrupole-surface, H2 induced dipole-surface (induction), and bulk energies, respectively. [kcal/mol] Height [Å] Etot ER ED EQels Eind Ebulk Above Mg2+ 2.51 -0.66427 2.14659 -1.77438 -0.52915 -0.39171 -0.11560 Heli Above Saddle 3.46 -0.36470 0.29975 -0.59855 0.00000 0.00000 -0.06589 Above O2- 3.75 -0.28791 0.17639 -0.44406 0.03839 -0.00206 -0.05657 Above Mg2+ 3.19 -0.42482 0.46667 -0.79176 0.00000 -0.02322 -0.07651 Para Above Saddle 3.46 -0.36470 0.29975 -0.59855 0.00000 0.00000 -0.06589 Above O2- 3.63 -0.33321 0.25630 -0.52425 0.00000 -0.00410 -0.06116 Above Mg2+ 3.70 -0.28042 0.14841 -0.45319 0.08535 -0.00295 -0.05805 Cart Above Saddle 3.46 -0.36470 0.29975 -0.59855 0.00000 0.00000 -0.06589 Above O2- 3.25 -0.48713 0.61412 -0.78631 -0.22115 -0.01983 -0.07396 137 Table 4-18 The Model I MiEB and MaEB data for the H2 species over MgO in the LBE case. The MiEB and MaEB terms were defined in section 4.3.1.1.1. H2 species Barrier Energy [kcal/mol] MiEB MaEB Heli 0.29957 0.37636 Para 0.06012 0.09161 Cart 0.20671 0.12243 4.3.1.2 The NaCl substrate Table 4-19 shows the minimum energies for the interaction of an H2 molecule with the NaCl surface. Although the binding energies for this substrate are stronger, they mostly follow the same trends as in the MgO case. For example, the minimum energies are the same for all H2 species over the saddle sites. Besides, EQels and Eind are zero over the saddle sites. Like the MgO substrate, the first and second most attractive sites for the Heli and Para molecules are in turn over the cation and saddle sites. Also, among all H2 species, the strongest binding energy occurs in the Heli/Cation (Na+) system10,16,20,22,23,26,29,37,53,56. That being said, there is an important difference between the H2 binding energies over MgO and NaCl substrate. As opposed to the MgO substrate, the second strongest binding energy for the H2 molecule over NaCl occurs in the system of Para/Na+. On MgO, however, the second strongest binding energy happens in a system of Cart/O2-. On NaCl, the strongest (Heli/Na+) binding energy is almost 1.4 and 2.1 times the second (Para/Na+) and third (Cart/Cl-) strongest binding energies, respectively. Table 4-20 shows the MiEB and MaEB energies for this substrate. These barriers were defined in section 4.3.1.1.1. Compared to the energy barriers over MgO, for both Para and Heli 138 molecules, the MiEB and MaEB energies over NaCl are much larger. For the Cart molecule, however, the MiEB is very small, consistent with a previous theoretical study by Briquez et al.26. This means that, compared to the MgO substrate, the Cart molecule can move more freely along the channels of cations over NaCl. Also, the data in Table 4-20 show that, over the NaCl substrate, the largest MiEB (Heli) is in turn 2.1 and 72 times larger than the second (Para) and third (Cart) largest MiEBs over this substrate. Table 4-19 The Model I H2-NaCl minimum interaction energies for nine different systems over this substrate. Etot, ER, ED, EQels, Eind, and Ebulk are the total, repulsion, dispersion, H2 quadrupole-surface, H2 induced dipole-surface (induction), and bulk energies, respectively. [kcal/mol] Height [Å] Etot ER ED EQels Eind Ebulk Above Na+ 2.78 -0.87561 0.91982 -1.27832 -0.27704 -0.19363 -0.04644 Heli Above Saddle 3.42 -0.40375 0.32843 -0.69862 0.00000 0.00000 -0.03357 Above Cl- 3.97 -0.28173 0.17385 -0.43449 0.03680 -0.00189 -0.05600 Above Na+ 2.91 -0.63344 0.63822 -1.09294 0.00000 -0.13537 -0.04335 Para Above Saddle 3.42 -0.40375 0.32843 -0.69862 0.00000 0.00000 -0.03357 Above Cl- 3.87 -0.29626 0.21568 -0.47812 0.00000 -0.00657 -0.02723 Above Na+ 3.35 -0.30420 0.20131 -0.65927 0.22571 -0.03725 -0.03472 Cart Above Saddle 3.42 -0.40375 0.32843 -0.69862 0.00000 0.00000 -0.03357 Above Cl- 3.71 -0.41030 0.34823 -0.58925 -0.12801 -0.01198 -0.02929 139 Table 4-20 The Model I MiEB and MaEB data for the H2 species over NaCl. The MiEB and MaEB terms were defined in section 4.3.1.1.1. H2 species Barrier Energy [kcal/mol] MiEB MaEB Heli 0.47186 0.59388 Para 0.22969 0.33718 Cart 0.00655 0.10610 4.3.2 Model II As explained in section 4.1.2, in Model II, the repulsion and dispersion parameters are adjusted by fitting the energy of the Heli/Cation system to the reference experimental values. These adjusted parameters are then used in the calculation of all H2 molecules regardless of their position over the surface. Therefore, based on the fitting procedure mentioned above, in both models, the energies of the Heli/Mg2+ systems are the same. However, unlike Model I, in Model II the repulsion and dispersion energies are treated anisotropically. Since in Model II, the bulk potential is a part of the dispersion-repulsion energy, it is also anisotropic in nature. Nonetheless, including the anisotropic effects in these energies only has a minor impact on the binding energies of the H2 specie. This means that, like Model I, the dispersion and repulsion energies are still more dependent on the position of the H2 molecule over the surface than on the type of molecule. Since in Model II, both dispersion and repulsion energies are adjusted in the parametrization procedure, even at the same heights, these energies are different from the corresponding dispersion and repulsion energies in Model I. 140 In addition, in Model II, higher order PT correction terms are also included in the calculation of the total energy. This will add new interactions to each type of the potential. For example, in Model I, EQels is the interaction of the H2 permanent quadrupole moment with the surface. In Model II, however, the interaction between the induced quadrupole moment of the H2 molecule with the surface is also included in EQels. Therefore, compared to Model I, shifts in all types of energies are expected. For example, in all H2/Saddle systems in Model II, EQels is a negative quantity (see Table 4-21), whereas in Model I it is a zero quantity (see Table 4-15Table 4-21). In the following paragraphs, the energies of a single H2 molecule with the MgO and NaCl substrates in Model II are presented. It should be mentioned that, here the data are only compared with those in Model I when both are in the same energy case (either UBE or LBE). 4.3.2.1 The MgO substrate In section 4.1.1, it was explained that for this substrate, two different values (UBE and LBE) are used in the fitting procedure for the interaction of the H2 molecule with the MgO surface28,67. Also, it was stated that the only difference between these two energy cases is the repulsion potential. In this section, the H2-surface interactions are discussed for both of these energy cases. 4.3.2.1.1 The UBE Case Table 4-21 shows the minimum energies of the H2 species over MgO in the UBE case when Model II is used in the calculations. As stated earlier, in both models I and II, the binding energies of the Heli/Mg2+ systems are fitted to the same value and hence, in Table 4-21 and Table 4-15, these binding energies are the same. Similar to Model I, the Heli/Mg2+ system has the strongest binding energy over the surface. But, unlike Model I, the second strongest binding 141 energy occurs in the Para/Mg2+ system. This is in fact the most important effect of including the higher-order PT correction terms into the potential energy. Comparing the binding energies of the Para/Mg2+ system in these models show that this energy in Model II is 1.2 times the corresponding energy in Model I. For the Cart/O2- system, however, the binding energy in Model II shows a decrease of 11% compared to Model I. Therefore, including higher-order PT terms results in a stronger binding energy in the Para/Mg2+ system compared to the Cart/O2- system. In addition, in Model II, the energy difference between Heli/Mg2+ (strongest binding energy) and Para/Mg2+ (second strongest binding energy) is smaller, which is about 47% of the corresponding difference in Model I. But, the difference between Heli/Mg2+ and the Cart/O2- binding energies is 1.4 times that in Model I. Furthermore, the comparison of the data between Table 4-21 and Table 4-15 shows that, except for the Heli/Mg2+ and Cart/O2- systems, adding higher-order PT terms to the potential in Model II results in stronger binding energies for the H2 molecules compared to Model I, regardless of the type of the molecule and its position over the surface. 142 Table 4-21 The Model II H2-MgO minimum interaction energies for nine different systems in the UBE case. Etot, ER, ED, EQels, and Eind are the total, repulsion, dispersion, H2 quadrupole-surface, and H2 induced dipole-surface (induction) energies, respectively. [kcal/mol] Height [Å] Etot ER ED EQels Eind Above Mg2+ 2.90 -0.60055 0.77822 -1.10471 -0.19910 -0.07520 Heli Above Saddle 3.45 -0.37682 0.24297 -0.61803 -0.00179 0.00000 Above O2- 3.75 -0.30134 0.13855 -0.46405 0.02618 -0.00206 Above Mg2+ 3.00 -0.51060 0.60176 -0.99228 -0.06875 -0.05188 Para Above Saddle 3.45 -0.38206 0.24639 -0.62431 -0.00435 0.00000 Above O2- 3.65 -0.33903 0.18989 -0.51991 -0.00587 -0.00331 Above Mg2+ 3.45 -0.33861 0.19347 -0.62070 0.09707 -0.00850 Cart Above Saddle 3.45 -0.38412 0.25342 -0.63652 -0.00112 0.00000 Above O2- 3.55 -0.41598 0.26768 -0.59078 -0.08745 -0.00557 Table 4-22 shows the MiEB and MaEB results for the H2/MgO system in the UBE case using Model II. Compared to the data in Model I (Table 4-16), the results here show a decrease in MiEB and MaEB quantities for both the Heli and Cart molecules. For Heli H2, for example, the MiEB in Model II is about 91% of the corresponding amount in Model I, whereas for the Cart molecules the shift is more dramatic and is about 19% of the MiEB in Model I. For the Para molecules, however, there is a considerable increase in both of these barriers. As an example, its MiEB in Model II is about 1.4 times that in Model I (Table 4-16). These results suggest that including higher-order PT terms in the potential in Model II results in smaller barriers against the 143 lateral motion of the Heli and Cart molecules over the surface, whereas for the Para molecules, these barriers shift to larger values. Table 4-22 The Model II MiEB and MaEB data for the H2 species over MgO in the UBE case. The MiEB and MaEB terms were defined in section 4.3.1.1.1. H2 species Barrier Energy [kcal/mol] MiEB MaEB Heli 0.22373 0.29921 Para 0.12854 0.17157 Cart 0.03186 0.07737 4.3.2.1.2 The LBE Case Table 4-23 shows Model II interaction energies between an H2 molecule and the MgO surface in the LBE case. Similar to Model I, the only difference from the UBE case is in the repulsion potential. But similar to the UBE case, the binding energy of the Heli/Mg2+ system is the same as the corresponding energy in Model I (Table 4-17). Therefore, the difference between the binding energies of Heli/Mg2+ systems in the UBE and LBE cases are expected to be the same in models I and II. In addition, as seen in Table 4-23, similar to Model I, the Heli/Mg2+ system has the strongest binding energy among all in the LBE case. But, unlike Model I, the second strongest binding energy occurs in the Para/Mg2+ system. For this system in the LBE case, the binding energy in Model II is 1.3 times the corresponding energy in Model I. For Cart/O2-, however, the binding 144 energy in Model II has decreased by 11%. This shift in energy between models I and II is the same as the corresponding shift in the UBE case. On the other hand, in Model II (Table 4-23), the binding energy of the Para/Mg2+ system in the LBE case is about 9.5% stronger than the corresponding energy in the UBE case. For the system of Cart/O2-, the binding energy is about 15% stronger. In Model I (Table 4-17), however, these shifts in energy were smaller. For example, in Model I, the Para/Mg2+ and Cart/O2- binding energies were in turn about 3.8% and 4.4% stronger in the LBE case. Once again, the difference between the UBE and LBE binding energies of the Heli/Mg2+ system remains the same in models I and II. Table 4-24 shows Model II MiEB and MaEB results for the H2/MgO system in the LBE case. On the one hand, compared to the LBE case in Model I (Table 4-20), the MiEB barriers for the Heli and Cart molecules are about 10% and 84% smaller. For Para, however, the MiEB (in Model II) is 2.6 times the corresponding amount in Model I. On the other hand, compared to the UBE case in Model II (Table 4-21), the MiEB for the Heli, Para, and Cart species are in turn 21%, 9% and 24% larger. 145 Table 4-23 The Model II H2-MgO minimum interaction energies for nine different systems over this substrate in the LBE case. Etot, ER, ED, EQels, and Eind are the total, repulsion, dispersion, H2 quadrupole-surface, and H2 induced dipole-surface (induction) energies, respectively. [kcal/mol] Height [Å] Etot ER ED EQels Eind Above Mg2+ 2.79 -0.66448 0.97483 -1.25995 -0.25989 -0.11980 Heli Above Saddle 3.42 -0.39453 0.24557 -0.63813 -0.00204 0.00000 Above O2- 3.69 -0.31266 0.15377 -0.49335 0.02953 -0.00266 Above Mg2+ 2.90 -0.55913 0.73069 -1.11450 -0.09679 -0.07924 Para Above Saddle 3.41 -0.40034 0.25616 -0.65157 -0.00515 0.00000 Above O2- 3.60 -0.35366 0.20564 -0.54812 -0.00729 -0.00409 Above Mg2+ 3.34 -0.35163 0.23690 -0.69551 0.12044 -0.01354 Cart Above Saddle 3.43 -0.40228 0.24906 -0.65021 -0.00122 0.00000 Above O2- 3.50 -0.43702 0.29155 -0.62415 -0.09768 -0.00688 Table 4-24 The Model II MiEB and MaEB data for the H2 molecules over MgO in the LBE case. The MiEB and MaEB terms were defined in section 4.3.1.1.1. H2 species Barrier Energy [kcal/mol] MiEB MaEB Heli 0.26995 0.31285 Para 0.15879 0.20547 Cart 0.03474 0.08539 146 4.3.2.2 The NaCl substrate Table 4-25 shows the Model II data for the interaction of a single H2 molecule with the NaCl substrate. Similar to Model I (Table 4-19), the Heli/Na+ system has the strongest binding energy of all. Although this binding energy is within the range of the experimental data16,22,26, it is different from the corresponding energy in Model I (about 10% smaller in Model II). This is due to the fact that, as explained in section 4.1, for NaCl, both Heli/Na+ and Para/Na+ energies are fitted at the same time to the experimental data that is associated with a large uncertainty. In fact, the binding energies in Table 4-19 and Table 4-25 for these two systems are the best energies that could be obtained in the fitting procedure. The comparison of the results between Table 4-25 and Table 4-19 show that, the binding energies of the Heli/Na+ and Cart/Cl- systems are in turn 10% and 19% weaker in Model II. For the system of Para/Na+, however, it is 6% stronger. On the other hand, in Model II, the strongest binding energy of an H2 molecule over NaCl (the Heli/Na+ system) is in turn about 1.2 and 2.4 times the second (Para/Na+) and third (Cart/Cl-) strongest binding energies over this substrate. These ratios were in turn 1.4 and 2.1 in Model I. In addition, one odd result in Model II is that, the strongest binding energy for the Cart molecule occurs over the saddle site. This is against the previous theoretical results by Briquez et al.26, in which the most attractive site for the “cartwheel-like” H2 molecule was found to be the anion site. It should be added that in our calculations, the barrier against the lateral motion of the Cart molecule is very small (< 0.5 meV) and this molecule can move freely over the surface. This last result, however, is consistent with the calculations of Briquez et al.26 In fact, this drawback in the results of Model II for the Cart molecule over NaCl originates from the parameters that were used in the fitting procedure. The consistent results with the results of Briquez et al. 26 can be obtained if a different set of adjustable parameters (C6 and C12 for the interaction between H2 and 147 NaCl) are used. But, compared to the findings of Briquez et al.26, this set of parameters gives weaker binding energies for all H2 species over the anion as well as the saddle site. It also results in stronger binding energies for all these H2 species over the cation site. In the lack of experimental data for the Cart molecule as well as the C6 and C12 parameters for H2-NaCl energies, it cannot be confirmed which one of these cases is the correct one. It should be mentioned though that, over NaCl and above the anion site, the Cart molecule sits far away from the surface (see Table 4-25) and hence, acquiring a weaker binding energy may be reasonable. As will be seen in Chapter 7, in the simulation results of Model II, only Para and Heli molecules are seen in the adsorbed layer and hence, these H2 layers on NaCl will not be affected by the uncertainties in the energy of the Cart molecule over this substrate. Moreover, in 0, it is shown that the Cart molecule may only be observed as one of the three states of an ortho-H2 molecule over an anion site. This may suggest that its odd behavior over the surface in Model II may be because of the assumption that forces the molecule to stay in this state at any positions over the surface. Table 4-26 presents the MiEB and MaEB results for H2/NaCl in Model II. Compared to Model I (Table 4-20), the MiEB data for the Heli molecules decreased by 6%. For the Para molecules, however, it has increased by 36%. If these two molecules are compared in Model II, the MiEB for Heli is 1.4 times the corresponding quantity for the Para molecule. For the Cart molecule, the MiEB in Model II is 1.6 times that in Model I, but once again in Model II the most attractive interacting sites are in a different order than in Model I. 148 Table 4-25 The Model II H2-NaCl minimum interaction energies for nine different systems over this substrate. Etot, ER, ED, EQels, and Eind are the total, repulsion, dispersion, H2 quadrupole-surface, and H2 induced dipole-surface (induction) energies, respectively. [kcal/mol] Height [Å] Etot ER ED EQels Eind Above Na+ 2.70 -0.78738 1.04046 -1.28986 -0.28898 -0.24928 Heli Above Saddle 3.57 -0.34518 0.24038 -0.57912 -0.00217 0.00000 Above Cl- 4.08 -0.26668 0.13155 -0.38901 0.02447 -0.00322 Above Na+ 2.77 -0.66809 0.87364 -1.19243 -0.13130 -0.21087 Para Above Saddle 3.56 -0.35070 0.25550 -0.60050 -0.00543 0.00000 Above Cl- 4.01 -0.26840 0.16511 -0.42481 -0.00414 -0.00423 Above Na+ 3.22 -0.34136 0.31687 -0.76917 0.17580 -0.05609 Cart Above Saddle 3.58 -0.35168 0.24801 -0.58707 -0.00131 0.00000 Above Cl- 3.92 -0.33131 0.21659 -0.47376 -0.06807 -0.00618 Table 4-26 The Model II MiEB and MaEB data for the H2 molecules over NaCl. These MiEB and MaEB terms were defined in section 4.3.1.1.1. H2 species Barrier Energy [kcal/mol] MiEB MaEB Heli 0.44220 0.52070 Para 0.31739 0.39969 Cart 0.01032 0.02038 149 4.4 Summary In this chapter, first, the fitting procedure to get an H2-surface energy comparable to the experimental data was discussed. It was mentioned that, in both models, the initial parameters were obtained from a previous study25 in our research group. Then, in Model I only the repulsion parameters were adjusted. In Model II , however, both the repulsion and dispersion parameters were adjusted to get an acceptable energy. Nonetheless, in both models, the Heli/cation system was used in the fitting procedure, and then the adjusted parameters in this system were used to calculate the interaction of all H2 species regardless of their rotational state as well as their positions over the surface. Then, the convergence of the H2-surface potential based on the number of Fourier terms used in the potential operators for each component of the potential was discussed. In Model I, these Fourier terms were only used in the electrostatic interactions. To calculate these interactions, the comments by Kroes and Mowrey53,54 was used for the number of the Fourier terms that must be used in the potential operators. But (unlike Model I) in Model II, the Fourier terms were used in the dispersion-repulsion interactions as well. The procedures for the convergence of the Fourier series were discussed. For the electrostatic interactions more Fourier terms than what was suggested by Kroes and Mowrey53,54 was used in the calculation of the potential operators, but it was found that the results are nearly the same. Also, for the dispersion-repulsion interactions, the number of layers of ions underneath the top layer that must be used in the calculations was also discussed. In general, five layers are used in the calculations, the top layer plus a total of four layers (the bulk potential) underneath. Also, the convergence of the PT series for each component of the potential was discussed. In general, a well-converged energy could be obtained after including the second-order PT 150 correction term in both the dispersion-repulsion and the induction energies. For the H2 quadrupole-surface interactions, however, going up to fourth-order correction terms was recommended. In some cases, the analysis of including the fifth-order PT terms was also discussed. 151 H2-H2 interactions In 0, the adsorbed H2 molecule-surface interactions were presented and it was discussed how the energy of different H2 species vary depending on the type of this molecule and its position over the ionic surface. In this chapter, the two-body H2-H2 interaction is presented when the molecules are in specific rotational states as well as a three-body H2-H2 interaction where the interaction between a pair of H2 molecules is affected by their individual interactions with the surface. Towards this end, models I and II of Chapter 3 for the H2-H2 interaction are discussed separately. These H2-H2 calculations were not done in Model III as it does not include these interactions (see section 3.2). As explained in Chapter 3 (sections 3.2.1 and 3.2.2), in both models I and II, the H2-H2 interaction is formulated by using the first-order perturbation to the energy (see Eq. (3-52)). Therefore, the formulation of the H2-H2 interactions is the same in models I and II. Since this formalism is similar to the H2-surface interaction in Model I (averaging over the wavefunctions), it is referred to as the RAP model. But there is a difference between the H2-H2 interactions in models I and II, which is in the adsorbed H2 wavefunction (rotational state) that is used in these models. As explained in Chapter 3, this wavefunction determines the distribution of the H2 molecular axis around a fixed axis in the molecule. This wavefunction, and hence distribution, may be affected by the presence of the external electric fields around the molecule. In Model I, it is assumed that the rotational state of the H2 molecule is the same as in the isolated system and is not affected by the ionic surface underneath this molecule. In Model II, however, the electric filed of the ionic surface affects the rotational state of the H2 molecule. Once again, as stated in Chapter 3, in both of these models, the assumption is 152 that only up to first-order PT energies are enough in the calculation of the H2-H2 interaction energy. In this section, the differences between the results of the H2-H2 interaction energies of models I and II are discussed. The importance of including the effects of the external fields on these interactions is also investigated. To do this analysis, first the interaction between classical two-atom H2 molecules (instantaneous molecular axes) is studied. Then this interaction energy is used as a reference for the interaction between H2 molecules in models I and II (RAP interactions). 5.1 Instantaneous interactions To provide a benchmark for the RAP interactions, the underlying instantaneous H2-H2 interactions (in the two-atom H2 model) for nine standard configurations (as illustrated in Figure 5-1) are studied here. Previous studies78,79,85 showed that the interaction between two-atom H2 molecules depend very strongly on the orientation of the H2 molecules sitting next to each other. Figure 5-1 shows nine of these configurations. The instantaneous interaction energy between the H2 molecules in these configurations is very sensitive to the molecular orientation due mainly to the anisotropic electrostatic interactions. Figure 5-2 shows these energies based on the distance 𝑟 between the centers of mass of molecules 1 and 2. To calculate these energies the potential model of Diep and Johnson78,79 is used. As seen in Figure 5-2, the “T” configuration (case 3) shows the strongest binding energy (-0.11310 kcal/mol) among the configurations presented in Figure 5-1, whereas the “end on” configuration (case 4) shows the weakest (-0.02688 kcal/mol). Also, as seen in Figure 5-2, cases 3 and 4 are in fact the two extreme cases in energy. In addition, the maximum shift in the 153 position of the minimum energies is from 3.36 Å in case 3 to 3.76 Å in case 4. For reference, Figure 5-2 also includes the Para-Para interaction from the upcoming RAP case. For the Para molecules, as one would expect for a distribution that is a uniform average over all instantaneous orientations, the interaction energy should lie between these two extremes as well as other cases shown in Figure 5-1. Figure 5-2 confirms this with the Para-Para interaction showing a minimum of -0.06883 kcal/mol at 3.44 Å. In the next section, this interaction energy will also be used as a reference for the interaction between different H2 species in the RAP model (models I and II). Figure 5-1 Different configurations for pair interactions between two-atom H2 molecules. The solid arrows indicate the molecular axes in each molecule. The dashed arrows ( 𝒓ሬ⃗ ) show the vector connecting the molecules’ centers of mass. ⦿ depicts a molecular axis perpendicular to the plane of the paper. 154 Figure 5-2 The H2-H2 interaction potentials as functions of distance r for the nine configurations shown in Figure 5-1 as well as the RAP Para-Para case. The energies for cases 6 and 8 are not shown here, because case 6 is very similar to case 1 and case 8 is similar to the Para-Para interaction. 5.2 RAP interactions 5.2.1 Model I Figure 5-3 shows the interaction energies between two H2 molecules in Model I. To calculate these interactions, the procedure explained in section 3.2.1 was used. As seen in the figure, the Heli-Heli interaction shows the strongest binding energy (-0.06987 kcal/mol at 3.47 Å), which is only slightly stronger than the reference Para-Para interactions (-0.06883 kcal/mol at 3.44 Å). -0.12-0.08-0.0400.042 4 6 8 10Energy [kcal/mol]r [Å]Case 1Case 2Case 3Case 4Case 5Case 7Case 9Para-Para155 The Cart-Cart interaction (-0.06250 kcal/mol at 3.47 Å), however, is slightly more repulsive compared to the Para-Para energy. Therefore, it is concluded that, in Model I, the H2-H2 interactions between two similar molecules is similar to the reference Para-Para interaction. Nevertheless, compared to the H2-surface binding energies in Table 4-15, the H2-H2 binding energies are an order of magnitude smaller. Figure 5-3 Model I H2-H2 RAP energies between a pair of like H2 molecules. The Para-Para interaction is the reference interaction to which the other energies are compared. To investigate how the interaction energy between unlike H2 molecules differ from the corresponding energy between two like H2 molecules (Figure 5-3), Figure 5-4 plots their energies. Again, the Para-Para interaction is used as the reference energy. The results show a negligible shift in the energies between unlike H2 molecules from the reference Para-Para interaction. Therefore, in Model I, all the RAP H2-H2 interactions show similar trends to the reference Para--0.08-0.040.000.042 3 4 5 6Energy[kcal/mol]r [Å]Para-ParaHeli-HeliCart-Cart156 Para interaction. The largest deviation is seen in the Cart-Cart interactions (see Figure 5-3) with almost a 9% shift in energy compared to the Para-Para interaction. Figure 5-4 Model I H2-H2 RAP energies between a pair of unlike H2 molecules. The Para-Para interaction is the reference interaction to which the other energies are compared. 5.2.2 Model II As explained in Chapter 3, in Model II an H2 molecule’s rotational wavefunction is not a single pure rotational state as it is in Model I, but is instead a mixture of rigid rotor states (see Eq. (3-61), Eq. (3-62), and Eq. (3-63)). In principle this mixture will vary with position over the surface and with the nature of the substrate. It should be mentioned that the effect of the substrate on the wavefunctions correlates with the strength of the binding energy (see Table 4-21, Table 4-23, and Table 4-25). The stronger the binding energy, the bigger the change to the rotational wavefunctions (degree of mixing) and their distributions compared to the isolated system. In -0.08-0.040.000.042 3 4 5 6Energy [kcal/mol]r [Å]Para-ParaHeli-ParaHeli-CartPara-Cart157 order to see how these wavefunctions affect the interactions between H2 molecules in the adsorbed layer, these interactions are plotted and compared to the ones obtained from Model I. One important notion here is that because the Model II wavefunctions change with position (whereas the Model I ones do not), a choice of location must be made when determining the wavefunction of the H2 molecule to calculate the H2-H2 interaction in this model and compare it with the results of Model I. Therefore, in Model II, the wavefunctions of Heli, Para, and Cart molecules are chosen at specific positions over the surface and their wavefunctions (Eq. (3-61), Eq. (3-62), and Eq. (3-63)) are then kept fixed during the calculation of the H2-H2 energies (which then will depend only on distance 𝑟). To do this, first, the x-y position of the H2 molecule is chosen so the molecule is affected by the surface to a maximal extent and hence, the biggest change in the H2-H2 interaction energies is obtained. Then two parameters are varied in the calculation of the H2-H2 energies; 1) the distance between the H2 molecules in the H2-H2 configuration, and 2) their z positions. In fact, the z position is varied in such a way to see how the strength of the H2-surf interaction affect the H2-H2 energies within Model II and in comparison with Model I results. Note that the identification of Para in Model II is that state whose dominant rotational contribution is from the Υ଴଴ spherical harmonic. Similar criteria are used for those states designated Heli and Cart in Model II. Therefore, to calculate the H2-H2 interactions in this section, the rotational states of the Heli and Para molecules are taken from the Heli/Cation and Para/Cation systems (where they are affected maximally) and at different heights. For the Cart molecules, the wavefunction (mixing) coefficients are taken from the Cart/Anion system at different heights. In the following sections, the interaction energies between different H2 species over the MgO and NaCl substrates for Model II is discussed. Since the Cart molecule sits far away from the surface (see Table 4-21, Table 4-23, and Table 4-25), its 158 wavefunction is almost the same as Model I and hence, the H2-H2 results in Model II for the Cart molecule will not be discussed in the following sections. 5.2.2.1 The MgO substrate As explained in Chapter 4, for the MgO substrate, two values were considered as possible binding energies of the H2 molecule with the surface, namely, the UBE and LBE cases. In the previous section, it was stated that, in Model II, the stronger the binding energy the bigger the shift in the H2-H2 interaction energy compared to Model I. This means that this shift will be bigger if the LBE case is used in the calculations. Therefore, in this section, only the data for the LBE case are presented. In the UBE case, the energies show similar trends to the LBE case, but the shifts in the energy between models II and I are smaller due to smaller binding energies in this case. Once again, it is emphasized that at distances far from the surface (at about 4 Å and higher), the results of Model II approach the results of Model I (i.e., the energies presented in Figure 5-3 and Figure 5-4). The reason is that at this height the molecule is only slightly affected by the surface and hence, the H2 wavefunction in Model II is very similar to the isolated H2 wavefunction used in Model I. Figure 5-5 shows the interaction energy between Heli-Heli, Para-Para and Heli-Para molecules over the MgO surface in the LBE case. The energies have been calculated using the wavefunctions at four different heights of the molecules. The results show that in these cases the energies shift to more repulsive values as the molecule gets closer to the surface (the same shift is observed for the Cart-Cart, Heli-Cart, and the Para-Cart interactions). The exception here is the Heli-Heli interaction, in which the energy shifts to more attractive quantities. Since the amounts of these shifts are very small, they may not be seen in Figure 5-5. For example, the 159 biggest difference between the H2-H2 energy calculated in Model II from the corresponding energy in Model I (at about 4 Å shown in the figure) occurs in the Para-Para interaction (about 3.2%). The second largest shift happens in the Para-Cart interaction and is about 2.8%. (a) (b) (c) Figure 5-5 Model II a. Heli-Heli, b. Para-Para, and c. Heli-Para interaction energies for the wavefunctions fixed at different heights over the MgO surface in the LBE case. -0.08-0.040.000.042 3 4 5 6Energy [kcal/mol]r [Å]2.50 Å2.90 Å3.20 Å4.00 Å-0.08-0.040.000.042 3 4 5 6Energy [kcal/mol]r [Å]2.50 Å2.90 Å3.20 Å4.00 Å-0.08-0.040.000.042 3 4 5 6Energy [kcal/mol]r [Å]2.50 Å2.90 Å3.20 Å4.00 Å160 5.2.2.2 The NaCl substrate Figure 5-6 illustrates the H2-H2 interaction energies between the Heli-Heli, Para-Para and Heli-Para molecules at different heights over the NaCl substrate. The same trends as in MgO are observed for these interactions in Model II. The only difference is that, the shifts in these energies are smaller compared to the corresponding values over MgO (in the UBE case). For example, similar to MgO, the maximum shift in the H2-H2 energies occurs in the Para-Para interaction. This shift is about 2.5% and is smaller than the corresponding shift over the MgO substrate. For the Para-Cart interaction, the shift is only about 0.9%. Nonetheless, similar to the MgO case, at 4.0 Å, the H2-H2 energies become equal to the H2-H2 energies in Model I (presented in Figure 5-3 and Figure 5-4). 161 (a) (b) (c) Figure 5-6 Model II a. Heli-Heli, b. Para-Para, and c. Heli-Para interaction energies for the wavefunctions fixed at different heights over the MgO surface in the LBE case. 5.3 Summary The H2-H2 interaction energies in the adsorbed phase were calculated and compared in models I and II. In both cases, the first-order PT energy was used to calculate the interaction energy between two H2 molecules. This interaction potential is, as was shown in Figure 5-2, a -0.08-0.040.000.042 3 4 5 6Energy [kcal/mol]r [Å]2.50 Å2.90 Å3.20 Å4.00 Å-0.08-0.040.000.042 3 4 5 6Energy [kcal/mol]r [Å]2.50 Å2.90 Å3.20 Å4.00 Å-0.08-0.040.000.042 3 4 5 6Energy [kcal/mol]r [Å]2.50 Å2.90 Å3.20 Å4.00 Å162 rotationally averaged potential (RAP) for all the orientations of the molecular axes of the two H2 molecules interacting with each other. It was also discussed that, even though the same formulation is used in models I and II, the wavefunctions of the H2 molecules used in the equation are different in these models. In Model I, these wavefunctions are the same as the wavefunctions of the H2 molecules in the reference isolated (gas-phase) system. In Model II, however, the wavefunctions are the first-order PT wavefunctions of the H2 molecules affected by the surface electric field. The results show that, in Model II, the impact of the surface on the wavefunction of the H2 molecule in the adsorbed phase will affect the H2-H2 interaction energies by at most 3% compared to Model I and so can usually be ignored. 163 Simulation results for Model I In order to examine the structures of the H2 films over the NaCl or MgO substrates MC-NVT simulations are used. The results are then compared with the experiments to: 1) examine the validity of the model, and 2) answer questions such as the composition and the stability of the layers as well as the external vibrations of the molecules in the layer. In section 1.3.2, it was explained that, in the previous experimental studies22,23,26,29,37, a p(1×1) structure is observed for a monolayer of n-H2 and para-H2 molecules over NaCl. Also, in section 1.3.1, it was explained that over MgO, the n-H2 molecules form a sequence of c(n×2) structures, whereas the para-H2 molecules form different structures at different coverages of the layer10,20,37,41. This means that the structures are not completely known. Besides, as was explained in section 1.3.1, the composition and stabilities of these structures are not fully known either. Based on these large uncertainties in the experimental data, in Chapter 1, it was hypothesized that these structures may depend on the thermodynamic conditions of the system as well as the rotational state of the H2 species. To examine this hypothesis, first Model I was developed and then used it in the MC-NVT simulations. In this chapter, it is fully investigate if this model is able to reproduce the experimental data and give a detailed picture of the observed H2 films over the ionic surfaces of NaCl and MgO. If successful, then in the next step, the questions regarding the composition and stability of the layers as well as the vibrations of the H2 molecules can be answered. In order to be able to compare the results with the experimental data, the first requirement is that the thermodynamic conditions used in the simulations match those in the experiments. This means that the temperature and pressure must be the same in both simulation and experiment. In general, the proper ensemble to study the adsorption process is the μVT ensemble. In this 164 ensemble, the pressure of the system is implicitly included in the simulations through the chemical potential, which affects the coverage, composition, and the structure of the adsorbed films. However, in order to be able to do such MC simulations (MC-μVT), the accurate pressure of the system must be known75,95. But, because of the errors in the experimental data, the pressure of the system is not accurately known, making any comparison between the simulations and experiments difficult if not impossible. In order to accurately validate our models against the experimental data, instead of the pressure, it is easier to control the surface coverage by fixing the number of molecules N in the adsorbed phase (NVT ensemble). This is feasible when the interactions between the molecules and the surface are strong enough make desorption a rare event. In fact, the NVT ensemble simulation must be stopped when the interaction between H2 molecules in the adsorbed phase becomes free of the influence of the surface. This happens when the H2 molecules in the adsorbed phase sit far enough from the surface20, where they can move freely along at least one of the channels of the surface. At low heights, the interaction energy with the surface is the most dominant force against the movements of the molecules11,17,20,41. Therefore, the condition under which the MC-NVT simulations give reliable results must be first specified. In order to find these conditions, the height at which the H2-H2 interactions become important must be specified. Since in the layer, most of the molecules sit over their most attractive sites in the surface, energetically, the most important barrier against the lateral motion of the molecule (or the interaction between these molecules) is the MiEB quantity defined in section 4.3.1.1. In fact, without this barrier, the molecules aggregate and form an hcp structure. Since the MiEB quantity is defined based on the optimum heights (over both the most and second most attractive sites for the H2 molecule in the surface), it is a constant quantity. But, in the simulations, the 165 molecules do not necessarily sit at their optimum heights over the surface. In order to include the factor of height in the calculation of the minimum energy barrier, a new quantity of this barrier is introduced here. It is called the Height Dependent Minimum Energy Barrier (HD-MiEB) and is defined as follows: HD − MiEB = 𝐸୦ (௢௩௘௥ ௠௢௦௧ ௔௧௧௥௔௖௧௜௩௘ ௦௜௧௘) − 𝐸୫୧୬ (௢௩௘௥ ௦௘௖௢௡ௗ ௠௢௦௧ ௔௧௧௥௔௖௧௜௩௘ ௦௜௧௘) (6-1) where 𝐸௛ is the binding energy of the molecule at height h over its most attractive site. The difference between the HD-MiEB quantity and the MiEB quantity in Eq. (4-1) is in the first term in Eq. (6-1), i.e., 𝐸. In fact, this energy in HD-MiEB is calculated based on the height of the molecule over the surface, whereas in MiEB, it is always the minimum binding energy over the surface. But, the barrier against the motion of the molecule in the layer (HD-MiEB) depends on the temperature. In fact, as the temperature of the system goes up, the molecules get further away from the surface and hence, the effect of the surface on the motion of the molecule becomes smaller. By increasing the temperature, the system will reach a point, where the molecule does not see any barrier against its lateral motion, i.e., the HD-MiEB becomes zero. This point is the height beyond which the number of particles in the adsorbed phase cannot be a good representative of the pressure of the system in the gas phase overhead. This height is called the Upper Limit Height (ULH). In fact, the NVT simulation results can only be discussed if the heights of the molecules in the layer are below this height. At higher heights, μVT simulations must be done to study the properties of the layer. Therefore, before doing the MC-NVT simulations, first, the ULH quantities must be specified over each substrate. 166 6.1 H2 films over NaCl As explained in the previous section, the ULH quantities for the layer are specified based on the HD-MiEB value. According to Eq. (6-1), this value is obtained from the binding energy of a single H2 molecule over its most and second most attractive sites in the surface, respectively. Also, as stated earlier, ULH is the height at which the HD-MiEB quantity becomes zero. Table 6.1 shows the height at which the energy of either a Heli or Para H2 species over cation (most attractive site, see Table 4-19) becomes equal to the minimum energy of the molecule over the saddle site (second most attractive site), which means the height at which HD-MiEB = 0. In this table, the third column lists the “actual” ULH values at which the HD-MiEB is zero. But to ensure that in the simulations enough molecules have gained access to the height of zero HD-MiEB, some slightly higher values are specified as the ULH quantities. These are called the “specified” ULH and are presented in the fourth column of Table 6.1. Then, in order to know when to stop the simulations, the temperature at which the H2 molecules in the layer gain access to the ULH heights is considered as the Upper Limit Temperature (ULT), beyond which the MC-NVT simulation results are likely not valid. This temperature (ULT) is determined based on the height probability distributions (or HDFs) obtained in the MC-NVT simulations at different temperatures. 167 Table 6-1 The “actual” and “specified” ULH quantities for a Heli or Para H2 molecule over NaCl when Model I is used in the calculation of the energy. The ULH term is defined in the text. System E(H₂/Na⁺) ≈ Emin (H₂/Saddle) [kcal/mol] Actual ULH [Å] Specified ULH [Å] Heli/Na+ -0.40345 3.88 4.00 Para/Na+ -0.40236 3.72 3.90 6.1.1 Monolayers of n-H2 Based on the previous experimental and theoretical studies22,23,26,29,37, a monolayer of n-H2 molecules forms a p(1×1) structure that is presumably formed by a layer of only Heli molecules over the NaCl surface. This may be because, as suggested by these studies, in a (1×1) structure the available ions (attractive sites) of the substrate are occupied by the same H2 species in the monolayer. As shown in Table 4-19, the Heli/Na+ system has the strongest binding energy among all and hence, a monolayer of n-H2 molecules over NaCl must be formed by a layer of only Heli molecules. In order to discuss the results of the MC-NVT simulations, the first step is to find ULT. To determine this quantity, the HDFs of the layer at different temperatures are used. In fact, this function (HDF) gives the probability of finding the H2 molecules in the layer at any height above the ionic surface. Figure 6-1 illustrates this probability distribution for a monolayer of Heli molecules over NaCl at different temperatures up to 20 K. In Fact, 20 K is the ULT quantity for the Heli molecules over NaCl. Once again, this is the temperature at which the probability of finding the Heli molecules at heights comparable to the ULH quantity of 4.00 Å (Table 6.1) becomes nonzero (or HD-MiEB becomes zero). Thus, the simulation results for the Heli H2 films 168 on NaCl using Model I, can only be discussed and compared with the experiment up to, but not including, 20 K. At ULT and higher temperatures, the best ensemble in the MC simulations is the μVT ensemble provided that the pressure of the system is accurately known. Figure 6-1 The HDFs of a monolayer of Heli molecules over NaCl at temperatures up to 20 K. At 20 K, the function becomes nonzero at the ULH quantity of 4.0 Å. To discuss the simulation results in this chapter as well as Chapter 7 diagrams like Figure 6-2 are used. In this diagram, each particle in the simulation box is shown in a different color. Although, the axis of quantization is always fixed along the z-axis perpendicular to the surface, it is shown by a black dot in the figure (and the simulated structures) to help distinguish between para- and ortho-H2 species. 05102.4 3.0 3.6 4.2HDFDistance [Å]3 K7 K14 K20 K169 Figure 6-2 The color key diagram for the simulated H2 layers over the NaCl and MgO substrates. Although, the axis of quantization is applied to both para- and ortho-H2 species, for visualization purposes, it is only used for the ortho species in the images of the simulated configurations. Figure 6-3 shows the simulation results for the monolayers of Heli molecules over NaCl at different temperatures up to 19 K. In these simulations, Model I was used for the interactions between the molecules. Also, in all these simulations the starting configuration was a p(1×1) structure. The results in Figure 6-3 show that this structure is stable and highly ordered at 3 K (indicated by a square in subfigure (a)). They also show that even increasing the temperature up to 19 K will not break the p(1×1) structure. 170 (a) (b) (c) (d) Figure 6-3 Snapshots of the final configurations of a simulated monolayer of Heli molecules over NaCl using Model I and at, a. 3 K, b. 7 K, c. 14 K, and d. 19 K. The square in the subfigure (a) demonstrates the p(1×1) structure. The arrow in subfigure (d) indicates the movement of one of the Heli molecules away from atop a cation. In order to make sure that the structures seen in Figure 6-3, are indeed the p(1×1) structure, the H2-H2 Radial Distribution Functions (RDFs) of the simulated monolayers are studied. These functions are illustrated in Figure 6-4 (Heli-Heli RDFs in this case). They show the probability of finding each pair of molecules at a specific distance from each other in the simulation box. For example, in Figure 6-4, the peaks observed at 3 K occur exactly as expected for a p(1×1) structure as determined by the substrate lattice spacings. As seen in Figure 6-4, at higher temperatures, the peaks occur at the same locations as at 3 K, indicating the stability of the p(1×1) slightly moved molecule 3.96 Å 171 p(1×1) structure up to 19 K. With increasing temperature, the peaks broaden and shorten, indicating small movements of some of the molecules around the cation sites. For example, in subfigure (d) of Figure 6-3, one of the Heli molecules that has moved from atop Na+ ion is shown by an arrow. These movements only slightly affect the arrangements of the molecules in the layer and the structure is still p(1×1), as confirmed by the Heli-Heli RDFs shown in Figure 6-4. Therefore, within the temperature range studied here, the p(1×1) structure is the most stable structure seen over the surface. This structure is formed only by the Heli H2 molecules. These results are in agreement with the previous experimental and theoretical studies22,23,26,29,37. Figure 6-4 Heli-Heli RDFs of the simulated monolayers of Heli molecules over NaCl at temperatures up to 19 K. 03692 4 6 8 10Heli-Heli RDFDistance [Å]3 K7 K14 K19 K172 In order to explain why these structures are formed over NaCl, in Table 6-2 the three important energies causing these structures are listed; 1) the binding energy of a Heli molecule over its most attractive site (Na+ cation), 2) the binding energy between two Heli molecules, and 3) the MiEB quantity of the Heli molecule. In fact, two factors play dominant roles in forming the p(1×1) structures over NaCl. The first and most important factor is the binding energy of a Heli molecule over a cation (first column in Table 6-2), which is almost more than 12 times the binding energy between two Heli molecules (second column in the table). The second factor is the large MiEB (third column). As seen in Table 6-2, the MiEB quantity is almost seven times the binding energy of the two Heli molecules. This means that the barrier against the lateral motion of the Heli molecules is much larger than the attraction between the Heli molecules and since the molecules are strongly bound to the Na+ cations, they stay localized over these ions which results in the formation of a p(1×1) structure. Table 6-2 Comparison of the binding energy of a Heli H2 molecule over the Na+ site, the binding energy between two Heli H2 molecules, as well as the MiEB quantity for this molecule over the NaCl surface when Model I is used in the calculations. The MiEB is defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Heli-Heli binding energy [kcal/mol] MiEB [kcal/mol] Heli -0.87561 -0.06987 0.47186 In addition to the structures, the energies of the simulated layers must also be compared with experiment. Table 6-3 summarizes these energies at different temperatures. These are the energies of the simulated Heli H2 monolayers as well as the energies from previous experimental studies22,23,26,29,37. As seen in the table, at all temperatures, the simulation energies are in good 173 agreement with the experimental data. It should be noted that, the differences between the monolayer energies at temperatures shown in Table 6-3 can only be seen in the simulation data. In the experimental data, the uncertainties in the measurements are so large that make such comparison between the energies difficult if not impossible. Table 6-3 Energies (per molecule) of a monolayer of Heli molecules over NaCl. The quantities in brackets are either the uncertainties in the experimental data (previous studies) or the standard deviations resulted from the statistical averaging of the energies in the MC-NVT simulations (present work using Model I). Study Type of the study Temperature [K] Energy [kcal/mol/molecule] This work Theory (Model I) 5 -0.968 (± 0.001) 8 -0.959 (± 0.002) 10 -0.953 (± 0.003) 14 -0.933 (± 0.003) Briquez et al. 26 Theory - -0.980 Grunwald and Ewing 29 Experiment 5.2 -0.80 (± 0.24) Traeger et al. 22,37 Experiment 8 -0.92 (± 0.11) Heidburg et al. 23 Experiment ≥ 10 K -0.88 (± 0.23) 6.1.2 Monolayers of Para molecules Like the Heli monolayer, the first step is to specify the temperature (ULT) at which the molecules in the layer can get access to the ULH height of 3.90 Å (specified in Table 6-1). Figure 6-5 illustrates the HDFs of a simulated monolayer of Para molecules at three different temperatures. The data show that at 11 K, the probability of finding the Para molecules at heights 174 comparable to ULH (3.90 Å) becomes nonzero (or HD-MiEB becomes zero, these probabilities may not be seen in the figure). This indicates that 11 K is the starting temperature at which the MC-NVT simulation results become unreliable and MC-μVT simulations should be used. Figure 6-5 The HDFs of a monolayer of Para molecules over NaCl at temperatures up to 11 K. At 11 K and at the ULH quantity of 3.90 Å the probability becomes nonzero. In order to see how the Para molecules, arrange themselves over NaCl, MC-NVT simulations were used at temperatures up to 10 K. In Figure 6-6, the final configurations of these simulated layers are shown at temperatures 3, 7, and 10 K. 0482.5 3.0 3.5 4.0HDFDistance [Å]3 K7 K11 K175 (a) (b) (c) Figure 6-6 Snapshots of the final configurations of a monolayer of Para molecules over NaCl at a. 3 K, b. 7 K, and c. 10 K. As shown by the square in subfigure (a), the layer forms a p(1×1) structure at all temperatures up to 10 K. The configurations in Figure 6-6 show that, the p(1×1) structure is formed (shown by a square in subfigure (a)) and is stable up to 10 K. The results also show that at temperatures of 7 and 10 K (subfigures (b) and (c)), the p(1×1) film is not as perfectly structured as the one at 3 K due to small movements of the molecules around the cation sites. In fact, at higher temperatures, the molecules can sit further away from the surface, as seen in the HDFs of Figure 6-7, so they have slightly more freedom to move over these interacting sites. These small movements, however, cannot break the p(1×1) structure and this structure remains stable over the range shown in these figures. This is also confirmed by the RDFs of these layers in Figure 6-7. As seen in this figure, 3.96 Å p(1×1) 176 at all temperatures, the positions of the peaks are the same. As explained previously in Figure 6-4, these are exactly the positions that are expected for a p(1×1) structure. The broader and shorter peaks at higher temperatures indicate small shifts in the positions of the Para molecules around the cations, but, on average, the H2 molecules sit over cations in a p(1×1) structure. This result is in good agreement with the previous experimental and theoretical studies22,23,26,29,37 on the adsorbed films of Para H2 over NaCl. Figure 6-7 The Para-Para RDFs of a monolayer of Para molecules over NaCl at 3, 7 and 10 K, respectively. Similar to the Heli monolayers, three energies cause the formation of the p(1×1) structure in a monolayer of Para molecules; 1) the binding energy of this molecule to the Na+ cation, 2) the binding energy between two Para molecules, and 3) the MiEB of this molecule over NaCl. These are all listed in Table 6-4. The data in this table show that the binding energy of a Para molecule 0362 4 6 8 10Para-Para RDFDistance [Å]3 K7 K10 K177 to the cation site is almost nine times the attractive energy between two Para molecules in the layer. Also, the MiEB of the para molecule is more than three times the binding energy between two Para molecules. These factors result in the localization of the Para molecules over the Na+ cations. They also show that, even though small movements of the H2 species around cations is possible, the p(1×1) structure remains stable over the temperature range shown in Figure 6-7. This is in accord with the previous experimental and theoretical studies22,23,26,29,37 that observed a p(1×1) structure over this temperature range. It should be noted that, compared to the Heli molecules, the binding energy of the Para molecules to the Na+ site, is almost 28% weaker. Besides, the MiEB of Para is almost 51% smaller than the Heli molecule. Since the Para-Para and Heli-Heli energies are almost the same (Table 6-1 and Table 6-4), the stronger surface binding energy of the Heli molecule over cation as well as its bigger MiEB quantity (discussed above) result in higher ULH and ULT values in a monolayer of Heli molecules and they are expected to be stable to higher temperatures than the Para molecules. Table 6-4 Comparison of the binding energy of a Para H2 molecule over the Na+ site, the MiEB quantity of this molecule over the NaCl surface, as well as the binding energy between two Para H2 molecules when Model I is used in the calculations. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Para-Para binding energy [kcal/mol] MiEB [kcal/mol] Para -0.63344 -0.06883 0.22969 A comparison of the simulation and experimental binding energies of a monolayer of Para molecules over NaCl is presented in Table 6-5. The data in the table show good agreement between the energies of the simulated monolayers using Model I and the experiments22,23,26,29,37. 178 Similar to the Heli monolayers in Table 6-3, an accurate comparison of the energies between the simulated layers is limited by the large uncertainties in the experimental energies. Table 6-5 The energies (per molecule) of a monolayer of Para molecules over NaCl. The quantities in brackets are either the uncertainties in the experimental data (previous studies) or the standard deviations resulted from the statistical averaging in the MC simulations (present work using Model I). Study Type of the study Temperature [K] Energy [kcal/mol/molecule] This work Theory (Model I) 3 -0.7306 (± 0.0007) 5 -0.725 (± 0.001) 8 -0.715 (± 0.002) 10 -0.709 (± 0.003) Briquez et al. 26 Theory - -0.795 Grunwald and Ewing 29 Experiment 5.2 -0.80 (± 0.24) Traeger et al. 22,37 Experiment 8 -0.85 (± 0.12) Heidburg et al. 23 Experiment ≥ 10 K -0.71 (± 0.23) 6.2 H2 films over MgO As discussed in Chapter 4, since no accurate experimental energies are available for the binding energies of an H2 molecule over the MgO surface, two different energies, namely UBE and LBE, have been considered as the binding energy. In this section, the simulation results for H2 films over MgO are discussed for both of these energy cases. Similar to the NaCl substrate, the first step is to find ULH for each of the H2 species over the MgO substrate. 179 Table 6-6 presents the ULH quantities for the Heli and Para layers over MgO in both the UBE and LBE cases. As seen in the table, these values are the same as those for the NaCl substrate (compare the last columns in Table 6-1 and Table 6-6). Also, in Table 6-6, for each H2 species, these ULHs are the same regardless of the H2-MgO binding energy (UBE or LBE). Table 6-6 The ULH quantities for a Para and Heli molecule over the MgO substrate and in the UBE and LBE cases when Model I is used in the calculation of the energies. The ULH term has been defined in the text (see Eq. (6-1) and the explanation underneath this equation). Minimum binding energy [kcal/mol] System E(H₂/Mg²⁺) = Emin (H₂/Saddle) [kcal/mol] Actual ULH [Å] Specified ULH [Å] - 0.60039 (UBE) Heli/Mg2+ - 0.35475 3.89 4.00 Para/Mg2+ - 0.35329 3.74 3.90 - 0.66427 (LBE) Heli/Mg2+ - 0.36582 3.86 4.00 Para/Mg2+ - 0.36550 3.70 3.90 On the other hand, as explained earlier in this chapter, H2 films may form different structures based on the type of the H2 species and the coverage of the films adsorbed over the MgO surface. In the following sections, these films will be examined and discussed in both the UBE and LBE cases. 6.2.1 Half coverage (coverage of 0.50) At this coverage, experiments show a c(2×2) structure for a layer of n-H2 molecules, which is a mixture of ortho- and para-H2 molecules. For a layer of Para molecules, however, the structure 180 of the layers is not perfectly known from the experiments. In these experiments and from the data, it was interpreted that the layers are a mixture of islands of c(2×2) alongside disordered patches or maybe c(4×2) or c(6×2) (or hcp) structures68. In this section, these structures are examined at different temperatures using MC-NVT simulations. 6.2.1.1 The UBE case 6.2.1.1.1 Films of n-H2 molecules Based on the definition of a c type structure in section 1.2.2.1, in a c(2×2) structure all the molecules in the unit cell must be of the same type. Also based on the geometry of this structure, all the molecules in the unit cell should sit on the same interacting site in the surface. The previous experimental10,37 and theoretical25 studies suggested that these molecules are in fact helicopter-like ortho-H2 molecules sitting over cations. It can also be confirmed based on our energy results for a single H2 molecule over the MgO surface in Table 4-15. As seen, the Heli/Mg2+ system has the strongest binding energy among all and hence, a c(2×2) structure should be formed by the Heli molecules sitting over every other cation site along the channels of cations in the surface. Similar to the NaCl case (see section 6.1.1), in order to be able to compare the simulation results with the experimental data, the first step is to find the ULT quantities for the H2 molecules over this substrate. This temperature is specified based on the height probability distributions (or HDFs) of the layer at different temperatures. Figure 6-8 illustrates these HDFs for a layer of Heli molecules over the MgO surface at half coverage. The results show that, at 8 K the Heli molecules gain access to heights of about 4.0 Å (the ULH value for a Heli molecule in Table 6-6). Therefore, in these films, 7 K is the maximum 181 temperature up to which the MC-NVT simulations can be done. For temperatures equal to 8 K and higher, the MC-μVT simulations should be used to examine the properties of the Heli films at half coverage. Figure 6-8 The HDFs for a simulated layer of Heli molecules over MgO at half coverage using Model I. The results at 8 K show a nonzero probability at the ULH of 4.0 Å, so the MC-NVT simulations are consistent with the experiment only up to this temperature. To investigate the structures of the Heli H2 films at half coverage, MC-NVT simulations of these layers have been done at different temperatures up to ULT. Figure 6-9 demonstrates the final configurations of these layers. In all these simulations, the starting configuration was a c(2×2) structure. The simulation results show that the c(2×2) structure is stable up to 5 K. At 6 K, two defects (one independent/single defect) appear(s) in the structure, which are indicated by arrows in subfigure (c) of Figure 6-9. These defects may expose the surface to the adsorption of 0123452.2 2.7 3.2 3.7 4.2HDFDistance [Å]3 K5 K8 K182 more Heli molecules from the gas phase. Therefore, 6K is the starting temperature beyond which the surface is not fully covered with the c(2×2) structure and islands of higher density structures may be observable. This can be confirmed further by the configuration of the film at 7 K (subfigure (d) in Figure 6-9), which shows that more defects appear in the structure. At 6 and 7 K, in turn 4% and 7% of the layer is covered by a defect (aggregated) structure. (a) (b) (c) (d) Figure 6-9 Final configurations of a layer of Heli molecules at a. 3 K, b. 5 K, c. 6 K, and d. 7 K using Model I. The square in subfigure (a) indicates a c(2×2) unit cell for the H2 film. The arrows in subfigures (c) and (d) show defects in the c(2×2) structure. The number of arrows show the number of defects formed in the film at each temperature. 2.96 Å c(2×2) defects defects 183 To confirm that the structures seen in Figure 6-9 are in fact the c(2×2) structure, the Heli-Heli RDFs of these layers are presented in Figure 6-10. Two large peaks appear in the graph at all temperatures shown in the figure. Based on the distances between the ions in the MgO crystal, the positions of these two peaks are exactly at distances expected for a c(2×2) structure. It should be added that, at temperatures up to 5 K, the layer is a perfect c(2×2) structure. At 6 and 7 K, however, two new bumps appear in the graphs, signaling the presence of defects in the layer (these defects are shown by arrows in subfigures (c) and (d) of Figure 6-9). These bumps grow larger as the temperature is raised from 6 to 7 K, indicating the formation of more defects in the structure. In fact, at 6 K, three defects (one of which is an independent defect) appear in the structure, which are shown by arrows in subfigure (c) of Figure 6-9. At 7 K, however, six defects are formed (shown by arrows in subfigure (d) of Figure 6-9). This means that, at 6 and 7 K, islands of aggregated molecules along with open sites occur within the c(2×2) structure instead of the c(2×2) structure covering the whole surface. Since, the positions of the shorter peaks (the two bumps in the graph) at 6 and 7 K are the same, the nature of the defects are the same at these temperatures. Besides, from Figure 6-10, the position of the first bump is at about 3.2 Å, which is only slightly larger than the distance between two neighboring cations (3.0 Å) along the channel of cations in the MgO surface. This may imply that the defects are formed by the Heli molecules jumping over the saddle sites to neighboring available/open cation sites. This observation suggests that even at lower temperatures such movements may be possible. Therefore, in the rest of this section, the possibility of the formation of defects in the c(2×2) structures at lower temperatures is fully analyzed. This analysis will finally lead to a statistical mechanics model that will be used to approximately predict the stability of the c(2×2) structure in the MC-NVT 184 simulations. It should be noted that to accurately determine the stability of the c(2×2) structure at temperatures higher than ULT, the MC-μVT simulations must be used. Figure 6-10 The Heli-Heli RDFs of a simulated layer of Heli molecules over MgO at half coverage using Model I. The larger peaks show the average positions of the molecules which are similar to the positions expected for a c(2×2) structure. The two bumps in the graphs at 6 and 7 K indicate the appearance of some defects in the c(2×2) structure. The larger bumps at 7 K compared to 6 K show the presence of more defects in the structure. So far, the simulations have shown that the Heli molecules over MgO and at half coverage form a perfect c(2×2) structure. In this structure, only half of the cation sites in the surface are occupied by the Heli molecules. This is not like the p(1×1) structure encountered on the NaCl substrate, in which all the cations are occupied by the H2 (Heli or Para) molecules. In fact, in the p(1×1) structure, the H2 molecules do not have any other interacting sites but the cations to sit onto and form a low-energy structure. In a c(2×2) structure, however, half of the cations are 07143 5 7Heli-Heli RDFDistance [Å]3 K5 K6 K7 K185 open, which means that the H2 molecules have the chance to jump to the neighboring cations and form defects which may result in a lower-energy structure. The formation of a sufficient number of these defects may consequently result in the disappearance of the c(2×2) structure. Since the MC-NVT simulations of this layer is done at very low temperatures, the molecules may not have enough energy to do these jumps and form defects in the layer. In order to make sure that the perfect c(2×2) structure is the most stable structure over the temperature range shown in Figure 6-9, some defects are artificially and systematically created in these layers. These defects are created by moving one of the H2 molecules in the c(2×2) unit cell over to the neighboring cation. Then MC simulations are done on these defect structures and the energies of these layers are compared with the energies of the perfect c(2×2) layers at the same temperature. The difference between these energies tells us which structure is more stable. This difference is also used in a statistical mechanics model to find the approximate temperature up to which the c(2×2) structure can be seen in the MC-NVT simulations. In order to create these defects in the structure, the important thing is that the energy of the defect structure must be different enough from the c(2×2) layer. Since creating only one defect in the structure results in a layer with an energy very close to the perfect c(2×2) layer, it was decided to create four defects in the structure. On the one hand, creating this number of defects gives a more efficient statistical average for the energy of the defect structure. On the other hand, this number of defects (and not more) will give a c(2×2) layer with defects nicely spaced out so they cannot interact with one another in the layer. Besides, four defects can be considered as an acceptable number of defects that can be created in the c(2×2) structure while still maintaining the coverage of 0.5 in the adsorbed layer. In fact, the four defects in the structure can be considered as small fluctuations (~5%) in the coverage that may also be observed in the 186 experiments. Also, the other purpose of forming these defects is to examine if these defects will result in the aggregation of the layer or even the formation of islands of defect structures. Figure 6-11 demonstrates how these four-defect structures are made in a c(2×2) layer. For the sake of the argument, this layer is called a 4d-c(2×2) structure (subfigure (b)). It should be mentioned that, in this section, all the simulations started with the 4d-c(2×2) structure. This structure then was equilibrated during the simulations, which means that it had the chance to either remain as a 4d-c(2×2) structure or form a c(2×2) structure, or even form an aggregated layer. (a) (b) Figure 6-11 The c(2×2) structure (subfigure (a)) as well as a 4d-c(2×2) structure (subfigure (b)), which is created by four defects in a c(2×2) layer. The circles in subfigure (b) indicates the location of each one of the defects in the layer and how they are spaced out in the layer. Figure 6-12 compares the energies of the perfect c(2×2) and the 4d-c(2×2) layers at different temperatures. It should be mentioned that, for the purposes of our defect analysis, these energies are reported for a layer of 72 H2 molecules and not on a per molecule basis. The same is true for the error bars, which show the standard deviations associated with the energy of the layer in the 187 simulations. As seen in the graph, at all temperatures the energy of the c(2×2) layer (even when the error bars are included in the comparison) is lower than the energy of the 4d-c(2×2) layer. This means that, at the temperatures shown in Figure 6-12, creating defects in the c(2×2) structure results in an increase in the energy. Therefore, the c(2×2) layer is the most stable structure at half coverage. It should be added that, in the course of the simulations, the 4d-c(2×2) structure stays stable up to 4 K. At 5 K, however, one of the defects goes back to the c(2×2) structure, so the final configuration at this temperature is in fact a 3d-c(2×2) layer. As seen in Figure 6-12, the energy of this layer is still distinguishable from the perfect c(2×2) layer. Therefore, from our defect analysis it is concluded that, in the UBE case, the c(2×2) structure is the most stable structure for a layer of Heli H2 molecules over MgO at half coverage and at temperatures from 0 to 5 K. 188 Figure 6-12 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 5 K. The defect c(2×2) layers contain 4 defects (4d-c(2×2) layer) at the temperature range of 1 to 4 K, whereas at 5 K, they only include three defects in the layer (3d-c(2×2) layer). The equations shown on each curve are only based on the data from 1 to 4 K. Also, they are the equations used for the statistical mechanics model explained in the text. The error bars in each case show the standard deviation in the energy. As mentioned earlier, in order to examine the thermal stability of the c(2×2) layer at temperatures higher than 6 K, MC-μVT simulations should be done. Using MC-NVT simulations, however, the thermal stability can be approximately determined, which is done by using a crude statistical mechanics model based on the energies of the 4d-c(2×2) and the perfect c(2×2) layers discussed in Figure 6-12. This model is, in fact, based on the probability of seeing each one of these structures at half coverage. According to the model, this probability, depends on two factors; 1) degeneracy of finding the layer in each of these structures, and 2) the energies of the perfect c(2×2) and the 4d-c(2×2) layers. The important notion here is that, these energies y = 0.2293x - 48.8680R² = 0.9998y = 0.2306x - 49.2303R² = 0.9998-49.0-48.5-48.0-47.50 1 2 3 4 5 6Energy [kcal/mol]Temperature [K]c(2×2)c(2×2) with defects189 must be the energies of the layers at 0 K, because at this temperature, there is no entropic contribution to the energy. Since the simulations cannot be done at 0 K, the linear equations fitted to the curves in Figure 6-12 are used to extrapolate the energies of these layers back to 0 K. It was mentioned earlier that the defect layer at 5 K is 3d-c(2×2) (and not 4d-c(2×2)) and hence, the data at 5 K have not been used in constructing the slope equations in Figure 6-12. As seen, the R2 values are close to 1, indicating that at temperatures close to zero, the energies of these layers change linearly. Also, the lines are parallel to each other, which means that, over the temperature range used for these equations, the shift in energies from one temperature to another is the same in the perfect c(2×2) and 4d-c(2×2) layers. By using the equations of the lines in Figure 6-12, the energies of the 4d-c(2×2) and perfect c(2×2) can be obtained at 0 K (the intercepts of the lines). These energies are then used in a statistical mechanics model to estimate the thermal stability of the layer. This model, in fact, gives the ratio between the probabilities of seeing either a 1d-c(2×2) structure or a perfect c(2×2) structure in the whole simulated layer. The 1d-c(2×2) layer is in fact a c(2×2) layer with only one defect in the whole structure and its energy is obtained from the 4d-c(2×2) layer as discussed below. The aforementioned statistical mechanics model has the following form51,96: 𝑃ଵௗି௖(ଶ×ଶ)𝑃௖(ଶ×ଶ)=𝑔ଵௗି௖(ଶ×ଶ). 𝑒ିாభ೏ష೎(మ×మ)/௞ಳ்𝑔௖(ଶ×ଶ). 𝑒ିா೎(మ×మ)/௞ಳ் (6-2) where P is the probability of finding the desired layer, 𝑔 is its degeneracy, 𝐸 is the energy of the layer, 𝑘஻ is the Boltzmann’s constant, and 𝑇 is the temperature of the system, and 𝑘஻𝑇 is the thermal energy available for the system. To calculate the probability of one of the structures, for example the 1d-c(2×2) structure, Eq. (6-2) can be re-written in the following form: 190 𝑃ଵௗି௖(ଶ×ଶ)𝑃ଵௗି௖(ଶ×ଶ) + 𝑃௖(ଶ×ଶ)=𝑔ଵௗି௖(ଶ×ଶ) × 𝑒ିாభ೏ష೎(మ×మ)/௞ಳ்𝑔ଵௗି௖(ଶ×ଶ). 𝑒ିாభ೏ష೎(మ×మ)/௞ಳ் + 𝑔௖(ଶ×ଶ). 𝑒ିா೎(మ×మ)/௞ಳ் (6-3) where 𝑃ଵௗି௖(ଶ×ଶ) + 𝑃௖(ଶ×ଶ) = 1. In order to find the degeneracy of each layer, the simplest method is to find the number of ways that one defect can be created in the c(2×2) unit cell (indicated by the square in subfigure (a) of Figure 6-9). To create one defect, there are four options, because there are four unoccupied neighboring cations in the unit cell. This means that 𝑔ଵௗି௖(ଶ×ଶ) = 4. But, the degeneracy of the c(2×2) structure is 1, because in the unit cell, there is only one way to create such structure. Thus Eq. (6-3) becomes: 𝑃ଵௗି௖(ଶ×ଶ) =4 × 𝑒ିாభ೏ష೎(మ×మ)/௞ಳ்4 × 𝑒ିாభ೏ష೎(మ×మ)/௞ಳ் + 1 × 𝑒ିா೎(మ×మ)/௞ಳ் (6-4) or 𝑃ଵௗି௖(ଶ×ଶ) =44 + 1 × 𝑒ା∆ாబ಼/௞ಳ்=44 + 1 × 𝑒ା൫ாభ೏ష೎(మ×మ)ିா೎(మ×మ)൯ |ೌ೟ బ಼/௞ಳ் (6-5) In Eq. (6-5), the difference in energy at 0 K (∆𝐸଴௄) is: ∆𝐸଴௄ = ൫𝐸ଵௗି௖(ଶ×ଶ) − 𝐸௖(ଶ×ଶ)൯ |௔௧ ଴௄ =14൫𝐸ସௗି௖(ଶ×ଶ) − 𝐸௖(ଶ×ଶ)൯ |௔௧ ଴௄ (6-6) This difference is, in fact, 1/4 of the difference in energy between the 4d-c(2×2) and the perfect c(2×2) layers at 0 K. As explained above, this difference is obtained from the intercepts of the two slope equations in Figure 6-12. From the intercepts and by including the errors, this 191 difference is about 0.091 ± 0.003 kcal/mol (or 45.6 ± 1.5 K). Thus, Eq. (6-5) may be written in the following form: 𝑃ଵௗି௖(ଶ×ଶ) =44 + 1 × 𝑒ାସହ.଺/் (6-7) Using Eq. (6-7), it can be determined how much of the structure (in %) will turn to a single-defect structure as a function of temperature. For example, this equation predicts that at 5 K, the probability of finding a single defect in the c(2×2) structure is 0.0004. This means that for the 72 Heli molecules in the structure, only 0.03 (≈0) of them will be single defects. This prediction from the statistical model is consistent with the simulation results in subfigure (b) of Figure 6-9 that show no defect in the structure. At 6 K, the model predicts 0.14 (≈0) for a single defect in the layer. The simulation, however, shows more than one defect (subfigure (c) in Figure 6-9), but they are not independent of each other. This shows that, our model is crude in estimating the number of single defects in the structure (it assumes defects do not interact), but it can be used to approximately determine the thermal stability of the layer in the MC-NVT simulations. For example, the model predicts that, at about 18 K, half (50%) of the layer is covered with single-defect c(2×2) unit cells (i.e., 18 out of 36 c(2×2) unit cells), which means that the perfect c(2×2) structure may not be seen in the MC-NVT simulations. Therefore, these results imply that experimentally the c(2×2) structure cannot be observed beyond this temperature. It should be added that this temperature is in fact an upper limit for the temperature up to which the c(2×2) structure may be seen in the MC-NVT simulations, because our model at 6 K predicts fewer defects than what is actually seen in the simulations. Nonetheless, these results are in accord with a previous experimental study in which a c(2×2) structure was observed for n-H2 at half coverage and at a temperature range of 5.8 to 9 K 10. 192 Similar to the NaCl substrate, three important energies play a role in observing the c(2×2) structure at this coverage; 1) the binding energy of the Heli H2 molecule over the cation site, 2) the binding energy between two Heli molecules in the layer, and 3) the MiEB of the Heli molecules over the MgO surface. As seen in Table 6-7, the binding energy of the Heli molecule above Mg2+ as well as the MiEB of this molecule are, in turn, 8.6 and 3.5 times the binding energy between two Heli molecules in the layer. These two factors help the molecules to remain localized atop cations in the form of a c(2×2) structure at the experimental temperatures discussed above. Table 6-7 The binding energy of a Heli molecule over the Mg2+ cation, the energy between two Heli molecules, as well as the MiEB of a Heli molecule over the MgO surface when Model I along with the UBE case are used in the calculations. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Heli-Heli binding energy [kcal/mol] MiEB [kcal/mol] Heli -0.60039 -0.06987 0.24603 Besides the structures, the energies of the H2 films must also be compared with the experiments. But, unlike the NaCl case, there is no experimental data available in the literature for the energy a layer of H2 molecules over MgO at half coverage. Thus, this comparison is not possible, but for future reference, the energies of a layer of Heli molecules at half coverage and at different temperatures are presented in Table 6-8. As seen in the table, at T ≤ 5 K, the total energy decreases with increasing the temperature, which is the result of the decrease in both H2-H2 and H2-surface interactions. In fact, according to Figure 6-8, by increasing the temperature up to 6 K, the molecules in the c(2×2) structure gain access to higher heights. This results in a 193 dramatic decrease in the H2-surface interactions. On the other hand, the molecules that now sit further away from the surface are no longer as strongly trapped over the cations and can move more freely over these sites. This may cause these molecules to sit a bit closer to the neighboring Heli molecules in the layer and decrease the H2-H2 interactions (stronger H2-H2 binding energies). But, this shift in the H2-H2 interaction energy is very small compared to the shift in the H2-surface energy. Therefore, at 6 and 7 K, even though defects are formed and the H2-H2 interactions decrease (stronger binding energies), the total energy increases (weaker binding energies) due to the large increase in the H2-surf interactions (weaker binding energy). Table 6-8 The energies (per molecule) of an MC-NVT simulated layer of Heli molecules over MgO at half coverage at temperatures up to the ULT quantity. The errors are the standard deviations obtained from the statistical averaging in the MC-NVT simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. Energy [kcal/mol/molecule] Temperature [K] Structure Total H2-surf H2-H2 1 c(2×2) -0.6805 ± 0.0003 -0.5957 ± 0.0003 -0.08300 ± 0.00002 3 c(2×2) -0.674 ± 0.001 -0.591 ± 0.001 -0.08308 ± 0.00006 5 c(2×2) -0.668 ± 0.002 -0.584 ± 0.002 -0.08319 ± 0.0001 6 c(2×2) with defects -0.663 ± 0.002 -0.578 ± 0.003 -0.0847 ± 0.0007 7 c(2×2) with defects -0.630 ± 0.002 -0.572 ± 0.003 -0.086 ± 0.001 194 6.2.1.1.2 Films of Para molecules As specified in Table 6-1, the ULH quantity for the Para molecules over MgO in the UBE case is 3.90 Å. Figure 6-13 illustrates the HDFs of the MC-NVT simulated layers of Para molecules over MgO at half coverage and at different temperatures. The results show that, based on the criterion explained in Table 6-1, the MC-NVT simulations can only be done up to, but not including, 3 K. At 3 K and higher temperatures, the MC-μVT simulations must be used to study the properties of the layer at half coverage. Figure 6-13 The HDFs for a layer of Para molecules over MgO at half coverage and at temperatures 1 to 3 K. In these simulations Model I is used along with the UBE case. At 3 K, some of the molecules gain access to the ULH quantity, so the ULT value is 3 K. Figure 6-14 shows the final configurations for a layer of Para molecules over MgO at half coverage. In these simulations, the starting configuration was a perfect c(2×2) structure. As seen in the figure, even at low temperatures of 1 and 2 K, the c(2×2) structure breaks down and the 012343.0 3.3 3.6 3.9HDFDistance [Å]1 K2 K3 K195 layer forms an aggregated structure along with patches of open surface. The aggregated structure is in the form of an hcp structure, which is depicted by a hexagon in subfigure (a) of Figure 6-14. This structure is close to a c(6×2) structure indicated by a rectangle in subfigure (b) of Figure 6-14, but it is not a perfect c(6×2) structure. The formation of these structures is confirmed by the Para-Para RDFs in Figure 6-15, where the peaks appear at positions expected for an hcp (or in this case nearly a c(6×2)) structure. Although as stated in section 1.3.1.2, from the experiments, the structures of para-H2 layers over MgO at half coverage are not very well known, it was interpreted that the layers are a mixture of islands of c(2×2) alongside disordered patches or maybe c(4×2) or c(6×2) (or hcp) structures68. Our simulation results are not in agreement with the experiments because there is no sign of a c(2×2) structure over the surface. Also, the patches of open surface may lead to the adsorption of more H2 molecules onto the surface that will change the coverage of the layer completely. On the other hand, these structures show how dominant the H2-H2 interaction energies are compared to the H2-surface energies and the molecules aggregate at such low temperatures. 196 (a) (b) Figure 6-14 Final configurations of the MC-NVT simulated layers of Para molecules over MgO at a. 1 K, and b. 2 K using Model I. The results show an aggregated structure, which is equivalent to a hexagonal closed pack (hcp) structure depicted by a hexagon in subfigure (a). In this case, this structure is very close to a c(6×2) structure shown by the rectangle in subfigure (b). Figure 6-15 The Para-Para RDFs of a film of Para molecules over MgO at half coverage and at temperatures of 1 and 2 K. In the MC-NVT simulation of these layers, Model I is used along with the UBE case. 0714213 4 5 6 7 8Para-Para RDFDistance [Å]1 K2 Kc(6×2) hcp 2.96 Å 197 To explain the formation of the hcp structures, three important energies presented in Table 6-9 are considered. As seen, although, the binding energy of the Para molecule to the surface is almost 6.9 times the attractive energy between two Para molecules, the MiEB for this molecule is less than the binding energy between two Para molecules. This means that the Para molecules, that see no barrier against their lateral motion over the surface, aggregate to lower the energy. Therefore, the aggregated (hcp) structure is formed in the simulations, because the very low MiEB of the Para molecule prevents the layer from forming even islands of c(2×2) structure over the surface. This process leaves some parts of the surface open to more adsorption of the H2 molecules from the gas phase. Table 6-9 The binding energy of a Para molecule over Mg2+, the binding energy between two Para molecules, as well as the MiEB of a Para molecule over the MgO surface when Model I along with the UBE case are used in the calculations. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Para-Para binding energy [kcal/mol] MiEB [kcal/mol] Para -0.40891 -0.06883 0.05455 As discussed above, the MC-NVT simulated structures of the layers of Para molecules over MgO and at half coverage are not in agreement with the experiment. It was seen that islands of the c(2×2) structures cannot be seen in the simulated layers, even though the main concern here are the patches of open surface which clearly show the dominance of the H2-H2 interactions. But, for a layer of n-H2 (Heli) molecules these c(2×2) structures were possible and it was shown that, at the investigated temperatures, the layer at half coverage was fully covered by these structures (see Figure 6-9). Comparing the energies in Table 6-9 and Table 6-7, gives more information 198 about why these differences are seen in the structures. These data show that the binding energy of the Para molecule to the surface or even the MiEB quantity for this molecule are not as strong as the corresponding energies for the Heli molecules. In fact, by using the binding energy in the UBE case, the energy might have been underestimated the binding energy of the H2 molecule to the surface (the H2-surface interaction). Since there is no accurate experimental data available for this energy, examining the structures at a stronger binding energy is necessary. This stronger binding energy was introduced in Chapter 4 and is called the LBE case. Also, as discussed in Chapter 4, lowering the energy will affect the binding energy of both Heli and Para molecules over the surface as well as the MiEBs of these molecules (see section 4.3). This may have an impact on the energies and structures of a layer of these molecules over the MgO surface at different coverages. To see how the properties of the layer change by changing the binding energy, in the next section, the MC-NVT simulations results are discussed for H2 layers over MgO at half coverage but in the LBE case. Then these results are compared with those already presented for the UBE case. 6.2.1.2 The LBE case 6.2.1.2.1 Films of n-H2 molecules Based on the discussion of section 6.2.1.1.1, for this layer, the c(2×2) structure is expected to be formed by only Heli molecules sitting over every other cation sites in the c(2×2) unit cell. As presented in Table 6-6, the ULH quantity for the Heli molecules over MgO in the LBE case is the same as the UBE case (4.0 Å in both cases). Figure 6-16 illustrates the HDFs of a layer of Heli molecules over MgO at half coverage and in the LBE case. The results show that at 10 K 199 and higher temperatures, the probability of finding the Heli molecules at heights of about 4.0 Å becomes nonzero. Therefore, the MC-NVT simulations can only be compared with the experimental data up to 10 K. Figure 6-16 The HDFs of a layer of Heli molecules over MgO at half coverage. In these simulations, Model I was used along with the LBE case. The data show that 10 K is the maximum temperature up to which the coverage can be kept consistent with the experiment. Figure 6-17 illustrates the final configurations of a simulated layer of Heli molecules over MgO at half coverage and in the LBE case. In these simulations, Model I was used in the calculation of the interactions between the molecules. Also, the starting configuration was a c(2×2) structure. The results show that, the c(2×2) structure stays stable up to 8 K (compared to 5 K in the UBE case in Figure 6-9). The unit cell of this structure is shown by a square in subfigure (a) of Figure 6-17. At 9 K, seven defects appear in the layer and it is no longer a perfect c(2×2) structure (indicated by arrows in subfigure (d)). 0.02.04.06.02.2 2.8 3.4 4HDFDistance [Å]3 K5 K8 K10 Kdefects 200 (a) (b) (c) (d) Figure 6-17 Final configurations of the simulated layers of Heli H2 molecules over MgO at half coverage when Model I along with the LBE case are used in the simulations. These configurations are at a. 3 K, b. 5 K, c. 8 K, and d. 9 K, respectively. The square in subfigure (a) indicates the c(2×2) unit cell in the layer. The arrows in subfigure (d) show the location of defects in the structure. Figure 6-18 confirms the formation of the c(2×2) structures in the layers shown in Figure 6-17. The two main peaks seen at 4.19 and 5.94 Å in Figure 6-18, are at positions expected for a perfect c(2×2) structure. At 9 K, two new smaller peaks (bumps) appear in the curve, indicating the formation of defects in the structure. From the positions of these new peaks (3.26 and 6.74 Å) it can be concluded that these defects are formed by some of the Heli molecules in the perfect c(2×2) structure jumping over to the neighboring cation sites in the layer. 2.96 Å c(2×2) defects 201 Figure 6-18 The Heli-Heli RDFs of a layer of Heli H2 molecules over MgO at half coverage. In these simulations, Model I along with the LBE case were used in the calculation of the energies. Similar to the UBE case, in order to make sure that the c(2×2) structure is the most stable structure for a layer of Heli molecules over MgO at half coverage and at temperatures shown in Figure 6-17 and Figure 6-18, the defect analysis similar to the procedure presented in section 6.2.1.1.1 is used. This is done because at low temperatures of 0 to 8 K, the molecules may not have enough energy to break the starting configuration of the c(2×2) structure in the course of the simulations. Based on the procedure explained in section 6.2.1.1.1, in the first step, four defects are created in the c(2×2) layer as shown in subfigure (b) of Figure 6-11 (the 4d-c(2×2) film). This is the starting configuration in the defect structure simulations. Then MC-NVT simulations are done on these films and their energies are compared with the energies of the perfect c(2×2) films at the same temperature. Figure 6-19 illustrates these energies. Once again, for the purpose of our analysis here, all the energies shown in this figure for a layer of 72 H2 08163 5 7Heli-Heli RDFDistance [Å]3 K5 K8 K9 K202 molecules and not a per molecule basis. It should also be noted that, during the simulations of the 4d-c(2×2) films, the starting 4d-c(2×2) layer did not remain stable at 5, 7, and 8 K, and fewer or more defects were formed in the layer. For example, at 5 K, one of the defects went back to its original spot in the c(2×2) structure and the final configuration is a c(2×2) structure with only three defects (a 3d-c(2×2) layer). At 6 K, the layer is still 4d-c(2×2), but at 7 K, more defects appear in the structure. At 8 K, only one defect survived and three of the defects went back to the c(2×2) structure (a 1d-c(2×2) layer). This is the reason why in Figure 6-19, the energy of the defect layer at 8 K is very close to the perfect c(2×2) film (the bars or standard deviations overlap). But, as was explained in section 6.2.1.1.1, for simplicity, in our defect analysis, only the 4d-c(2×2) layers as well as the perfect c(2×2) layers are used. Therefore, only the simulated layers at temperatures between 1 and 4 K are used in forming the statistical mechanics model of Eq. (6-7). Nevertheless, the data in Figure 6-19 show that, creating defects in the structure results in an increase in the energy at all the temperatures presented in this figure. This means that, the perfect c(2×2) layer is the most stable structure over this temperature range. 203 Figure 6-19 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 8 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at the temperature range of 1 to 4 K, whereas at higher temperatures, either fewer or more defects appear in the layer. The equations shown on each curve are only based on the data from 1 to 4 K and are used for the statistical mechanics model explained in the text. The bars in each case show the standard deviation in the energy. Similar to the UBE case, in the MC-NVT simulations, the thermal stability of the c(2×2) film can be approximately determined at temperatures higher than 8 K. By using the intercepts of the slope equations shown in Figure 6-19, ∆𝐸଴௄ in Eq. (6-5) can be calculated. This difference is about 0.134 ± 0.003 kcal/mol (or 67.4 ± 1.5 K). By substituting this energy into Eq. (6-5): 𝑃ଵௗି௖(ଶ×ଶ) =44 + 1 × 𝑒ା଺଻.ସ/் (6-8) Now, by using Eq. (6-7), at 9 K, no single defects are expected to be seen in the structure. The simulation results in Figure 6-17 show three defects at this temperature, some of which are y = 0.2336x - 53.8841R² = 0.9998y = 0.2322x - 53.3484R² = 0.9999-53.7-53.2-52.7-52.2-51.7-51.20 3 6 9Energy [kcal/mol]Temperature [K]c(2×2)c(2×2) with defects204 multiple (dependent) defects. This comparison alone shows that the model is crude, but it may give an approximate value for the temperature up to which the c(2×2) can be seen in the simulations. Based on the model, in the LBE case, at about 27 K, (50%) of the c(2×2) film is covered with single-defect c(2×2) unit cells (i.e., 18 out of 36 c(2×2) unit cells) and the perfect c(2×2) layer can no longer be seen in the simulations (compared to 18 K in the UBE case). Similar to the UBE case, these results are also in agreement with the previous experimental findings at temperatures of 5.8 to 9 K, in which a c(2×2) film was observed for a layer of n-H2 over MgO at half coverage10. To explain why these structures are formed, the energies shown in Table 6-10 are used. As seen, in the LBE case, the binding energy and the MiEB of the Heli molecule over the MgO surface are in turn, 9.6 and 4.3 times the attractive energy between two Heli molecules. In fact, as explained in section 6.2.1.1.1, it is the combination of these three energies that forces the Heli molecules to sit over cations in the form of a c(2×2) structure. But, to understand why in the LBE case the c(2×2) structure is stable up to a higher temperature, the energies in Table 6-10 (the LBE case) are compared with the corresponding energies in Table 6-7 (the UBE case). Comparison of the data show that in the LBE case, the binding energy is 11% stronger than the UBE case. Also, the MiEB value in the LBE case is about 22% larger than the UBE case (see also section 4.3.1.1.2). These two factors are in fact the reason why in the LBE case the simulated c(2×2) structures are stable up to higher temperatures. These results also stress the importance of: 1) the binding energy of the H2 molecule to the surface, 2) the MiEB of the H2 molecule, and 3) the H2-H2 interaction energy, in the formation of the c(2×2) structures over the surface. 205 Table 6-10. The binding energy of a Heli molecule over Mg2+, the binding energy between two Heli molecules, as well as the MiEB of a Heli molecule over the MgO surface when Model I along with the LBE case are used in the calculations. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Heli-Heli binding energy [kcal/mol] MiEB [kcal/mol] Heli -0.66427 -0.06987 0.29957 Similar to the UBE case, the energies of the c(2×2) films obtained from the simulations are listed in Table 6-11. As seen in the table, by increasing the temperature, the H2-surface interactions increase (weaker binding) which is a result of the molecules getting further away from the surface. Although, there is a slight decrease (stronger binding) in the H2-H2 interactions by increasing the temperature, the shift in H2-surface interactions is dominant and hence, the total energy of the c(2×2) layers increases (weaker binding) by increasing the temperature. It is interesting to note that, at the same temperature, the only difference between the energies of the simulated layers in Table 6-8 and Table 6-11 is in the H2-surface interactions. This is because, in the LBE and the UBE cases, the H2-H2 interaction does not depend on the position (height) of the molecule over the surface whereas the H2-surface interaction does. 206 Table 6-11 The energies of the MC-NVT simulated c(2×2) films of Heli molecules over MgO when Model I along with the LBE case are used in the simulations. The numbers after the ± sign are the MC-NVT simulation standard deviations for the energies. Energy [kcal/mol/molecule] Temperature [K] Structure Total H2-surf H2-H2 1 c(2×2) -0.7451 ± 0.0003 -0.6621 ± 0.0003 -0.08298 ± 0.00001 3 c(2×2) -0.7387 ± 0.0009 -0.6557 ± 0.0009 -0.08301 ± 0.00004 5 c(2×2) -0.732 ± 0.002 -0.648 ± 0.002 -0.08304 ± 0.00007 8 c(2×2) -0.721 ± 0.003 -0.638 ± 0.003 -0.0831 ± 0.0001 9 c(2×2) with defects -0.712 ± 0.003 -0.627 ± 0.004 -0.085 ± 0.001 6.2.1.2.2 Films of Para molecules In section 6.2.1.1.2, it was discussed that a layer of Para molecules at half coverage cannot form the c(2×2) structure, even in the form of islands. Therefore, based on the energies that play role in forming these structures (Table 6-9), a stronger binding energy (the LBE case) to the surface was suggested. Similar to the UBE case, before examining the layers of Para molecules over MgO at half coverage, the first step is to find the ULT quantity based on ULH (Table 6-6). Once again, to determine ULH the HDFs of the layers are used. Figure 6-20 shows these functions for a layer of Para molecules over MgO at half coverage, when Model I along with the LBE case are used in the simulations. Based on these and the procedure explained in section 6.1, the temperature up to which the coverage can be controlled in the MC-NVT simulations is 2 K. 207 At 3 K, some of the molecules are found at heights equal to the ULH quantity of 4.0 Å. Therefore, for temperatures higher than 2 K, the MC-μVT simulations should be used. Figure 6-20 The HDFs of a layer of Para molecules over MgO at different temperatures when the coverage is kept fixed at half. In these simulations, Model I along with the LBE case were used in the calculation of the energies. At 3 K, some of the molecules gain access to the ULH quantity and hence, 3 K is the maximum temperature up to which the MC-NVT simulation conditions are consistent with the experiment. Figure 6-21 illustrates the final configurations of the simulated layers of Para molecules at half coverage and at 1 and 2 K (Model I, the LBE case). The results show that, similar to the UBE case, the molecules also tend to aggregate and form an hcp structure. Similar to the UBE case, these structures are close to a c(6×2) structure. Although, because of the arrangement of the molecules in the final configurations of Figure 6-21, a full unit cell of these structures cannot be shown on the figure, the formation of an hcp (or maybe c(6×2)) structure in Figure 6-21 can be 0.02.04.02.8 3.1 3.4 3.7 4HDFDistance [Å]1 K2 K3 K208 confirmed by the Para-Para RDFs of these layers obtained from the MC-NVT simulations. These functions are shown in Figure 6-22. As seen, the peaks of the curves in in the figure are at positions expected for an hcp (nearly a c(6×2)) structure. Therefore, based on the simulation results in the LBE case, it is concluded that the c(2×2) (or even islands of c(2×2)) structure cannot be obtained for layers of Para molecules over MgO and at half coverage. This is not in agreement with the experiment, in which islands of the c(2×2) structures along with disordered patches or hcp structures were suggested68. Once again, the patches of open surface show the dominance of the H2-H2 interactions compared to the H2-surface interaction energies. (a) (b) Figure 6-21 The final configurations of a layer of Para molecules over MgO at half coverage and at a. 1 K, and b. 2 K. In these simulations, Model I along with the LBE case were used in the calculation of the energies. The hexagon shows an hcp structure. hcp 2.96 Å 209 Figure 6-22 The Para-Para RDFs of a layer of Para molecules at half coverage when Model I along with the LBE case are used in the calculation of the interaction between the molecules in the system. To explain why the c(2×2) structure is not seen in these layers, the three important energies presented in Table 6-12 are used. Compared to the UBE case, the binding energy is only slightly stronger (3.8%). Same is true for the MiEB of the molecule in this case (LBE), which is only about 10% stronger, but it is still lower than the Para-Para binding energy. Therefore, even in the LBE case, the Para molecules tend to aggregate over the surface to form a structure (hcp) with a lower energy and hence, the c(2×2) structure cannot be seen in the simulated layers. 0714213 4 5 6 7 8Para-Para RDFDistance [Å]1 K2 K210 Table 6-12 The binding energy of a Para molecule over Mg2+, the binding energy between two Para molecules, as well as the MiEB of a Para molecule over the MgO surface when Model I along with the LBE case are used in the calculations. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Para-Para binding energy [kcal/mol] MiEB [kcal/mol] Para -0.42482 -0.06883 0.06012 As discussed in the last paragraph of section 6.2.1.1.2, although there is uncertainty in the experimental data, the formation of hcp structures plus patches of open surface (instead of islands of c(2×2)) for Para molecules over MgO and at half coverage, may imply that the binding energies of these molecules to the surface are not strong enough. Two different energies were tested for the binding energies of the H2 molecules to the surface, but as was seen in the simulation results in the LBE case, it could not solve the problem. Although, there were shifts in the binding energy of the Para molecule to the surface as well as its MiEB, but these shifts (especially in MiEB) do not seem to be large enough to cause the c(2×2) structures to form. This could be because of the form of the interaction potentials that are used in Model I, which is mainly because of the assumptions that are made in the development of these potentials. This may suggest that Model I is not able to accurately describe the H2-surface (or even H2-H2) interactions and hence, the assumptions behind this model must be revisited. Therefore, Model II was developed which is an improved version of Model I and is fully explained in Chapter 3. In Chapter 7, the MC-NVT simulation results of Model II are discussed and compared with the experimental data as well as the results of Model I for both of the NaCl and MgO substrates. 211 6.3 Summary 6.3.1 The NaCl results Using Model I, the structures and stabilities of monolayers of n-H2 and para-H2 molecules over the NaCl surface were examined. The structures of the simulated monolayers of both the n-H2 and para-H2 molecules show that these layers form a p(1×1) structure in great agreement with the experiment. Interestingly enough, in the n-H2 monolayers, only the Heli molecules are responsible for forming the p(1×1) structure over the surface. Also, the energies of monolayers of n-H2 and para-H2 were examined over NaCl and the results are in accord with the experimental data. Table 6-13 summarizes the results of Model I. Table 6-13 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over NaCl. In the MC-NVT simulations, Model I was used in the calculation of the energies. The results for the energy are only shown at a specific temperature. The experimental references have been listed right below the table. Energy [kcal/mol/molecule] (at 8 K) Structure (up to 11 K) H2 species Coverage Simulation Experiment1 Simulation Experiment1,2 n-H2 full layer -0.959 ± 0.002 -0.92 ± 0.11 p(1×1) p(1×1) para-H2 full layer -0.715 ± 0.002 -0.85 ± 0.12 p(1×1) p(1×1) 1 Refs. 22,37 2 Refs. 22,23,26,29,37 212 6.3.2 The MgO results 6.3.2.1 The UBE case The structure and stability of the layers of n-H2 molecules over the MgO surface were also investigated when the UBE case was used in the calculation of the H2-surface binding energies. At half coverage, the MC-NVT results showed that the experimentally observed c(2×2) structure can be obtained in the simulations if only the Heli molecules are used in the layer. But, to make sure that the MC-NVT simulation results are reliable even at the low temperatures used in the MC-NVT simulations, a defect analysis was done on these layers. This analysis showed that the c(2×2) structure is indeed the most stable structure at the temperatures used in these simulations. But because reliable data could not be obtained at higher temperatures using MC-NVT simulations, a crude statistical mechanical model wax used to predict the approximate thermal stability of the layer. Using this model, the estimated temperature up to which the c(2×2) structure may be seen in the simulations is 18 K, consistent with experiments. In addition to the n-H2 molecules, the structures of the para-H2 molecules were also examined over the MgO surface at half coverage and in the UBE case. The results showed that the Para molecules aggregate and cannot form islands of c(2×2) structures over the surface, which is in contrast with the experimental findings. This suggested that the para H2-surface interaction energies might had been underestimated in the UBE case and hence, the LBE case was suggested. The results of the UBE case of Model I are summarized in Table 6-14. 213 Table 6-14 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over MgO. In the MC-NVT simulations, Model I along with the UBE case were used in the calculation of the energies. The results for the energy are only shown at a specific temperature. “NA” means that the data are not available, neither from simulations nor from experiments. Energy [kcal/mol/molecule] (at 5 K) Structure (up to 6 K) H2 species Coverage Simulation Experiment Simulation Experimen1 n-H2 0.5 -0.668 ± 0.002 NA c(2×2) c(2×2) para-H2 0.5 NA NA aggregated (islands of hcp) structure plus patches of open surface islands of c(2×2) alongside patches of disordered structures or hcp structure 1 Ref. 68 6.3.2.2 The LBE case In the next step, by using Model I and the LBE case, a more thermally stable c(2×2) structure was obtained in the MC-NVT simulations of a layer of n-H2 (only Heli) molecules at half coverage. Based on our crude statistical mechanical model mentioned above, the predicted temperature for observing either the c(2×2) structure or islands of this structure in the simulations is 27 K, consistent with experiments. But again, the MC-NVT simulations of the para-H2 molecules showed that these molecules cannot form the c(2×2) structure even when a stronger binding energy (LBE) is used in the simulations and aggregated layers along with 214 patches of open surface are formed instead in contrast with experiments. This suggested that Model I may not be good enough in the calculation of the energies and hence, Model II was proposed. A summary of the LBE results of Model I is presented in Table 6-15. Also, the results of Model II will be presented in the next chapter. Table 6-15 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over MgO. In the MC-NVT simulations, Model I along with the LBE case were used in the calculation of the energies. The results for the energy are only shown at a specific temperature. “NA” means that the data are not available, neither from simulations nor from experiments. Energy [kcal/mol/molecule] (at 8 K) Structure (up to 9 K) H2 species Coverage Simulation Experiment Simulation Experiment1 n-H2 0.5 -0.721 ± 0.003 NA c(2×2) c(2×2) para-H2 0.5 NA NA aggregated (islands of hcp) structure plus patches of open surface islands of c(2×2) alongside patches of disordered structures or hcp structure 1 Ref. 68 215 Simulation results for Model II In Chapter 6, the structures and stabilities of monolayers of para- and ortho-H2 molecules on NaCl as well as the sub-monolayers of these H2 species over MgO were presented. It was shown that the H2 molecules form layers on NaCl that are in accord with the experiment in terms of both structure and energy. For MgO, however, the simulation results for the para-H2 layers at half coverage showed that these molecules form structures that are not in agreement with the experiment. These results suggested that the energies in Model I may had been underestimated and hence, Model II was proposed. This model, as presented in section 4.3, causes shifts in the energy of the single H2 molecule over the surface. It also causes slight shifts in the energies between different pairs of H2 molecules, as discussed in section 5.2.2. To see how these shifts in energies affect the structures and stabilities of the H2 layers over NaCl and MgO, in this chapter, the results of these layers are presented for Model II using the 𝑉෠௦௘௣ case (see section 3.1.2.1) and compared to the experiments. In addition, a comparison between these data and the results of Chapter 6 (Model I) is done. Similar to Chapter 6, first the results of monolayers of H2 species over the NaCl substrate are presented. 7.1 H2 films over NaCl Similar to Chapter 6, the first step is to specify the ULH quantity for each H2 species over this substrate. By definition, this quantity is the height of the molecule over its most attractive site at which the HD-MiEB quantity becomes zero (see Eq. (6-1)). Table 7-1 lists the actual as well as specified ULH data for both Para and Heli molecules over NaCl when Model II is used in the calculation of the energies. Comparison of the data in Table 7-1 and Table 6-1 shows that, in 216 Model II, the energies over Na+ and the saddle site become equal at lower values compared to the corresponding energies in Model I. This is because the binding energies of both Para and Heli molecules over the saddle sites are weaker in Model II compared to Model I (Compare the data in Table 4-19 and Table 4-25). Because of that, more caution should be taken to make sure that the MC-NVT simulation still gives reliable results. Therefore, it was decided to specify the ULH quantities closer to the “actual” values in Model II (Table 7-1). In Model I (Table 6-1), the “specified” and the “actual” ULHs were slightly more distant. Nonetheless, the specified ULHs are the same in models I and II. Table 7-1 The “actual” and “specified” ULH quantities for Heli or Para H2 molecule over NaCl when Model I is used in the calculation of the energy. The ULH term has been defined in the text (see Eq. (6-1) and the explanation underneath this equation). System E(H₂/Na⁺) ≈ Emin (H₂/Saddle) [kcal/mol] Actual ULH [Å] Specified ULH [Å] Heli/Na+ -0.34581 3.99 4.00 Para/Na+ -0.35047 3.87 3.90 7.1.1 Monolayers of n-H2 molecules As explained in Chapter 6, the simulation results are only discussed up to ULT. To find this value, HDFs of the H2 molecules in the layer are used. Figure 7-1 shows these HDFs for a layer of Heli molecules over NaCl at temperatures up to 20 K (see section 6.1.1 for why a layer of only Heli molecules is used). According to the data, at 20 K the probability of finding the molecules at heights equal to the ULH quantity of 4.0 Å becomes nonzero (or HD-MiEB = 0). 217 This is in fact the temperature at which the results start to become unreliable in the MC-NVT simulations. In fact, to study the properties of the layers at 20 K and higher, MC-μVT simulations must be used. Figure 7-1 The HDFs of a monolayer of Heli molecules over NaCl at temperatures up to 20 K when Model II is used in the simulations. At 20 K, the function becomes nonzero at the ULH quantity of 4.0 Å. Figure 7-2 illustrates the final configurations of simulated monolayers of Heli molecules over NaCl when Model II is used in the calculation of the energies. The results are shown only up to 19 K, which is the maximum temperature at which the MC-NVT results are discussed. These configurations show that, in the course of the simulations, the p(1×1) structure is stable over the temperature range indicated in the figure. 05102.4 3.0 3.6 4.2HDFDistance [Å]3 K7 K14 K20 K218 (a) (b) (c) (d) Figure 7-2 Snapshots of the final configurations of a simulated monolayer of Heli molecules over NaCl using Model II and at, a. 3 K, b. 7 K, c. 14 K, and d. 19 K. The square in subfigure (a) shows the p(1×1) structure. The arrow in subfigure (d) indicates the start of the movement of Heli molecules away from atop cations. To confirm the formation of the p(1×1) structures at temperatures shown in Figure 7-2, in Figure 7-3, the RDFs of these structures are illustrated. It is seen that, at the temperatures shown in the figure, the peaks appear at the same positions as expected for a p(1×1) structure. Although, small movements of the Heli molecules around cation sites are observed at 19 K (shown by an arrow in subfigure (d) of Figure 7-2), the p(1×1) structure is stable up to this temperature, which is confirmed by the positions of the peaks in the Heli-Heli RDFs at this temperature. These results are in agreement with the previous experimental and theoretical studies22,23,26,29,37. slightly moved molecule 3.96 Å p(1×1) 219 Figure 7-3 Heli-Heli RDFs of the simulated monolayers of Heli molecules over NaCl at temperatures up to 19 K when Model II is used in the simulations. The data for the monolayers of Heli molecules over NaCl show that, Model II predicts almost the same results as Model I. In fact, the binding energy of the Heli molecule to the cation in Model II is 5.4% weaker than the binding energy in Model I (see Table 4-19 and Table 4-25). On the other hand, the MiEB quantity in Model I is only 6% larger. As explained in Chapter 6, the MiEB quantity as well as the Heli-surface and the Heli-Heli binding energies are the most important factors in forming the p(1×1) structures over the NaCl substrate. Table 7-2 presents all these energies in Model II, which can be compared with the corresponding energies in Model I illustrated in Table 6-2. It should be noted that, to simplify the comparisons, the binding energy between two Heli molecules in Table 7-2 is assumed to be the same as that in Model I (compare 0369122 4 6 8 10Heli-Heli RDFDistance [Å]3 K7 K14 K19 K220 it with the energy in Table 6-2). With these energies being the same (or quite similar) for models I and II, the same p(1×1) structures occur. Nonetheless, the lower optimum height for the Heli/Na+ system in Model II (compare Table 4-19 and Table 4-25) results in peaks at lower positions in the HDFs of Model II (compare Figure 7-1 and Figure 6-1). Also, the slightly more repulsive Heli-Heli interactions in Model II result in sharper and higher peaks in the RDFs of the H2 layers (Figure 7-3) compared to Model I (Figure 6-3). Table 7-2 The binding energy of a Heli H2 molecule over the Na+ site, the binding energy between two Heli H2 molecules, as well as the MiEB quantity for this molecule over the NaCl surface using Model II. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Heli-Heli binding energy [kcal/mol] MiEB [kcal/mol] Heli -0.78738 -0.06987 0.44220 As well as the structures, the energies of the Heli layers over NaCl have also been examined. Table 7-3 lists these energies, which are different from the ones obtained in Model I, but because of the large uncertainties in the experimental data, they are still in good agreement with the experiment22,23,26,29,37. It is also seen that, the energies of the MC-NVT simulated monolayers using Model II are consistently higher than the corresponding energies using Model I (Table 6-3). This is mainly because of the slightly weaker binding energies of the Heli molecules over Na+ ions in Model II (see the Heli/Na+ energies in Table 4-19 and Table 4-25). 221 Table 7-3 Energies of a monolayer of Heli molecules over NaCl. The quantities in brackets are either the uncertainties in the experimental data (previous studies) or the standard deviations resulted from the statistical averaging in the MC-NVT simulations (present work using Model II). Study Type of the study Temperature [K] Energy [kcal/mol/molecule] This work Theory (Model II) 5 -0.883 (± 0.001) 8 -0.873 (± 0.002) 10 -0.866 (± 0.003) 14 -0.852 (± 0.004) Briquez et al. 26 Theory - -0.980 Grunwald and Ewing 29 Experiment 5.2 -0.80 (± 0.24) Traeger et al. 22,37 Experiment 8 -0.92 (± 0.11) Heidburg et al. 23 Experiment ≥ 10 K -0.88 (± 0.23) So far, the simulated energies and structures of a layer of Heli molecules over NaCl are in great agreement with the experiment. However, the question remains as to what effect the convergence of the Heli-NaCl energies obtained from Model II has on the energies of the simulated Heli layers. In Chapter 4, it was explained that the energy of a single Heli molecule over NaCl is very well-converged when the molecule sits far enough from the surface (at heights 0.3 Å below the optimum and higher). Therefore, to see the effect of the Heli-NaCl convergence on the energy of the layer, the average height of the layer is calculated as follows: 〈ℎ〉 =∑ 𝑃௜ℎ௜௜∑ 𝑃௜௜ (7-1) where 𝑃௜ is the probability of finding molecule i at height ℎ௜ in the layer. The angled brackets in this equation show the weighted average quantity. Based on this equation and the HDF data in 222 Figure 7-1, the average heights of the H2 monolayers at temperatures of 3, 7, 14, and 20 K are 2.72, 2.74, 2.78, and 2.81 Å, respectively. These heights are greater than the optimum height of the Heli molecule in the Heli/Na+ system (2.70 Å in Model II) and hence, the effect of the Heli-NaCl convergence on the energy of these layers should be insignificant. 7.1.2 Monolayers of Para molecules In order to study these layers, the first step is to find the ULT quantity using the HDFs of the layer. Figure 7-4 shows these HDFs for a layer of Para molecules at full coverage. Based on the data in this figure, 13 K is the temperature at which some of the molecules get to heights comparable to the ULH quantity of 3.90 Å, and hence, 12 K is the maximum temperature up to which the MC-NVT simulation results are reliable. 223 Figure 7-4 The HDFs of a monolayer of Heli molecules over NaCl at temperatures up to 13 K using Model II. At 13 K, the HDF becomes nonzero at the ULH quantity of 3.90 Å. Figure 7-5 shows the final configurations of a simulated monolayer of Para molecules over NaCl when Model II is used in the calculation of the energies. As seen by the square in subfigure (a), the p(1×1) structure is stable up to 12 K. This is confirmed by the Para-Para RDFs of these layers as shown in Figure 7-6. At all temperatures, the positions of the peaks are the same and occur at distances expected for a p(1×1) layer. These results are in great agreement with the previous experimental and theoretical studies22,23,26,29,37. 03692.5 3.0 3.5 4.0HDFDistance [Å]3 K7 K13 K224 (a) (b) (c) Figure 7-5 Snapshots of the final configurations of a monolayer of Para molecules over NaCl at a. 3 K, b. 7 K, and c. 12 K when Model II is used in the simulations. As shown by the square in subfigure (a), the layer forms a p(1×1) structure at all temperatures shown here. Table 7-4 lists the three important energies in the layer that help forming the p(1×1) structure. Since the Para-Para binding energy in Model II is only slightly different from the corresponding energy in Model I (see Figure 5-5 and Figure 5-6), it is assumed that they are the same (like the Heli case in Table 7-2). As seen in Table 7-4, the strong binding energy of the Para molecule to the cation (9.7 times the Para-Para binding energy) as well as the large MiEB of this molecule (4.6 times the Para-Para binding energy) result in the localization of the Para molecules over the cation sites and consequently, the formation of the p(1×1) structure. 3.96 Å p(1×1) 225 Figure 7-6 The Para-Para RDFs of a simulated monolayer of Para molecules over NaCl at 3, 7 and 10 K, respectively, when Model II is used in the simulations. Table 7-4 The binding energy of a Para molecule over the Na+ site, the Para-Para binding energy, as well as the MiEB quantity for this molecule over the NaCl surface using Model II. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Para-Para binding energy [kcal/mol] MiEB [kcal/mol] Para -0.66809 -0.06883 0.31739 Like the Heli monolayers over NaCl, for Para, Model II predicts the same structure for the layer as Model I. But for Para, Model II results in a higher ULT. Although, the binding energy is only 5.5% stronger in Model II, a higher ULT in this model is mainly because of its 38.1% larger MiEB compared to Model I (compare Table 4-20 and Table 4-26). These two factors also result 03692 4 6 8 10Para-Para RDFDistance [Å]3 K7 K12 K226 in considerably sharper and higher peaks in the Para-Para RDFs of Model II (compare Figure 7-5 and Figure 6-7). Also, since the optimum height of the Para/Na+ system is lower in Model II (compare Table 4-19 and Table 4-25), the peaks in the HDFs of Model II occur at lower heights. So far, it has been shown that the structures obtained from the MC-NVT simulations using Model II are in good agreement with the experiments. It was also shown that, despite differences between the energies in models I and II, the structures obtained in these models are the same. To see how the energies of the para monolayers over NaCl (using Model II) compare with the experiment, in Table 7-5, these energies are listed. The results show good agreement with the experimental data. These energies can also be compared with the energies of the same layers in Model I (Table 6-5). As seen, at all temperatures, the energies of the simulated layers in Model II are consistently lower than the corresponding energies in Model I (Table 6-5), which is because of the stronger binding energy of Para/Na+ in Model II (see this energy in Table 4-19 and Table 4-25). 227 Table 7-5 Energies of a monolayer of Para molecules over NaCl when Model II is used in the simulations. The quantities in brackets are either the uncertainties in the experimental data (previous studies) or the standard deviations resulted from the statistical averaging in the MC-NVT simulations (present work using Model II). Study Type of the study Temperature [K] Energy [kcal/mol/molecule] This work Theory (Model II) 3 -0.7661 (± 0.0009) 5 -0.760 (± 0.001) 8 -0.751 (± 0.002) 10 -0.744 (± 0.003) Briquez et al. 26 Theory - -0.795 Grunwald and Ewing 29 Experiment 5.2 -0.80 (± 0.24) Traeger et al. 22,37 Experiment 8 -0.85 (± 0.12) Heidburg et al. 23 Experiment ≥ 10 K -0.71 (± 0.23) Although, the energies and structures of the simulated layers of Para molecules over NaCl are in good agreement with the experiments when Model II is used in the simulations, the impact of the Para-NaCl interaction energy convergence on the energies of these layers has yet to be discussed. Using Eq. (7-1) along with the HDFs of the monolayers in Figure 7-4, the average heights of the Para molecules in the layers can be calculated. These heights are 2.80, 2.82, and 2.86 Å at temperatures of 3, 5, and 13 K, respectively. Since the optimum height of the Para molecule over the Na+ site is 2.77 Å (see Table 4-25), therefore at all temperatures the average height of the layer is greater than this value. Thus, based on the energy convergence analysis 228 discussed in Chapter 4, the effect of the single H2-surface energy convergence on the energy of the layer is negligible. It is also worth discussing the effect of using a different set of adjustable parameters (C6 and C12 ) for the H2-NaCl interactions as discussed in section 4.3.2.2. It was explained there that, in Model II, using this new set will result in stronger binding energies of the Heli and Para molecules over Na+ compared to what was presented in Table 4-25. It was also stated that using this new set of parameters results in weaker binding energies for the Heli and Para molecules over the saddle and anion sites compared to the data shown in Table 4-25. According to Eq. (6-1), a stronger binding energy over cation (most attractive site) and a weaker binding energy over the saddle site (second most attractive site) results in larger MiEBs for both of the Para and Heli molecules. Thus, a stronger binding energy over cation as well as a larger MiEB will result in an increase in the thermal stabilities of the Heli and Para layers over NaCl compared to the results presented in this chapter. They will also result in lower energies for these layers (~0.05 kcal/mol/molecule) compared to those presented in Table 7-3 and Table 7-5. Since the uncertainties in the experimental data are larger than this possible shift in the energy, its effect is considered to be insignificant. Once again it should be stressed that the Cart molecules were not used in the simulation of a monolayer of ortho-H2 molecules over NaCl, which was because of the considerably weaker binding energy of this molecule over its most attractive site (Cl-) compared to the Heli and Para molecules over Na+. 7.2 H2 films over MgO In this section, the MC-NVT simulation results for the sub-monolayers and monolayers of ortho- and para-H2 molecules over the MgO substrate in both the UBE and LBE cases are 229 discussed. Similar to Model I, the first step is to find the ULH quantity for different H2 species over this substrate. Table 7-6 lists these data. Since in Model II (second column in Table 7-6), the binding energies of the Para or Heli molecules over the saddle sites are weaker compared to Model I (second column in Table 6-6), in Model II the “specified” ULHs are chosen to be closer to the “actual” ULHs. In fact, in Model I, because of the stronger binding energies over the saddle sites, it was safe to choose “specified” heights much higher than the “actual” heights. But here, a lower height have to be chosen to make sure that the MC-NVT simulation can be trusted. Based on these, unlike the NaCl substrate, in the MgO substrate, the ULHs are different in models I and II. Nonetheless, the ULHs are the same in the UBE and LBE cases of MgO. It should be noted that, the ULHs are only used to find the ULT quantity. Beyond this temperature, the MC-NVT simulation results are unreliable and MC-μVT simulations should be used. In the following sections, the simulation results for the layers of H2 molecules over MgO are presented. Based on the experimental findings, the results are discussed based on the coverage and the type of these H2 species. Table 7-6 The ULH quantities for Para and Heli molecules over the MgO substrate for both the UBE and LBE cases when Model II is used in the calculation of the energies. Minimum binding energy [kcal/mol] System E(H₂/Mg²⁺) = Emin (H₂/Saddle) [kcal/mol] Actual ULH [Å] Specified ULH [Å] - 0.60039 (UBE) Heli/Mg2+ - 0.37584 3.80 3.80 Para/Mg2+ - 0.38260 3.69 3.70 - 0.66427 (LBE) Heli/Mg2+ - 0.39321 3.76 3.80 Para/Mg2+ - 0.40041 3.65 3.70 230 7.2.1 Half coverage (coverage of 0.50) Based on the experiments, at this coverage, a c(2×2) structure is expected for a layer of n-H2 molecules over the surface. For the para-H2 molecules, however, the structures are not well-known, and it was interpreted to be islands of c(2×2) along with a mixture of disordered patches or hcp structures. In Chapter 6, these structures were examined in the MC-NVT simulations using Model I. Although the c(2×2) structure was seen for the n-H2 molecules, for the para-H2 molecules there was no sign of these structures over the surface. This led us to the development of Model II, a more complete and sophisticated version of Model I. Based on its complexity, this new model should give a better description of the system. In this section, these layers are examined at different temperatures using MC-NVT simulations along with Model II. 7.2.1.1 The UBE case 7.2.1.1.1 Films of n-H2 molecules As explained in section 6.2.1.1.1, using Model I, a layer of n-H2 molecules at half coverage forms a c(2×2) structure. In this structure, the Heli molecules sit over every other cation sites in the unit cell. Here it is investigated if these layers are still formed using Model II. Similar to Model I, the first step is to find the ULT quantities. As explained in detail previously, these quantities are obtained based on the HDFs of the layers at different temperatures. Figure 7-7 illustrates the HDFs for a layer of Heli molecules over MgO at half coverage when Model II along with the UBE case are used in the calculation of the energies. The results show that at 8 K, the probability of finding the particles at heights comparable to ULH of 3.80 Å 231 becomes nonzero and hence, 7 K is the maximum temperature up to which the MC-NVT simulation results can be discussed. Figure 7-7 The HDFs for a simulated layer of Heli molecules over MgO at half coverage using Model II and the UBE case. The data at 8 K show a nonzero probability at the ULH of 3.80 Å, so the MC-NVT simulations are consistent with the experiment only up to this temperature. Figure 7-8 shows the final configurations of a layer of Heli molecules over MgO at half coverage and at different temperatures up to ULT. In all these simulations, Model II as well as the UBE case are used. The starting configuration in all these simulations is the c(2×2) structure. The results in this figure show that, the c(2×2) structure is stable up to 6 K (compared to 5 K in Model I in Figure 6-9). At 7 K, four defects appear in the structure (compared to five in Model I), which are shown by arrows in subfigure (d) of Figure 7-8. It should be noted that these are not single defects, but multiple (dependent) defects appearing in the same spot which are likely 012345672.5 3.0 3.5 4.0HDFDistance [Å]3 K5 K8 K232 nucleated by the first defect in that spot since these defects are within the interaction range of one another. (a) (b) (c) (d) Figure 7-8 Final configurations of a layer of Heli molecules at a. 3 K, b. 5 K, c. 6 K, and d. 7 K using Model II and the UBE case. The square in subfigure (a) indicates a c(2×2) unit cell over the surface. The four arrows in subfigure (d) show the four defects formed in the c(2×2) layer. Figure 7-9 demonstrates the Heli-Heli RDFs of the layers shown in Figure 7-8. The stability of the c(2×2) structure at temperatures up to 6 K as well as the formation of defects at 7 K are all supported by the Heli-Heli RDFs in Figure 7-9. For example, the larger peaks in this figure are at the positions expected for a c(2×2) structure. The bumps at 7 K show the appearance of the defects in the structure. These bumps are exactly at the same positions as the bumps in Figure c(2×2) 2.96 Å defects defects 233 6-10. This indicates that changing the model did not affect the mechanism of forming the defects in the structure, namely, Heli molecules jumping over the saddle point to neighboring cations. Figure 7-9 The Heli-Heli RDFs of a simulated layer of Heli molecules over MgO at half coverage using Model II as well as the UBE case. The larger peaks appear at positions expected for a c(2×2) structure. The two bumps in the graphs at 7 K indicate the appearance of some defects in the c(2×2) structure. In order to investigate how these defects can affect the c(2×2) structure at lower temperatures of 1 to 6 K, and to approximately determine the stability of the c(2×2) layer at higher temperatures, the defect analysis of section 6.2.1.1.1 is used. Based on the procedure explained previously, first, four defects are created in the c(2×2) layer (the 4d-c(2×2) layer in Figure 6-11) and then this structure is used as the initial configuration in the MC-NVT simulations. In the next step, the energy of the simulated layer is compared with the energy of a perfect c(2×2) layer, and finally, based on the procedure explained for Figure 6-12, the difference between the energies of these two layers at 0 K is used in the statistical mechanical model of Eq. (6-4). 05103 5 7Heli-HeliRDFDistance [Å]3 K5 K6 K7 K234 Figure 7-10 illustrates the energies of these simulated layers (defect and perfect c(2×2)) at temperatures of 1 to 6 K. These energies are for a layer of 72 H2 molecules as opposed to a per molecule basis. The results show that at temperatures of 1 to 4 K, the layer remained at 4d-c(2×2) structure, whereas at 5 and 6 K, more defects appear in the layer. Therefore, only the data from 1 to 4 K were used to extract the slope equations in this figure. Using the intercepts of these equations, the difference between the energies of a 1d-c(2×2) and a perfect c(2×2) layer can be obtained based on the procedure explained in section 6.2.1.1.1. This difference is about 0.053 ± 0.007 kcal/mol (or 26.7 ± 3.5 K). Now, if this value is substituted into Eq. (6-5), the following equation is obtained: 𝑃ଵௗି௖(ଶ×ଶ) =44 + 1 × 𝑒ାଶ଺.଻/் (7-2) By using Eq. (7-2), the percentage of the layer that turns into a single-defect structure can be obtained at each temperature. For example, this equation predicts that at 5 K, the probability of converting a molecule in the structure to a defect is 0.02. This means that among 72 molecules in the layer, only 1.44 (≈1) of them will be a single defect. This prediction is close to our simulation result that shows no defects in the structure (subfigure (b) of Figure 7-8). At 6 K, the model predicts three single defects in the layer, which is different from the simulation results that show no defect in the structure (subfigure (c) of Figure 7-8). At 7 K, our model predicts six single defects in the layer, whereas in the simulated layer four dependent defects are seen (subfigure (b) of Figure 7-8). But, as explained in Chapter 6, the statistical mechanical model proposed here is a crude estimation of the number of single defects in the structure. Nonetheless, this model predicts that at 11 K, about half (50%) of the layer is covered by single-defect c(2×2) unit cells (i.e., 18 out of 36 c(2×2) unit cells) and the perfect c(2×2) structure may no longer be seen in the 235 MC-NVT simulations. But this temperature might be a lower limit for the actual temperature up to which the c(2×2) structure may be obtainable in the simulations, because the model consistently predicts more defects than what is seen in the simulations. It should also be mentioned that, as seen in Figure 7-10, at 3 and 4 K, the energy of the 4d-c(2×2) layer overlaps with the energy of the c(2×2) structure. This means that at these temperatures and probably up to the limit of 11 K discussed above, defects are likely to appear in the structure, but on average, they have a higher (weaker) energy compared to the c(2×2) structure. Therefore, at these temperatures, islands of defect structures are probable. By increasing the temperature, these islands cover more and more parts of the surface until half of the surface is covered by these single-defect c(2×2) unit cells and the c(2×2) structure is no longer seen in the simulations. Nonetheless, the simulation results in this section and at low temperatures of 0 to 10 K are in accord with the previous experimental study in which a c(2×2) structure was observed for n-H2 at half coverage and at the temperature range of 5.8 to 9 K 10. 236 Figure 7-10 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 6 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at the temperature range of 1 to 4 K. At 5 K and 6 K, more defects appear in the structure and they are no longer 4d-c(2×2) layers. The equations shown on each curve are only based on the data from 1 to 4 K and are used for the statistical model explained in the text. The bars in each case show the standard deviation in the energy. To explain why the c(2×2) structure is seen for a layer of Heli molecules over MgO at half coverage, three energies presented in Table 7-7 are used. Again, the Heli-Heli interaction is considered to be the same as Model I. According to the data in this table, the binding energy of the Heli molecule to the surface is 8.6 times the binding energy between two Heli molecules in the layer. Also, the MiEB of the Heli molecule is 3.2 times the Heli-Heli binding energy. Therefore, these two factors force the Heli molecules to stay localized on top of the cations in the form of a c(2×2) structure (probably with islands of defect structures at temperatures higher than 5 K). y = 0.2224x - 51.1077R² = 0.9986y = 0.1964x - 50.8955R² = 0.9991-51.0-50.5-50.0-49.5-49.00 2 4 6Energy [kcal/mol]Temperature [K]c(2×2)c(2×2) with defects237 Table 7-7 The energy of a Heli molecule over cation, the attractive energy between two of these species, as well as the MiEB of this molecule over MgO using Model II and the UBE case. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Heli-Heli binding energy [kcal/mol] MiEB [kcal/mol] Heli -0.60055 -0.06987 0.22373 On the other hand, comparing the defect analysis results of models I and II shows that the c(2×2) layers are probably seen up to higher temperatures in Model I. For example, the difference between the energies of a 1d-c(2×2) structure and perfect c(2×2) structure at 0 K ((∆𝐸଴௄) in Eq. (6-6)) is about 45.6 ± 1.5 K in Model I, whereas in Model II this quantity is around 26.7 ± 3.5 K. This difference can be explained by comparing the energies in Table 6-7 and Table 7-7. These data show that, with exactly the same binding energies of the Heli molecules over the cation sites as well as the similar Heli-Heli interactions in models I and II, the larger ∆𝐸଴௄ in Model I is mainly due to the 9.1% larger MiEB in this model. Structures aside, the energies of the layers of Heli molecules over MgO at half coverage have also been examined. As mentioned in Chapter 6, there is no experimental data available for the energies of these layers. But, the results of Model II can be compared with those from Model I in Table 6-8. Table 7-8 presents the energies of these simulated layers when Model II is used in the simulations. Comparing the energies in this table with the corresponding energies in Table 6-8 shows that, at all temperatures, the total energies are lower (stronger binding) in Model II. This difference in energy is mainly because of the slightly lower Heli-Heli interaction energies (stronger binding) in Model II. For example, at temperatures of 1, 3, and 5 K, the H2-surf 238 interactions obtained from both models are the same, but the lower H2-H2 interactions in Model II resulted in lower total energies for the layers in Table 7-8. At 6 K, the configuration resulted from Model II is still a perfect c(2×2) structure (Figure 7-8), whereas in Model I it is c(2×2) with three defects in the layer (Figure 6-9). In fact, in the defect structures, some of the Heli molecules jump to the neighboring cations and since these molecules do not sit right over the cation sites, their binding energies with the surface decreases. For example, at 6 K, a lower (stronger) H2-surf interaction energy is seen for the simulated layer obtained from Model II (Table 7-8). The same comparison can be made for the simulated layers at 7 K based on the number of defects that appear in these layers. Table 7-8 The energies of MC-NVT simulated layers of Heli molecules over MgO at half coverage at temperatures up to the ULT quantity when Model I along with the UBE case were used in the calculation of the energies. The errors are the standard deviations obtained from the statistical averaging in the MC simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. Energy [kcal/mol/molecule] Temperature [K] Structure Total H2-surf H2-H2 1 c(2×2) -0.7068 ± 0.0004 -0.5975 ± 0.0003 -0.1093 ± 0.0004 3 c(2×2) -0.701 ± 0.001 -0.5910 ± 0.0009 -0.1097 ± 0.0005 5 c(2×2) -0.694 ± 0.002 -0.584 ± 0.002 -0.1098 ± 0.0007 6 c(2×2) -0.692 ± 0.002 -0.581 ± 0.002 -0.1104 ± 0.0008 7 c(2×2) with defects -0.687 ± 0.002 -0.575 ± 0.002 -0.112 ± 0.001 239 Last but not list, is the impact of the H2-MgO interaction energy convergence on the energies of the simulated layers. In Table 4-21, it was seen that the optimum height of a Heli molecule over the Mg2+ site is about 2.90 Å when Model II along with the UBE case are used in the calculation of the energy. It was also explained that the energies are well-converged when the molecule sits over the surface at heights higher than about 0.3 Å below the optimum height. In order to investigate the impact of the convergence of the H2-surface interaction energy on the energy of the layer, the average heights of the molecules in the layer must be calculated. Using the HDF data in Figure 7-7 and Eq. (7-1), the average heights of 2.93, 2.95, and 3.03 Å are obtained for the Heli molecules in the layer at temperatures of 3, 5, and 8 K, respectively. This means that, at all temperatures presented in Figure 7-7, the average heights are above the optimum height of a Heli molecule over the cation sites (2.90 Å). Therefore, based on the energy convergence analysis explained in Chapter 4, the impact of the single H2-surface energy convergence on the energy of the layer is negligible. 7.2.1.1.2 Films of Para molecules In Chapter 6, the layers of Para molecules over MgO and at half coverage were discussed for the case where Model I along with the UBE case are used in the simulations. It was shown that these layers cannot form islands of c(2×2) structure, and hence, Model II was developed. In this section, the results of this model for these layers are discussed. Similar to the previous cases, the first step is to find the ULT quantity. Figure 7-11 shows the HDF results for these layers. According to this data, the probability of finding some of the Para molecules in the layer at heights comparable to the ULH quantity of 3.70 Å becomes nonzero and hence, 4 K is the maximum temperature up to which the MC-NVT simulations can be used. 240 Figure 7-11 The HDFs for a layer of Para molecules over MgO at half coverage and at temperatures 1 to 4 K when Model II is used in the simulations along with the UBE case. At 4 K, some of the molecules gain access to the ULH quantity, so 4 K is the ULT value for this layer. Figure 7-12 shows snapshots of the final configurations of the simulated layers of Para molecules over MgO and at half coverage when Model II along with the UBE case are used in the calculation of the energies. Once again, the starting configuration in all these simulations is a c(2×2) layer. As seen in the figure, at temperatures up to 3 K, the c(2×2) structures stay stable in the course of the simulations, which is confirmed by the Para-Para RDFs shown in Figure 7-13. The peaks in this figure appear at the same positions seen in Figure 7-9, which are the positions expected for a c(2×2) layer. This is in contrast with the results of Model I (Figure 6-14). 05102.6 2.9 3.2 3.5 3.8HDFDistance [Å]1 K2 K3 K4 K241 (a) (b) (c) Figure 7-12 Final configurations of the simulated layers of Para molecules over MgO at a. 1 K, and b. 2 K, and c. 3 K, using Model II along with the UBE case. The results show a c(2×2) structure shown by a square in subfigure (a). 2.96 Å c(2×2) 242 Figure 7-13 The Para-Para RDFs of a layer of Para molecules at half coverage when Model II along with the UBE case are used in the calculation of the interaction between the molecules in the system. To make sure that the structures seen in the simulations are the most stable structure at temperatures shown in Figure 7-12, the defect analysis explained in section 6.2.1.1.1 is utilized. Figure 7-14 shows the energies of the defect as well as perfect c(2×2) layers at temperatures up to 3.5 K. All these energies are for a layer of 72 H2 molecules and not a per molecule basis. In this figure, the results only up to 3.5 K are shown because the c(2×2) layers are not stable beyond this temperature and whose data cannot be used in the analysis. Also, only the data up to 2 K are used in extracting the equations of the straight lines because at temperatures higher than 2 K, more defects appear in the 4d-c(2×2) layers and hence, their energies will not be of help in our defect analysis. Nevertheless, Figure 7-14 shows that up to 2 K, although the energies of the 4d-c(2×2) layer overlaps with the energy of the perfect c(2×2) layer, on average, the 4d-c(2×2) layer 048123 4 5 6 7Para-Para RDFDistance [Å]1 K2 K3 K243 has a lower energy. This means that, although coexistence of these layers is possible, the formation of defect or aggregated layers is more probable. These results also show that the c(2×2) layers at temperatures up to 3 K (Figure 7-12) probably remain stable during the MC-NVT simulations because of the very low temperatures of these layers. In fact, the molecules in these c(2×2) layers do not have enough energy to overcome the MiEB barriers and form defect or aggregated layers over the surface. Figure 7-14 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 3.5 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at the temperature range of 1 to 2 K. At higher temperatures, more defects appear in the structure and they are no longer 4d-c(2×2) layers. The equations shown on each curve are only based on the data from 1 to 2 K and are used for the statistical model explained in the text. The bars in each case show the standard deviation in the energy. y = 0.1923x - 44.5291R² = 0.9964y = 0.1715x - 44.5598R² = 0.9904-44.5-44.1-43.70 1 2 3 4Energy [kcal/mol]Temperature [K]c(2×2)c(2×2) with four defects244 Comparison of the simulation results of Para molecules in models I and II show that in Model II, the formation of the c(2×2) layer over the surface is possible, whereas in Model I, the layer aggregates even at temperatures below 2 K. This metastability in Model II results may be explained by comparing the three important energies presented in Table 7-9 (Model II) and Table 6-9 (Model I). Once again, despite small differences, the Para-Para interactions are assumed to be the same in models I and II. Comparing the data in these tables show that, the binding energy of the Para molecule over cation is 24.9% stronger in Model II. Also, the MiEB of the Para molecule in Model II is 2.4 times the corresponding quantity in Model I. The stronger binding energy of the Heli molecule to the cation site as well as the larger MiEB of this molecule in Model II are probably the reasons that help the c(2×2) layers to stay stable in the MC-NVT simulations using Model II. Nonetheless, the data in Table 7-9 suggest that, although the binding energy of the Para molecule to the cation site and the MiEB of this molecule are in turn 7.4 and 1.9 times the binding energy between Para molecules, they are not strong enough to help the Para molecules form the c(2×2) layers. But since the energies of the c(2×2) and 4d-c(2×2) structures overlap, the c(2×2) structure might be metastable provided that it is formed upon adsorption of the Para molecules over the surface. Nonetheless, the energies of the 4d-c(2×2) structures are lower than the c(2×2) structures indicating that finding the defect (or even aggregated) structures over the surface is more probable. Using the MC-NVT simulation results in this section, it is cannot be said with certainty what these defect (or aggregated) structures look like. It is thought that these structures are a combination of defect and hcp structures but MC-μVT simulations must be done to answer this question. Therefore, the results in this section for the Para layers over MgO at half coverage are in agreement with the previous experimental studies in which islands of c(2×2) as well as disordered patches or hcp (maybe c(4×2) or c(6×2)) structures were 245 suggested for a layer of Para molecules over the MgO surface at half coverage68. It should be added that, due to the metastability of the layer (energy overlap of the defect and perfect c(2×2) layers), the uncertainty in ∆𝐸଴௄ is larger than its value and hence, the statistical mechanical model proposed in section 6.2.1.1.1 cannot be used to predict the thermal stability of these layers. Table 7-9 The energy of a Para molecule over cation, the attractive energy between two of these species, as well as the MiEB of this molecule over MgO using Model II and the UBE case. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Para-Para binding energy [kcal/mol] MiEB [kcal/mol] Para -0.51060 -0.06883 0.12854 Although metastable, for future reference, the energies of the simulated layers of Figure 7-12 are presented in Table 7-10. The results show that by increasing the temperature, the H2-surf interactions decrease, which is the result of the molecules sitting at higher heights above cations in the c(2×2) structure. The higher heights of these molecules in the layer also result in the interaction energies between these molecules similar to those of non-adsorbed molecules. This means that the binding energies between these molecules slightly increase, as explained in detail in Chapter 5 (subfigure (b) of Figure 5-5). 246 Table 7-10 The energies of MC simulated layers of Heli molecules over MgO at half coverage at temperatures up to the ULT quantity when Model II along with the LBE case are used in the calculation of the energies. The errors are the standard deviations obtained from the statistical averaging in the MC simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. Energy [kcal/mol/molecule] Temperature [K] Structure Total H2-surf H2-H2 1 c(2×2) -0.6158 ± 0.0006 -0.5072 ± 0.0003 -0.1086 ± 0.0006 2 c(2×2) -0.6132 ± 0.0009 -0.5039 ± 0.0007 -0.1093 ± 0.0007 3 c(2×2) -0.611 ± 0.001 -0.5008 ± 0.0009 -0.1102 ± 0.0008 Similar to the Heli molecules, in order to investigate the impact of the convergence of the Para-MgO interaction energy (presented in Chapter 4) on the energies of the layers (presented in Table 7-10), the average heights of the molecules in these layers must be calculated. Using Eq. (7-1) and the data in Figure 7-11, the average heights of the layers are 3.02, 3.03, 3.05, and 3.07 at temperatures of 1, 2, 3, and 4 K, respectively. Since the optimum height of the Para molecule over the cation site in Model II and in the UBE case is 3.0 Å (see Table 4-21), at all temperatures shown in Figure 7-11, the average heights are above the optimum height and based on the analyses given in Chapter 4, the impact of the single H2-surface energy convergence on the energy of the layer is negligible. 247 7.2.1.2 The LBE case 7.2.1.2.1 Films of n-H2 molecules In these films, the c(2×2) structure is expected to be seen by only a layer of Heli H2 molecules sitting over the cation sites in the unit cell (see section 6.2.1.1.1). Based on the ULH quantity in Table 7-6 and based on the HDFs in Figure 7-15, the probability of finding the Heli molecules in the layer at heights comparable to the ULH becomes nonzero at 10 K. Thus, this temperature is the maximum temperature up to which the MC-NVT simulation results on these layers can be discussed. Figure 7-15 The HDFs for a simulated layer of Heli molecules over MgO at half coverage using Model II and the LBE case. The data at 10 K show a nonzero probability at the height of 3.80 Å (ULH), so the MC-NVT simulations are only consistent with the experiment up to this temperature. 012345672.4 2.9 3.4 3.9HDFDistance [Å]3 K5 K8 K10 K248 (a) (b) (c) (d) Figure 7-16 Final configurations of a layer of Heli molecules at a. 3 K, b. 5 K, c. 8 K, and d. 9 K using Model II as well as the LBE case. The square in subfigure (a) indicates a c(2×2) unit cell over the surface. The arrows in subfigure (d) show the defects appearing in the c(2×2) structure. The number of arrows in this subfigure show the number of defects formed in the layer at 9 K. Figure 7-16 shows the final configurations of the Heli molecules at half coverage when Model II along with the LBE case are used in the calculation of the interaction energies. Similar to the UBE case, all these simulations started with a c(2×2) structure. The results show that, in the course of the simulations, the c(2×2) structure remains stable up to 8 K (similar to Model I in the LBE case). At 9 K, however, many defects (some of which are shown by arrows in subfigure (d) of Figure 7-16) appear in the structure, forming patches of aggregated layers while leaving parts of the surface open for the adsorption of more molecules. The formation of the c(2×2) structure c(2×2) 2.96 Å defects defects 249 as well as the defect layers are confirmed in the Heli-Heli RDFs shown in Figure 7-17. Based on the positions of the smaller peaks (bumps) in this figure, the nature of the defects in the layers are the same as those in the UBE case (Figure 7-9). Figure 7-17 The Heli-Heli RDFs of a simulated layer of Heli molecules over MgO at half coverage using Model II as well as the LBE case. The larger peaks appear at positions expected for a c(2×2) structure. The two bumps in the graphs at 9 K indicate the appearance of some defects in the c(2×2) structure. To make sure that at low temperatures of 0 to 8 K, the c(2×2) structure is the most stable structure over the surface, the defect analysis of section 6.2.1.1.1 is used. The results are presented in Figure 7-18. Before discussing the results, it should be mentioned that, for the purposes of our analysis, the energies in this figure are reported for a layer of 72 H2 molecules. In Figure 7-18, the c(2×2) structure remains stable in the simulations only up to 8 K. On the other hand, the 4d-c(2×2) structure stays stable only up to 6 K. This means that the data for up to 048123 5 7Heli-Heli RDFDistance [Å]3 K5 K8 K9 K250 6 K can be used in our defect analysis. However, for consistency and in order for a better comparison with the results of Model I in Figure 7-10, the data only up to 4 K were used in the fitting procedure to find the equations of the straight lines in Figure 7-18. Figure 7-18 The energies of the perfect and defect c(2×2) layers at temperatures of 1 to 8 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at the temperature range of 1 to 6 K. At higher temperatures, more defects, some of which are independent of each other, appear in the structure and they are no longer 4d-c(2×2) layers. For consistency purposes, the equations of the straight lines are extracted based on the data from 1 to 4 K. The bars in each case show the standard deviation in the energy. Following the procedure explained in section 6.2.1.1.1, ∆𝐸଴௄ can be obtained. By using the intercepts of the slope equations in Figure 7-18, this amount is about 0.054±0.012 kcal/mol (or 27.4 ± 5.8 K). This difference is very similar to the corresponding amount in the UBE case (in y = 0.2211x - 55.6622R² = 0.9991y = 0.2270x - 55.4443R² = 0.9944-55.5-55.0-54.5-54.0-53.5-53.00 3 6 9Energy [kcal/mol]Temperature [K]c(2×2)c(2×2) with defects251 Model I they were different). It is, however, significantly smaller than the one obtained in the LBE case of Model I. Substituting the quantity obtained for ∆𝐸଴௄ into Eq. (6-5) gives: 𝑃ଵௗି௖(ଶ×ଶ) =44 + 1 × 𝑒ାଶ଻.ସ/௞ಳ் (7-3) By using Eq. (7-3), at 5 K, the probability of finding a single defect in the layer is 0.02, which is only slightly higher than the corresponding temperature in the UBE case. Based on this value for the probability, at 5 K, only one the of the molecules form a single defect, which is close to our simulation result that shows no defects in the layer (subfigure (b) of Figure 7-16). At 9 K, the model predicts eleven single (independent) defects in the layer. This prediction is different from what obtained in the simulations (subfigure (d) of Figure 7-16). It should be mentioned that, in the simulated layer, the defects are not single defects and they depend on each other (multiple defects). As explained earlier, this model is crude in estimating the number of single defects in the layer. Nonetheless, this model predicts that at 11 K, about 50% of the layer turns to single-defect c(2×2) unit cells (i.e., 18 out of 36 c(2×2) unit cells) and the perfect c(2×2) structure may not be seen in the simulations. Similar to the UBE case, this temperature is probably a lower limit for the actual temperature up to which the c(2×2) structure may be observable in the MC-NVT simulations. In addition, the data in Figure 7-18 show that at temperatures up to 6 K, on average, the energy of the c(2×2) structure is lower than the 4d-c(2×2) structure. Besides, although there is a small overlap of the energies at 5 K, the data up to 6 K show that the energy of the c(2×2) structure is lower than the 4d-c(2×2) structure. Therefore, based on these data, it can be concluded that in Model II and in the LBE case, the starting temperature for observing the coexistence of the perfect c(2×2) and the 4d-c(2×2) structures starts at temperatures higher than 252 6 K. This temperature was 3 K in the UBE case (Figure 7-10). These results are in agreement with the previous experimental study which reported c(2×2) structures for the layers of n-H2 at half coverage and at the temperature range of 5.8 to 9 K 10. The defect analyses in the LBE and the UBE cases of Model II have shown that the temperature at which the perfect c(2×2) structure disappears is the same for these cases (11 K). This may imply that, the stability of the c(2×2) layers are the same in the UBE and LBE case. But, based on the MC-NVT simulation results, the c(2×2) layer seems to be stable up to higher temperatures in the LBE case. As explained earlier, the statistical mechanical model used to predict the stability of the layer in the MC-NVT simulations is just a crude estimation and cannot accurately predict the stability. In order to explain why the c(2×2) layer is stable up to higher temperatures in the LBE case, it is suggested to get help from the comparison between the three energies presented in Table 7-7 for the UBE case with the corresponding energies in Table 7-11 for the LBE case. Once again, despite slight differences in the Heli-Heli interaction energies in the LBE and the UBE cases, for simplicity, this energy is assumed to be the same in these cases. Based on the data in Table 7-7 and Table 7-11, the binding energy of the Heli molecule to the cation site is about 11% stronger in the LBE case. Also, the MiEB quantity is 20.7% larger in the LBE case. Therefore, based on these two energies and the fact that the Heli-Heli energies are almost the same in the UBE and LBE cases, the c(2×2) structure is expected to be stable up to higher temperatures in the LBE case. For example, as seen in Figure 7-8 and Figure 7-16, the perfect c(2×2) layers are stable up to higher temperatures in the LBE case and the defects start to form at higher temperatures in this case. 253 Table 7-11 The energy of a Heli molecule when sits over cation, the attractive energy between two of these species, as well as the MiEB of this molecule over MgO using Model II and the LBE case. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol] Heli-Heli binding energy [kcal/mol] MiEB [kcal/mol] Heli -0.66448 -0.06987 0.26995 On the other hand, compared to the LBE case in Model I (Figure 6-17), the c(2×2) layers in Model II are stable up to lower temperatures (Figure 7-16). For example, based on our statistical mechanical model, ∆𝐸଴௄ (Eq. (6-6)) is about 67.4 ± 1.5 K in Model I, whereas in Model II it is about 27.4 ± 5.8 K. Similar to the UBE case, since the binding energies are the same, this large difference in the stability of the c(2×2) layer between models I and II is mainly because of the 11% larger MiEB in Model I. In addition to the structures, the energies of the layers of Heli molecules over MgO at half coverage were also investigated. In the lack of experimental data, the results here are compared with the results of Model I in Table 6-11. Table 7-12 lists the energies of the simulated layers using Model II in the LBE case. Comparing the energies show that at temperatures up to 8 K, the H2-surf interactions are the same. The H2-H2 interactions, however, are lower in Model II, which is because the slightly more attractive Heli-Heli interactions in this model (when both Heli molecules are sitting over the cations, see section 5.2.2.1). In fact, this lower H2-H2 interactions gives rise to lower total energies over this temperature range in Model II. 254 Table 7-12 The energies of MC simulated layers of Heli molecules over MgO at half coverage at temperatures up to the ULT quantity when Model II along with the LBE case were used in the calculation of the energies. The errors are the standard deviations obtained from the statistical averaging in the MC simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. Energy [kcal/mol/molecule] Temperature [K] Structure Total H2-surf H2-H2 1 c(2×2) -0.7670 ± 0.0004 -0.6615 ± 0.0003 -0.1085 ± 0.0003 3 c(2×2) -0.764 ± 0.001 -0.6551 ± 0.0009 -0.1089 ± 0.0004 5 c(2×2) -0.758 ± 0.002 -0.648 ± 0.002 -0.1094 ± 0.0006 8 c(2×2) -0.748 ± 0.003 -0.638 ± 0.003 -0.1094 ± 0.0007 9 c(2×2) with defects -0.742 ± 0.003 -0.629 ± 0.004 -0.114 ± 0.005 Finally, the impact of the convergence of the energy of a single H2 molecule with the surface on the energy of the layers was also investigated. Using Eq. (7-1) and the HDF curves in Figure 7-15, the average heights of the molecules in the layer are 2.82, 2.84, 2.86, and 2.93 Å at temperatures of 3, 5, 8, and 10 K, respectively. This means that, at all these temperatures, the average heights of the molecules are above the optimum height (2.79 Å) of a Heli molecule over the Mg2+ site (see Table 4-23). Therefore, based on the energy convergence analysis presented in Chapter 4, the impact of the convergence of the energy of a single H2 molecule with the surface on the energy of the layers is negligible. 255 7.2.1.2.2 Films of Para molecules Based on the HDFs shown in Figure 7-19, the ULT value up to which the MC-NVT simulations can be done on a layer of Para molecules on MgO at half coverage and in the LBE case is 6 K. Figure 7-19 The HDFs for a layer of Para molecules over MgO at half coverage and at temperatures 1 to 6 K when Model II along with the LBE case is used in the simulations. At 6 K, some of the molecules gain access to the ULH quantity, so 6 K is the ULT quantity for this layer. Figure 7-20 illustrates the final configurations of the simulated layers of Para molecules over MgO and at half coverage when Model II along with the LBE case are used for the interaction between the molecules in the system. Similar to the previous cases, in all these simulations, the c(2×2) structure was used as the starting configuration. The results show that, at temperatures up 048122.4 2.9 3.4 3.9HDFDistance [Å]1 K3 K5 K6 K256 to 5 K, the c(2×2) structure stays stable in the course of the simulations, as confirmed by the Para-Para RDF curves in Figure 7-21. (a) (b) (c) (d) Figure 7-20 Final configurations of simulated layers of Para molecules over MgO at a. 1 K, and b. 3 K, c. 4 K, and d. 5 K using Model II along with the LBE case. The c(2×2) structure is shown by a square in subfigure (a). 2.96 Å c(2×2) 257 Figure 7-21 The Para-Para RDFs of a layer of Para molecules at half coverage when Model II along with the LBE case are used in the calculation of the interactions between the molecules in the system. In order to make sure that the c(2×2) structure is the most stable structure at temperatures up to 5 K (Figure 7-20), the defect analysis of section 6.2.1.1.1 must be done. Figure 7-22 illustrates the energies of 4d-c(2×2) as well as perfect c(2×2) layers at temperatures 1.5 up to 5 K. All these energies are for a layer of 72 H2 molecules. Also, the results at 1 K is not used here due to the strong correlation between the configurations in the production run of the MC-NVT simulation. The same thing is true for the data at 3.5 and 4 K. This forced us to only use the data from 1.5 up to 3 K for extracting the slop equations. The results in this figure show that at temperatures up to 3.5 K, the energy of the perfect c(2×2) layers are, on average, lower than the 4d-c(2×2) layers. This shift in energies is in contrast with what was seen for these layers in the UBE case (Figure 7-14). But, similar to the UBE case (Figure 7-14), the energies of the perfect c(2×2) and the 4d-c(2×2) layers overlap. This means that the coexistence of these layers is possible even though, on 07143 5 7Para-Para RDFDistance [Å]1 K3 K4 K5 K258 average, observing a perfect c(2×2) structure is more likely in the simulations and at temperatures up to 3.5 K. Also, similar to the UBE case, the structure of the defect or aggregated structures cannot be determined using the MC-NVT simulations and instead, MC-μVT simulation must be used. Moreover, since the perfect c(2×2) and the 4d-c(2×2) layers are metastable and because of the large uncertainties in the energies of these layers, the thermal stability of the c(2×2) structure using the statistical mechanics model proposed in section 6.2.1.1.1 cannot be discussed. In fact, because of the overlap of the energies of these layers, the uncertainty in ∆𝐸଴௄ is larger than its value, making the model impracticable. Figure 7-22 The energies of the perfect and defect c(2×2) layers at temperatures of 1.5 to 3 K. The defect c(2×2) structures contain 4 defects (4d-c(2×2) structure) at these temperatures. At 1, 3.5 and 4 K, the correlation of the data is very high, so these data were not used in the analysis here. The bars in each case show the standard deviation in the energy obtained in the MC-NVT simulations. y = 0.2323x - 48.061R² = 0.9869y = 0.2469x - 48.022R² = 0.9535-47.8-47.3-46.80 1 2 3 4 5Energy [kcal/mol]Temperature [K]c(2×2)4d-c(2×2)259 As seen in the defect analysis results, in the simulations using Model II and in the LBE case the perfect c(2×2) layers are slightly more stable than the defect layers and are more likely to be observed in the simulations. This is in contrast with the results of the UBE case of Model II for a layer of Para molecules over MgO and at half coverage. In order to understand this difference, the three energies that are listed in Table 7-13 must be used. This case is called “case 1”, which is the case in which Model II along with the LBE case are used in the calculation of the energies. Then, in the next step, the energies of this case are compared with two other cases. “Case 2” uses Model I and the LBE in the calculation of the energies (Table 6-12), whereas “case 3” uses Model II and the UBE (Table 7-9). It should be emphasized that all of these cases are for the Para molecules over the surface. Once again, despite differences between the Para-Para interaction energies in models I and II, it is assumed that they are the same in all the three cases mentioned above. The comparison of the energies in these cases show that, in case 1 (Model II, LBE), the binding energy of the Para molecule to the cation site is 31.6% stronger than case 2 (Model I, LBE). Also, the MiEB in case 1 (Model II, LBE) is about 2.6 times the corresponding quantity in case 2 (Model I, LBE). On the other hand, compared to case 3 (Model II, UBE), the binding energy of the Para over Mg2+ in case 1 (Model II, LBE) is 9.5% stronger. Besides, the MiEB of case 1 (Model II, LBE) is also larger than the MiEB of case 3 (Model II, UBE) by 23.5%. Since the difference in the binding energy of Para over Mg2+ as well as the difference in the MiEB of this molecule between cases 1 (Model II, LBE) and 3 (Model II, UBE) are smaller than the corresponding differences between cases 1 (Model II, LBE) and 2 (Model I, LBE), the simulation results are more similar in cases 1 (Model II, LBE) and 3 (Model II, UBE). For example, in both cases 1 (Model II, LBE) and 3 (Model II, UBE), the c(2×2) layers may be seen 260 in the simulations (metastable), even though these c(2×2) structures are more stable in case 1 (Model II, LBE). But, in case 2 (Model I, LBE), these c(2×2) structures can never be seen in the layer and instead, an hcp structure along with patches of open surface are obtained as explained in section 6.2.1.2.2. Therefore, it is concluded that, similar to case 3 (Model II, UBE), the MC-NVT simulation results of case 1 (Model II, LBE) are in agreement with the experimental data68. Table 7-13 The energy of a Para molecule when sits over cation, the attractive energy between two of these species, as well as the MiEB of this molecule over MgO using Model II and the LBE case. The MiEB has been defined in section 4.3.1.1.1. H2 species Binding energy over cation [kcal/mol/molecule] Para-Para binding energy [kcal/mol] MiEB [kcal/mol] Para -0.55913 -0.06883 0.15879 Structures aside, the energies of the c(2×2) layers have also been investigated. Table 7-14 lists these energies over the temperature range of 1 to 5 K. As was seen in Figure 7-20, at all these temperatures and in the course of the simulations, the simulated layers remain stable in the form of c(2×2) structures. In the lack of experimental data, the simulation results of the LBE case (Table 7-14) are only compared with the results of the UBE case (Table 7-10). Once again, in both of these tables, Model II was used in the calculation of the energies. Comparing the data in these tables show that, at the same temperatures of 1, 2, or 3 K, the Para-Para interactions are slightly more repulsive in the LBE case. This is because, in this case, the molecule sits closer to the surface. Since all the Para molecules sit over the cation sites, the closer the molecule gets to the surface, the more repulsive the interaction between molecules becomes (see subfigure (b) of Figure 5-5). Nonetheless, the more attractive total energy in the LBE case is due to the stronger 261 H2-surf binding energy in this case (compare the total energies at the same temperature in Table 7-10 and Table 7-14). Table 7-14 The energies of MC-NVT simulated layers of Para molecules over MgO at half coverage at temperatures up to the ULT quantity when Model II along with the LBE case are used in the calculation of the energies. The errors are the standard deviations obtained from the statistical averaging in the MC-NVT simulations. The total energy is the sum of the H2-surface and the H2-H2 interactions. Energy [kcal/mol/molecule] Temperature [K] Structure Total H2-surf H2-H2 1 c(2×2) -0.6640 ± 0.0005 -0.5559 ± 0.0003 -0.1081 ± 0.0004 2 c(2×2) -0.6607 ± 0.0008 -0.5526 ± 0.0006 -0.1081 ± 0.0005 3 c(2×2) -0.658 ± 0.001 -0.549 ± 0.001 -0.1085 ± 0.0006 4 c(2×2) -0.655 ± 0.001 -0.546 ± 0.001 -0.1087 ± 0.0007 5 c(2×2) -0.652 ± 0.002 -0.543 ± 0.002 -0.1092 ± 0.0008 Finally, the impact of the convergence of the Para-MgO interaction energy on the energies of the layers is discussed here. Similar to the previous cases, this effect can be investigated by calculating the average heights of the molecules in the layer by using Eq. (7-1). Based on the HDF curves in Figure 7-19, the average heights of the molecules are 2.93, 2.95, 2.97, and 3.06 Å at temperatures of 1, 3, 5, and 6 K, respectively. This means that at all these temperatures the average heights of the Para molecules in the layer are higher than the optimum height of this molecule over the cation site (2.90 Å in Table 4-23). Therefore, at all the temperatures presented 262 in this section, the impact of the convergence of the Para-MgO interaction energy on the energy of the layer is negligible. 7.2.2 Coverage of 0.75 At this coverage, experiments reported a c(4×2) structure for a layer of n-H2 molecules over MgO 10,37. For a layer of para-H2 molecules, however, an hcp structure was reported, which could be a c(n×2) structure, but from the data it could not be determined what the value of n was68. For a c(4×2) structure in the n-H2 layer, according to Figure 1-3, only the molecules that are at the corners and the center of the unit cell must be of the same type while the other molecules can be different from these molecules (unlike c(2×2) in which all the molecules must be the same). This means that the c(4×2) structure can also be formed by mixtures of ortho- and para-H2 molecules. In this section, MC-NVT simulations are used for a thorough investigation of all the possible combinations of these molecules in the n-H2 layers over the surface. Before doing the simulations, as explained in the introductory part of Chapter 6, one restriction is to keep the coverage of the layer in these simulations consistent with the experiment. To that purpose, the ULT quantity is used, which is the temperature up to which the coverage of the layer in the simulations can be kept fixed and consistent with the experiment. This quantity is specified based on the ULH quantity, as explained in section 6.1. But, finding the ULH in the H2 layers at this coverage, is not as simple as the 0.5 coverage. In fact, the problem in the 0.75 case is that, combinations of different H2 species over different interacting sites are possible. This makes specifying the ULH and ULT quantities more difficult. But, based on the preliminary simulation results and the fact that, for all H2 species, this height is close to 263 the height at which the MiEB against the lateral motion of the H2 molecule becomes zero, it was decided to specify this quantity as 4.0 Å. At heights greater than this value, the molecules can freely move over the cation-saddle-cation channels and this can cause many uncertainties in the simulation results. Based on this value for the ULH quantity and the preliminary HDF results obtained from the simulations of different compositions of the H2 layers over MgO at 0.75 coverage, the ULT quantity is obtained as 5 K. In fact, it is only up to this temperature that the simulation results of the H2 layers for all combinations of H2 molecules are reliable. Therefore, in this section, all the simulation results are only discussed up to, and including, 5 K. In addition to that, there is also another change in the simulations of the layers at 0.75 coverage compared to 0.5. Since the coverage of the layer has increased, in order to show the results better, the size of the simulation box is also increased from a 12 ion×12 ion simulation box at 0.5 coverage to a 24 ion×24 ion simulation box at 0.75 coverage. In the following paragraphs, first, the possible combinations of ortho- and para-H2 molecules in an n-H2 layer that can form a c(4×2) structure at 0.75 coverage are discussed and then the results of a layer of only Para molecules at this coverage are presented. It should also be added that, since both the UBE and the LBE cases show very similar properties, the conclusions that will be made are the same for these cases and hence, in this section, only the results of the UBE case are presented. 7.2.2.1 A mixture of Heli and Para molecules (the HP layer) The possibility of forming a c(4×2) structure has been examined for many combinations of ortho- and para-H2 species in an n-H2 layer. It was found that only three of these combinations may form the c(4×2) structure at 0.75 coverage. One of these candidates is a layer of a mixture of Heli and Para H2 molecules, with the number of Para molecules twice the number of Heli 264 molecules in the layer. This layer is called the HP layer. Figure 7-23 shows the simulation results for this layer at 5 K (subfigure (a)). As indicated by the square in subfigure (a), this layer can form a c(4×2) structure. The formation of this structure is also confirmed by the Heli-Heli RDF curves in subfigure (b). As seen, the positions of the peaks are exactly at distances expected for a perfect c(4×2) layer. These distances can be calculated based on the positions of the ions at the corners and centers of the unit cells underneath the adsorbed layer. As seen in subfigure (a), in this layer, the Heli molecules are pinned to the cations because of their strong binding energies to these sites as well as their large MiEB over the MgO surface (see also the blue H2 molecules in the c(4×2) unit cell of Figure 1-3). On the other hand, the Para molecules may be more mobile over the surface due to their slightly weaker binding energy to the surface as well as their considerably smaller MiEB compared to the Heli molecules (see Table 4-21 and Table 4-22). But, the movements of the Para molecules are limited to between the neighboring cation and saddle sites in the unit cell (the light-brown H2 molecules in the c(4×2) unit cell of Figure 1-3). In fact, it is the presence of these Para molecules as well as their slight movements over the surface that prevents the Heli molecules from jumping to neighboring cations and breaking the c(4×2) structure. It should be added that, since this layer forms a perfect c(4×2) structure, its Heli-Heli RDF curve will be used as the reference for the rest of the layers discussed in the following sub-sections. 265 (a) (b) Figure 7-23 a. The final configuration of an MC-NVT simulated HP layer at the coverage of 0.75 and at 5 K, and b. the Heli-Heli RDF data for this layer. In these simulations, Model II along with the UBE case were used in the calculation of the energies. The square in subfigure (a) indicates a c(4×2) unit cell. In subfigure (b), the peaks appear at positions expected for a perfect c(4×2) layer. 0482 4 6 8 10 12 14Heli-Heli RDFDistance [Å]2.96 Å 266 7.2.2.2 A mixture of Heli and Cart molecules (the HC layer) The second possible combination of the H2 molecules in an n-H2 layer that may form a c(4×2) structure is a layer of only ortho-H2 molecules, for example, a layer of Heli and Cart molecules. This layer is called the HC layer. In fact, this layer can be made by replacing the Para molecules in the HP layer with the Cart molecules. Figure 7-24 shows the MC-NVT simulation results for this layer at 5 K. The results show that the c(4×2) structure is stable up to 5 K as indicated by the square in subfigure (a). The formation of this structure is also confirmed by the Heli-Heli RDF data illustrated in subfigure (b) of Figure 7-24. Similar to the HP layer, in the HC layer, the Cart molecules are slightly mobile over the surface. But, their lower binding energy as well as their smaller MiEB compared to the Para molecules (see Table 4-22), makes the Cart molecules more mobile over the surface. Also, like the HP layer, the Heli molecules remain localized over the cation sites which is mainly due to their strong binding energy to this site as well as their large MiEB over the surface. 267 (a) (b) Figure 7-24 a. The final configuration of an MC-NVT simulated HC layer (using Model II and UBE) at the coverage of 0.75 and at 5 K, and b. the RDF data for this layer. In subfigure (b), the c(4×2) RDF graph is obtained from Figure 7-23. The square in subfigure (a) indicates a c(4×2) unit cell. In subfigure (b), the Heli-Heli RDF peaks appear at positions expected for a c(4×2) structure. 05102 4 6 8 10 12 14RDFDistance [Å]Heli-Helic(4×2)2.96 Å 268 7.2.2.3 A layer of only Heli molecules (the H layer) Another possible combination of n-H2 molecules that may form a c(4×2) layer is a layer of only Heli molecules, which is called the H layer. This layer is constructed by replacing the Para molecules in the HP layer by the Heli molecules. Using the MC-NVT simulations, the possibility of forming the c(4×2) structure by this layer was examined. Figure 7-25 presents these results for this layer at 5 K. As seen in subfigure (a), the results do not show any indication of an ordered structure in the layer, which is confirmed by the Heli-Heli RDF data in subfigure (b). In fact, the first peak of this curve appears at around 3.23 Å, which is close to the distance between two neighboring cations in the MgO surface (~3 Å). This means that not only the Heli molecules tend to sit close to the cation sites to minimize the H2-surface energy, but they also sit close to each other to lower the H2-H2 energy. This formation of the Heli molecules is a disordered layer with no sign of the c(4×2) structure in any directions over the surface. 269 (a) (b) Figure 7-25 a. The final configuration of an MC-NVT simulated H layer at the coverage of 0.75 and at 5 K, and b. the RDF data for this layer. In these simulations, Model II along with the UBE case were used in the calculation of the energies. In subfigure (b), the c(4×2) RDF graph is obtained from Figure 7-23. 0482 4 6 8 10 12 14RDFDistance [Å]Heli-Helic(4×2)2.96 Å 270 7.2.2.4 A layer of only Para molecules (the P layer) Also, the possibility of forming the c(4×2) structure by a layer of only Para molecules over MgO and at 0.75 coverage was examined. Figure 7-26 shows the MC-NVT simulation results for this layer at 5 K. As seen in the figure, this layer cannot form a uniform layer of the c(4×2) structure similar to the uniform c(4×2) structures in the HP and HC layers. In fact, the layer formed over the surface is a mixture of islands of hcp structures (shown by a hexagon in subfigure (a)) as well as some disordered structures. This hcp structure may be a c(6×2) layer (as shown by the rectangle in subfigure (a)), but some c(4×2) and p(5×2) unit cells is seen in the layer as well (as shown in subfigure (a)). It should also be added that at a lower temperature of 3 K, a similar structure to 5 K is seen over the surface. Therefore, based on the results presented here, at 0.75 coverage, the P layer forms a mixture of hcp structures and some disordered structures over the surface (and not a uniform c(4×2) structure). This result can explain the experimental findings68 for these layers in which indications of (n×2) structures, occurring in islands or as multiple small domains were reported. It can also explain why in the experiments68 the number n in in the c(n×2) unit cell could not be determined with certainty. 271 (a) (b) Figure 7-26 The final configuration of an MC-NVT simulated P layer at the coverage of 0.75 and at 5 K, and b. the RDF data for this layer. In these simulations, Model II along with the UBE case were used in the calculation of the energies. In subfigure (a), the hcp and c(6×2) structures are shown by a hexagon and a rectangle, respectively. In subfigure (b), the c(4×2) RDF graph is obtained from Figure 7-23. 03692 4 6 8 10 12 14RDFDistance [Å]Para-Parac(4×2) c(6×2) hcp c(4×2) p(5×2) 2.96 Å 272 The simulation results of the possible compositions of a layer of n-H2 molecules over the MgO surface at 0.75 coverage and at 5 K show that only two of these layers can form a uniform c(4×2) structure; the HP as well as the HC layers. In order to see if these layers are the most stable structures at this coverage, in Table 7-15 the energies of all these layers are compared. But, before comparing these energies, the impact of the convergence of the H2-surface energies on the total energies of these layers must be investigated. From the simulation results of the H2 layers over NaCl and MgO, it was seen that, on average, the H2 molecules sit at heights higher than their optimum values over each interacting site (for example, see the last paragraph of sections 7.1.1, 7.1.2, 7.2.1.1.1, and 7.2.1.1.2, etc.). The same analysis must also be done on all the layers presented in this section for the 0.75 coverage. Based on our calculations, same conclusion can be made here as well. For example, based on the HDF data for the HC layers, the average heights of the Heli and Cart molecules in the layer are in turn 2.96 and 3.50 Å. As explained in section 7.2.2.2, in the HC layer, the Heli molecules sit over cations, whereas the Cart molecules sit over the saddle sites. Therefore, based on the optimum heights of these molecules in Table 4-21 and the convergence analysis presented in Chapter 4, the impact of the single molecule energy convergence on the energy of the layer is negligible. Similar conclusions can be made for the HP, H and P layers as well. 273 Table 7-15 The energies of the HP, HC, and the H layers over MgO at 0.75 coverage and at 5 K when Model II along with the UBE case are used in the calculation of the energies. The errors are the standard deviations in the MC-NVT simulation energies for each layer. Energy [kcal/mol/molecule] Layer (composition) Total H2-surf H2-H2 HP -0.6614 ± 0.0009 -0.4782 ± 0.0008 -0.183 ± 0.001 HC -0.6263 ± 0.0008 -0.4433 ± 0.0005 -0.1829 ± 0.0007 H -0.6998 ± 0.0009 -0.533 ± 0.001 -0.167 ± 0.002 P -0.635 ± 0.001 -0.449 ± 0.001 -0.186 ± 0.001 Comparing the energies of the layers in Table 7-15 show that, for example, the energy of the HP layer is lower than the HC layer. Since the H2-H2 interactions are the same in these layers, the difference in the total energy is only because of the stronger H2-surface interactions in the H layer. The reason is that for a Para molecule, cations are the most favorable interacting site in the surface and in the HP layer these molecules prefer to sit over cations in order to lower the energy. For the Cart molecules, however, the most attractive site is the anion, but in the HC layer, these molecules do not have enough room in the unit cell to move over these sites, and instead they sit on the saddle sites (second most attractive site) in order to minimize the energy. But, the Cart/Saddle binding energy (in the HC layer) is much weaker than the binding energy of Para/Mg2+ (in the HP layer, see Table 4-21). In fact, it is this difference in energy that results in a weaker H2-surf energy in the HC layer. In addition, the H2-H2 interaction energy in the HP or HC or the P layers is stronger than the corresponding energy in the H layer. This is because of the formation of the hcp structure in the HP, HC, and P layers which lowers the H2-H2 interaction 274 energies. In the H layer, however, this structure cannot be formed due to the strong binding energy of the Heli molecules to the cations as well as their large MiEB over the surface. Nonetheless, as seen in Table 7-15, the total energy of the H layer is considerably lower than the total energies of the HC, HP, and P layers, which is due to the significantly stronger H2-surf interactions in this layer. Therefore, at 0.75 coverage and based on the simulation results presented in this section, the H layer is the most stable structure over the surface. But, this causes a problem, that is, this layer does not show any structures. In fact, this structure is in contrast with the experimental findings that suggested a c(4×2) structure for a layer of n-H2 molecules at this coverage 10,37. The discrepancy between the simulation results and the experimental data for a layer of n-H2 molecules may be explained in two ways. Firstly, as stated by Larese et al.11,17,20,41, the structures at this coverage are complex and cannot be determined with certainty. Secondly, Model II may not be able to predict the correct structure at this coverage. This drawback might be due to the assumptions that are made in this model. For example, one of the main assumptions that was made in Model II was to use the NDPT formulation in the calculation of the single H2 species with the surface (for more detail see Chapter 3). In this method, the ortho-H2 molecule is either in the so-called Heli state (ϒଵଵഥ or ϒଵଵ) or the Cart state (ϒଵ଴)) and it always stays in either of these two states over the surface. But, the problem with the NDPT formulation is that it can only be used when the energies of the eigenstates in the reference system are not the same. In fact, as explained in section 2.2.2, when the energies of these eigenstates (Heli and Cart) in the reference system (isolated H2 states of Heli (ϒଵଵഥ or ϒଵଵ), and Cart (ϒଵ଴)) are the same, the correct way of calculating the energies and wavefunctions of the ortho-H2 molecules on the ionic surface is to use the DPT method. In this method, in the adsorbed layer, the ortho-H2 species must be treated 275 as a linear combination (mixture) of the isolated eigenstates (Heli and Cart). The contribution of each one of these eigenstates to the total eigenstate of the ortho-H2 molecule in the adsorbed layer is then calculated based on the type of the substrate and the position of the molecule over the surface. By doing this, the ortho-H2 species in the adsorbed phase can be differentiated based on the contributions of each one of the isolated H2 eigenstates to the total eigenstate of the molecule. Consequently, this new eigenstate of the ortho-H2 molecule may show a totally different energy over the surface compared to the assumed Cart and Heli molecules (eigenstates) in Model II. In 0, the impact of this new form of the ortho-H2 molecule (Model III) is investigated in detail on both the H2-MgO and H2-NaCl energies as well as the adsorbed layers of these systems. 7.2.3 Coverage of 0.83 At this coverage, experiments reported c(6×2) structures for layers of n-H2 and hcp (or maybe c(6×2)) structures for para-H2 over the MgO surface. Similar to the 0.75 coverage, different compositions of the ortho- and para-H2 can form these layers. However, in section 7.2.2, it was concluded that the assumptions that were made in developing the model for the interactions of ortho-H2 molecules over the ionic surface (Model II), may have been too idealistic and they may need to be revisited. Thus, Model II may not be good enough in describing the interactions of these molecules. However, in Model II and for para-H2, no such assumptions were made. Therefore, in this section, only the structure of a layer of Para molecules is investigated at the coverage of 0.83. But before discussing the simulation results, again the first step is to specify the ULH quantity due to the restriction in the MC-NVT simulation that was discussed in sections 6.1 and 7.2.2. That is, to make sure that the conditions in the MC-NVT simulations are consistent 276 with the experiments. According to our H2-surface energy data, the best ULH value for the layers at 0.83 coverage is the ULH quantity that was used for the layer at 0.75 coverage, i.e., 4.0 Å. 7.2.3.1 A layer of only Para molecules (the UBE case) Based on the HDF data for this layer, the maximum temperature up to which the MC-NVT simulation results can be discussed is 5 K (=ULT). Figure 7-27 presents these simulation results for the layer at 5 K. As shown by the hexagon in subfigure (a), this layer forms an hcp structure. The RDF data in subfigure (b) confirm this, but they also suggest that the structure could be a c(6×2) structure, which is in agreement with the experiment10,37. In fact, as shown in subfigure (b), there is some slight mismatch between the peaks of the hcp structure (the dashed curve) and the positions expected for a perfect c(6×2) structure (the dotted curve). A closer look at subfigure (a) of Figure 7-27 may answer why this mismatch is seen here. In fact, at this coverage some p(5×2) and p(7×2) unit cells can also be seen in the layer (as shown in subfigure (a)). The presence of these sub-structures in the layer can shift the peaks from positions expected for a perfect c(6×2) structure. Therefore, it can be concluded that this structure is in fact an hcp structure (and not c(6×2)). This result is in agreement with the experiments68 in which indications of (n×2) structures (with no definitive assignment of “n”), occurring in islands or as multiple small domains were reported. Moreover, based on the first peak of the RDF data, the lattice constant of the hcp lattice (hexagon in subfigure (a)) is 3.40 Å in very good agreement with the experimental data of 3.46 Å 17,20. Also, the area covered by the Para molecules per unit cell in this layer was calculated. Using the calculated lattice constant mentioned above, the area per unit cell of 10.01 Å2/molecule is obtained which is in good agreement with the experimental data of 10.37 277 Å2/molecule17. It should be noted that, based on the experiments, 0.83 is the coverage at which the H2 monolayer fully covers the MgO surface (monolayer completion)17, no matter what the rotational state of the H2 molecules is (ortho or para). On the other hand, in subfigure (b) of Figure 7-27, the RDF data for a layer of Para molecules over MgO at the coverage of 0.75 (the P layer in section 7.2.2.4 at 5 K) is also compared with the corresponding data for a layer of Para molecules at the coverage of 0.83 (also at 5 K). This is done to see if the layer at 0.75 is the same as 0.83. The results show that these structures are similar and are both hcp structures, although the sharper peaks at the coverage of 0.83 show that it is slightly better structured. This may be because one structure, i.e., the c(6×2) structure, is the most dominant. Based on the positions of the first peaks in the RDF graphs, the lattice constants of the hcp structures in both coverages of 0.75 and 0.83 are the same (3.40 Å). It must be mentioned that, here only the results of the UBE case are shown. For the LBE case, the structures are very similar to the UBE case, but the energies of the layer are stronger. Also, the heights of the molecules in the layer are, on average, lower in the LBE case. 278 (a) (b) Figure 7-27 a. The final configuration of an MC-NVT simulated layer of Para molecules at the coverage of 0.83 and at 5 K, and b. the RDF data for this layer. In these simulations, Model II along with the UBE case were used in the calculation of the energies. In subfigure (b), the c(4×2) RDF is obtained from Figure 7-23. 048122 4 6 8 10 12 14RDFDistance [Å]Para-Para at 0.75c(6×2)Para-Para at 0.83[100] [110] 2.96 Å c(6×2) hcp p(5×2) p(7×2) 279 7.2.3.2 External vibrations in a layer of Para over MgO at 0.83 coverage The similarities in the structures of 0.75 and 0.83 coverage may also be supported by the experimental findings that reported similar external vibrations for these layers10,37. It should be added that, experimentally, these vibrations were observed either along the [110] or [100] directions of the crystal surface. These directions are depicted by the arrows in subfigure (a) of Figure 7-27. Table 7-16 presents these experimentally observed external vibrations as well as the calculated vibrations based on our potential model. It should be added that the calculated vibrations in this table are estimations of the vibrations that are seen experimentally. Since the experimental data are all in meV, the calculated vibrations are all presented in meV as well. Moreover, as seen in subfigure (a) of Figure 7-27 (or in the schematic diagram of Figure 7-28), at the coverage of 0.83, the Para molecules in the layer sit either on the cations or near the saddle sites. Therefore, the calculated vibrations are always centered on these sites. For example, in the hcp layer of Figure 7-27, the only possible vibration for a Para molecule overtop of the Mg2+site along the [100] direction is from anion to cation to the opposite anion site. This vibration is denoted by O2--Mg2+-O2- in Table 7-16 and only has a value along the [100] direction. The same is true for the O2--Sa-O2 and Sa-Mg2+-Sa along the [110] direction, with “Sa” referring to the saddle site. These vibrational energies are in fact the “T-Mode” energies, because they are vibrations of the molecule parallel to the MgO surface. Another vibrational mode involves the vibration of the molecule perpendicular to the surface. This vibrational energy is called the “S-Mode” energy and is always considered over a fixed site on the surface. In principle, both vibrations can be dispersive or non-dispersive (dispersionless). By definition, a dispersive mode is a wave that travels over the whole surface due to the collective motion of particles. It will have a number of 280 possible wavelengths with correspondingly different energies. In contrast, a non-dispersive mode is a vibration of a single molecule over the surface with a constant energy Table 7-16 The calculated (this work) and experimentally measured (previous studies) vibrational energies in a layer of Para molecules at both 0.75 and 0.83 coverages over the MgO surface. In the calculation of all these energies Model II along with the UBE case were used. “disp. E” in brackets refers to the “dispersive energy” that is the vibrational energy of a collection of molecules over the surface. T-Mode Energy [meV] S-Mode Energy [meV] Experiment [meV] 10,37 Direction O2--Mg2+-O2- O2--Sa-O2- Sa-Mg2+-Sa Above Mg2+ Above Saddle Along [100] 8.0 - - 10.8 10.1 10.1 and 8.1 Along [110] (Disp. E) - 3.7 8.2 10.8 10.1 10.1 and 8.1, (2.5 – 7) In order to calculate these vibrational energies, the Para-surface interaction energy curves of Model II are used along with the harmonic oscillator model that is fitted to these curves. Using this fitting procedure, the force constant 𝑘 of the potential energy can be obtained. In the next step, the vibrational frequency 𝜔 of the oscillator (𝜔 = ට ௞௠ where m is the mass of the para-H2 molecule) is calculated and then this frequency is used to calculate the energy spacing between the vibrational states (𝐸 = ħ𝜔, where ħ = ௛ଶగ). Since, the harmonic oscillator is an idealized model of the actual vibration, all the data reported in Table 7-16 are approximate vibrational frequencies expected for this structure. They are also an upper limit to the actual values since the anharmonic effects were not included in the calculations. 281 As may be seen in Table 7-16, experiments reveal two sets of dispersionless modes in the [100] and [110] directions and a dispersive mode in the [110] direction. The dispersionless modes have energies of 8.1 and 10.1 meV in both directions. The dispersive mode has a nonzero minimum value for the vibrational energy of 2.5 meV, which with increasing wavevector rises to a maximum of 7 meV and then drops to a value of 5 meV at the Brillouin zone boundary. The nonzero minimum value is somewhat unusual for a dispersion curve. As discussed above, the calculated vibrational energies are based on the structures determined from the simulations, where it was found that para molecules were located at cation sites or near saddle sites. The perpendicular or S-modes were found to have vibrational energies of 10.8 and 10.1 meV above the cation and saddle sites respectively. This compares quite favorably with the experimental value of 10.1 meV. No error bars were reported in the experimental work so the observed 10.1 value might be valid for both of the calculated S-modes. Since the T-modes correspond to motion parallel to the surface, the calculations are necessarily direction dependent. In the [100] direction, only the cation site will give rise to a T-mode and yield a calculated value of 8.0 meV which compares favorably with the second dispersionless mode. The calculations therefore indicate that the experimental 8.1 meV vibration can be assigned to the T-mode vibration of an H2 molecule absorbed on the cation site. The T-mode calculation above the cation in the [110] direction yields a vibrational energy of 8.2 meV that matches the second dispersionless mode, again leading to the assignment of the observed energy to a cation T-mode. Thus, the two experimentally observed dispersionless modes at 10.1 meV and 8.1 meV are accounted for by S-modes (saddle and possibly cation sites) and T-modes (cation site). Assuming the S-modes are fully accounted for, the dispersive mode consists solely of motion parallel to the surface. The geometry of the simulation layer indicates that one more T-282 mode is possible in the [110] direction, namely molecules vibrating laterally above the saddle site. The calculations for this site yield a much lower vibrational energy of 3.7 meV suggesting it is likely a component of the dispersion mode. It should be noted that anharmonic terms in the surface potential could significantly lower this value. A dispersive mode in the H2 overlayer involves the collective motion of molecules in close proximity and able to interact with one another. Since the molecules on the cation-sites have high vibrational energies, it seems unlikely that they would participate in such a mode leaving only the molecules around saddle sites as potential participants. Figure 7-28 shows a diagram of the approximate positions of the Para molecules over MgO at 0.75 and 0.83 coverages based on the simulation results of Figure 7-26 and Figure 7-27. Examining this figure and by excluding the cation sites, it is seen that the zig-zag chains of saddle-site molecules run vertically. These structures are likely candidates for supporting the dispersive modes. With an underlying surface potential present, it is important that the molecules are able to move. Because of the quantized nature of the molecular motion on the surface, a molecule must possess a quantum of lateral energy in order to support lateral motion. Thus, a minimum requirement for dispersive wave propagation is the excitation of a saddle T-mode. This accounts for the nonzero minimum energy. Note that this T-mode allows for motion in the [110] direction and supports a longitudinal wave. The calculated value of this T-mode is 3.7 meV, which is higher than the observed value of 2.5 meV. As noted earlier the value of 3.7 meV is likely an overestimate due to anharmonic effects. Another possibility is that there is a saddle T-mode in the [100] direction with a characteristic energy closer to the experimental value of 2.5 meV. Such a mode could exist if the molecules in the cation sites were effectively fixed in position, as they would be at energies below the cation T-mode value of 8.2 meV. It should be noted that a second saddle T-283 mode in the [11ത0] direction would support motion perpendicular to the direction of propagation allowing for a transverse wave. The existence of both transverse and longitudinal waves is hinted at by the existence of a maximum in the dispersion curve. Such a feature can occur when there is mixing between longitudinal and transverse waves. Figure 7-28 Schematic diagram for the vibration of a number of Para molecules along the [110] direction. The green dashed arrow indicates the maximum vibration of the Para molecules along the [110] direction. The green solid arrow shows the vibration due to the interaction between two neighboring Para molecules. The solid wine arrows show the directions along the surface of the crystal. “‒“ is an anion site whereas “+” is a cation site in the surface. The blue circles depict the Para molecules in the adsorbed layer. The main component that forms the basis of dispersive waves is the molecule-molecule interaction. The characteristic force constant of those interactions determines the value of the wave’s energy at the Brillouin zone boundary. For the dispersive phonon modes in the hydrogen film on MgO the experimental value is 5 meV. This value is likely due to a transverse mode. To 284 calculate this quantity requires the existence of detailed vibrational model which has yet to be worked out. One can, however, estimate the energy of a longitudinal mode which will yield an upper limit for the energy. The experimental value is 5 meV. On the diagram of Figure 7-28, the experimental dispersive energy was measured along the dashed green arrow. The maximum vibration between two Para molecules can be estimated along this direction. According to the figure, the nearest-neighbor Para molecules interact with one another along the green solid arrow. The vibration between these two molecules can be calculated using the potential energy between these molecules and the harmonic oscillator model. This energy is about 8.9 meV, the maximum vibrational energy occurring between two isolated Para molecules. But, to calculate the longitudinal vibration along the [110] direction, the motion along the green dashed arrow must be considered. This vibration is calculated by multiplying the para-para vibrational energy, as calculated along the solid green arrow, by cos θ, where θ is about 35°, to yield a value of 7.3 meV. Furthermore, the energy must be corrected from a pair of isolated molecules to a chain of linked molecule by multiplying by √2. This yields an energy of 10.4 meV for a longitudinal mode at the zone boundary, just over double the experimental observation of 5 meV. Thus, the observed dispersive mode is most likely transverse in nature at the zone boundary. If it is a transverse mode, then the value would be estimated using θ~35° yielding an energy of 5.1 meV. This is in good agreement with the experimental value of 5 meV. To test the sensitivity of the results, the vibrational energies in the LBE case were also calculated. The results show that the T-Mode O2--Sa-O2- energy remains unchanged (see Table D-1 in Appendix D). All the other vibrational energies, however, are consistently larger than the corresponding ones in the UBE case. Based on the agreement between the UBE vibrational frequencies and the experimental data, this may suggest that the UBE case is a more realistic 285 case for the binding energy of the layer. It should also be noted that, the estimates of the vibrational frequencies are limited by the potential used. In particular, it was constructed for the simulations by matching the binding energy at the cation site. Consequently, the zero-point energy has been incorporated into the binding energy yielding a shallower potential. Also, the fitting procedure favours the cation sites so that the frequencies extracted for the anion and saddle sites might be less reliable. The model does not calculate the location and value of the dispersion curve maximum. This would require a more detailed model of the dispersive modes. Nonetheless, this simple model based on the simulation result, seems to account for a number of the features and values observed experimentally. The ability to reproduce the experimental quantities is striking. 7.2.3.3 Final considerations To close this chapter, the UBE or LBE values are further tested for Heli/Mg2+ binding energies by looking at some other important properties of H2/MgO systems reported in the previous experimental and theoretical studies 11,20,41. As seen in Chapter 4 for Model II, the calculated binding energies of the para-H2 molecule over the cation site are 22.14 and 24.24 meV in the UBE and LBE cases, respectively (Table 4-21 and Table 4-23). Larese et al. used a semi-empirical quantum dynamical model to calculate the binding energy of a Para molecule over the Mg2+ site as well as the splitting energy between the ortho- and para-H2 rotational energies in the adsorbed phase. To obtain these values they used the ionic charge of the MgO crystal (𝑞) as an adjustable parameter in their model. They have found that the splitting energy in their model can obtained to be equal to their experimental value (11.25 meV) only when the electric charge is 𝑞 = 1, instead of 2 (see section 1.3.1.1). Using this procedure, they obtained 286 the binding energy of Para/Mg2+ as 22 meV and the splitting between ortho- and para-H2 in the adsorbed phase as 10.8 meV. They also calculated the ortho-para splitting energy in the isolated phase which is 14.7 meV20. Using this energy, this splitting energy can be estimated in the adsorbed phase in both the UBE and LBE cases. Note that here the assumption is that the Heli and Para molecules both sit over cation sites. This splitting energy is calculated as follows: 𝑆𝑝𝑙𝑖𝑡𝑡𝑖𝑛𝑔 (𝑎𝑑𝑠𝑜𝑟𝑏𝑒𝑑 𝑝ℎ𝑎𝑠𝑒)= 𝑆𝑝𝑙𝑖𝑡𝑡𝑖𝑛𝑔 (𝑖𝑠𝑜𝑙𝑎𝑡𝑒𝑑 𝑝ℎ𝑎𝑠𝑒)– 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑏𝑖𝑛𝑑𝑖𝑛𝑔 𝑒𝑛𝑒𝑟𝑔𝑖𝑒𝑠 Therefore, for the UBE case: 𝑆𝑝𝑙𝑖𝑡𝑡𝑖𝑛𝑔 (𝑎𝑑𝑠𝑜𝑟𝑏𝑒𝑑 𝑝ℎ𝑎𝑠𝑒) = 14.7 𝑚𝑒𝑉– 3.9 meV = 10.8 meV and for the LBE case: 𝑆𝑝𝑙𝑖𝑡𝑡𝑖𝑛𝑔 (𝑎𝑑𝑠𝑜𝑟𝑏𝑒𝑑 𝑝ℎ𝑎𝑠𝑒) = 14.7 𝑚𝑒𝑉– 4.6 meV = 10.1 meV Therefore, beside the vibrational frequencies in the previous section, the calculated splitting energies here may also imply that the UBE case is a closer energy to the “actual” binding energy of the ortho-H2 molecule over Mg2+. It is noted, however, that careful experimental studies are required to obtain the accurate binding energy of an H2 molecule to the surface. For example, to measure experimentally the binding energy of a Heli molecule to the Mg2+ site, a film of n-H2 molecules at half coverage must be studied. This is because previous experimental studies10,37 as well as our calculations show that, at this coverage, all the Heli molecules sit atop Mg2+ cations. 287 7.3 Summary 7.3.1 The NaCl results The structures and stabilities of monolayers of n-H2 and para-H2 molecules over the NaCl surface were examined in the MC-NVT simulations using Model II. For the n-H2 monolayers, the results show that the p(1×1) structure is formed and only by a layer of Heli molecules. The same structure was obtained for monolayers of para-H2 molecules over the surface. For both n- and para-H2 molecules the energies are in agreement with the experimental data. It was also discussed that the results of models I and II are almost the same for monolayers of n-H2 molecules. This is because, in these models, the Heli-Heli interactions are almost the same, and that the Heli-Na+ binding energy (~10%) as well as the MiEB of the Heli molecule over the surface are only slightly different (~6%). For para-H2, however, the MiEB is considerably larger (24.9%) in Model II and hence, a thermally-more-stable p(1×1) structure is obtained in the simulations. Note that the Para-Na+ binding energy as well as Para-Para interactions are similar in models I and II. These results are summarized in Table 7-17. 288 Table 7-17 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over NaCl. In the MC-NVT simulations, Model II was used in the calculation of the energies. The results for the energy are only shown at a specific temperature. The experimental references have been listed right below the table. Energy [kcal/mol/molecule] (at 8 K) Structure (up to 11 K) H2 species Coverage Simulation Experiment1 Simulation Experiment1,2 n-H2 Full layer -0.873 ± 0.002 -0.92 ± 0.11 p(1×1) p(1×1) para-H2 Full layer -0.751 ± 0.002 -0.85 ± 0.12 p(1×1) p(1×1) 1 Refs. 22,37 2 Refs. 22,23,26,29,37 7.3.2 The MgO results 7.3.2.1 Coverage of 0.5 7.3.2.1.1 The UBE case It is also investigated that the structures and stabilities of sub-monolayers of n-H2 over MgO using the MC-NVT simulations when the UBE case was used in the calculation of the energy. For n-H2 and at half coverage, the c(2×2) structure was seen over MgO. A defect analysis showed that this structure is the most stable structure at the temperatures used in the MC-NVT simulations. Also, a statistical mechanical model was developed in 6.2.1.1.1 to predict the thermal stability of the c(2×2) layer. The results showed that the full c(2×2) structure or islands of this structure may be observable in the simulations up to 11 K, in agreement with the 289 experiments. This temperature is much lower than the temperature (18 K) predicted in Model I for these layers. While the Heli-Mg2+ binding energy as well as the Heli-Heli interactions are the same in models I and II, the large difference in the temperatures obtained from the statistical mechanical model predictions in models I (11 K) and II is because of the 9.1% larger MiEB in Model I. In addition to the n-H2 layers, the possibility of forming the c(2×2) structure was also examined by a layer of para-H2 molecules at half coverage. The results showed that, although the defect structures are more probable, a metastable c(2×2) structure may be formed. These results are in contrast with the results of Model I in which aggregated (hcp) structures with patches of open surface were seen over the surface (see section 6.2.1.1.2), but they are in agreement with the experiments. It should be mentioned that compared to Model I, in Model II, the binding energy of Para-Mg2+ is 38.1% stronger. Also, the MiEB of Para in Model II is 2.4 times the corresponding quantity in Model I. Nonetheless, due to the energy overlap between the defect and perfect c(2×2) layers, the temperature up to which the c(2×2) layers may be observed experimentally could not be predicted by the statistical mechanical model. The long-term average structure of defect (or aggregated) structures cannot be determined using MC-NVT simulations and MC-μVT simulations must be used. The results of the MC-NVT simulations using Model I have been summarized in Table 7-18. 290 Table 7-18 The summary of the MC-NVT simulation results as well as the experimental data for the monolayers of H2 molecules over MgO. In the MC-NVT simulations, Model II along with the UBE case were used in the calculation of the energies. The results for each H2 species are only shown at a specific temperature. “NA” means that the data are not available from the experiments. Energy [kcal/mol/molecule] Structure H2 species Coverage Simulation Experiment Simulation Experiment1 n-H2 (at 5 K) 0.5 -0.694 ± 0.002 NA c(2×2) c(2×2) para-H2 (at 3 K) 0.5 -0.611 ± 0.001 NA patches of defect structure plus islands of c(2×2) islands of c(2×2) alongside patches of disordered structures or hcp structure 1 Ref. 68 7.3.2.1.2 The LBE case For a layer of n-H2 molecules at half coverage and in the LBE case, a more stable c(2×2) layer was obtained compared to the UBE case. Also, the statistical mechanical model predicted a temperature of 11 K for the maximum temperature up to which the c(2×2) structure may be observable in the simulations, in agreement with experiments. Although this estimated temperature is similar to the UBE case, a stronger Heli-Mg2+ (11%) as well as a larger MiEB (20.7%) in the LBE case result in slightly higher thermal stabilities for the simulated c(2×2) layers in the LBE case, as was seen in the MC-NVT simulation results. 291 For para-H2 molecules, the simulation results of the LBE case showed a metastable c(2×2) structure, similar to the UBE case. However, contrary to the UBE case, the perfect c(2×2) layers are more probable than the defect c(2×2) layers in the LBE case. These results are also in contrast with the results of the LBE case of Model I in which aggregated layers along with patches of open surface were predicted for the layer. But they are in agreement with the experimental data. It should be added that the defect (or even aggregated structure) cannot be determined using the MC-NVT simulations and MC-μVT simulations must be done to determine those structures. The results of this case have been summarized in Table 7-19. Table 7-19 The summary of the MC-NVT simulation results as well as the experimental data for the layers of H2 molecules over MgO at half coverage. In the MC-NVT simulations, Model II along with the LBE case were used in the calculation of the energies. The results for each H2 species are only shown at a specific temperature. “NA” means that the data are not available from the experiments. Energy [kcal/mol/molecule] Structure H2 species Coverage Simulation Experiment Simulation Experiment1 n-H2 (at 5 K) 0.5 -0.758 ± 0.002 NA c(2×2) c(2×2) para-H2 (at 5 K) 0.5 -0.652 ± 0.002 NA islands of c(2×2) plus patches of defect structure islands of c(2×2) alongside patches of disordered structures or hcp structure 1 Ref. 68 292 7.3.2.2 Coverage of 0.75 The possibility of forming the c(4×2) structure was examined for different compositions of n-H2 molecules over the MgO surface at 5 K and at 0.75 coverage. It was found that only the HC and HP layers could form this structure. However, the most stable layer over the surface at this coverage is the H layer, which is a disordered structure in contrast with experiments. Thus, it was concluded that the problem is either due to the uncertainties in the experimental data or the inability of our model in predicting these structures. Since for the ortho-H2 molecules, Model II may not be good enough in describing the interactions between these molecules and the surface, Model III was proposed. Whether or not Model III can solve the problem will be discussed in detail in Chapter 8. Also, the structures formed by the para-H2 molecules at 5 K and at 0.75 coverage were also investigated. The results show islands of hcp structure along with patches of disordered structures (not a uniform c(4×2) layer). In fact, these layers are hcp-structured layers that consist of mixtures of c(4×2), p(5×2), and c(6×2) unit cells. These results are consistent with the experimental data. The results of this section have been summarized in Table 7-20. 7.3.2.3 Coverage of 0.83 First of all, since Model II could not predict the experimentally observed c(4×2) structure for a layer of n-H2 (in fact ortho-H2) molecules at 0.75 coverage, it was concluded that Model III must be used for these H2 species (ortho-H2). Therefore, MC-NVT simulations was used only to predict the structure of a layer of para-H2 molecules at 0.83 coverage and at 5 K. The results showed a full hcp structure, which seems to be mainly covered by the c(6×2) unit cells. The layer also includes some p(5×2) and p(7×2) unit cells (see Figure 7-27a). These results are in 293 agreement with experiments. Also, the external vibrations of this layer were calculated using the harmonic oscillator model. The results can assign the types of vibrational modes to the observed vibrational frequencies in the non-dispersive experimental external vibrations. Also, the minimum and Brillouin zone boundary frequencies of the experimentally observed dispersive mode could be approximately calculated using a simple model. As explained, the calculated frequencies obtained in the harmonic oscillator model are always an upper limit to the actual vibrations seen in the experiment. The results of this section have also been summarized in Table 7-20. 294 Table 7-20 The summary of the MC-NVT simulation results as well as the experimental data for the layers of H2 molecules over MgO at the coverages of 0.75 and 0.83. In the MC-NVT simulations, Model II along with the UBE case were used in the calculation of the energies. “ND” means that the simulations were not done for that part of the study so the structures were not determined for that part. Structure (at 5 K) H2 species Coverage Simulation Experiment1 n-H2 0.75 disordered structure c(4×2) 0.83 ND c(6×2) para-H2 0.75 mixture of hcp structure [c(4×2) and p(5×2) and c(6×2) unit cells] and disordered structures indications of (n×2) structures, occurring in islands or as multiple small domains. No definitive assignment for n. 0.83 hcp structure [mainly covered by c(6×2) unit cells but also mixtures of p(5×2) and p(7×2) structures] indications of (n×2) structures, occurring in islands or as multiple small domains. No definitive assignment for n. 1 Ref. 68 295 Model III and its impact on the simulation results The simulation results in Chapter 7 showed that although Model II may be good enough to describe the interactions between para-H2 molecules and the ionic surface, it may not be good for the ortho-H2 molecules. This conclusion was mainly based on the simulated structures of the ortho-H2 layers over MgO at the coverage of 0.75, where inconsistency with experimental data is observed. Therefore, it was suggested that, to get a better description of these interactions, a more complete model must be used. This model is called Model III and was presented in Chapter 3 (see section 3.1.3). In fact, in Model III, the ortho-H2 molecule on the surface is always a mixture of J=1, mJ=0,±1 eigenstates. In this model, the ortho-H2 molecule always appears in three different states, each of which are a linear combination of the J=1, mJ=0,±1 eigenstates. These states are called Mixed State (MS) 1, MS 2, and MS 3. In this chapter, it is first discussed how these mixed states of the ortho-H2 molecule affect the H2-surface energies over the MgO and NaCl surfaces. Then the impact of these shifts in the energy of the single ortho-H2 molecules on the structures of the adsorbed ortho-H2 layers will be investigated. 8.1 Ortho-H2 molecule-ionic surface interactions 8.1.1 The MgO substrate (the UBE case) In order to compare the results of Model III with Model II, it must first be noted that, the calculations of Model III were only done up to first order DPT terms in energy (see section 3.1.3). Therefore, the energies of Model III can only be compared with the corresponding first-order energies of Model II (in the 𝑉෠௦௘௣ case, see section 3.1.2.1). Remember though that, in the 296 simulations using Model II, the energies of the ortho-H2 molecules were calculated up to fourth order. Thus, in this section, first the differences between up-to-first-order (1st) and up-to-fourth-order (4th) energies of the ortho-H2 molecules in Model II and in the 𝑉෠௦௘௣ case are investigated. Figure 8-1 shows this comparison for the UBE case over the anion-cation-anion (subfigure (a)) and the saddle-cation-saddle (subfigure (b)) channels. As seen, the 1st and the reference 4th energies are the same at all positions over the surface, except where the molecule sits very close to the cation site. Above this site, the deviation from the 4th energy is about 5.6% (0.03 kcal/mol or 1.5 meV), which is the highest deviation seen along this channel. This similarity in the 4th and 1st energies in Model II (the 𝑉෠௦௘௣ case) is mainly because of the cancellation of the energies in the higher-order terms in the PT series. Therefore, it is assumed that for the ortho-H2 molecules in Model II, going up to first-order energy can be a reasonably good approximation in the calculation of the binding energy of an ortho-H2 to the MgO surface. It should be added that along the saddle-anion-saddle channels the maximum difference occurs over the anion site, which is about 1% (0.004 kcal/mol or 0.17 meV, see Figure D-1 in Appendix D). 297 (a) (b) Figure 8-1 The minimum energy of the Heli and Cart molecules when up-to-first-order (1st) and up-to-fourth-order (4th) PT terms are used in the calculation of the total energy in the 𝑽෡𝒔𝒆𝒑 case of Model II. The data are for: a. along the anion-cation-anion channel, and b. along the saddle-cation-saddle channel of the MgO substrate. -0.65-0.45-0.25Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)Heli (4th)Cart (4th)Anion AnionCation-0.65-0.40-0.15Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)Heli (4th)Cart (4th)SaddleCationSaddle298 On the other hand, in the 𝑉෠௦௘௣ case of Model II and for the 4th energy, the assumption was that the energies obtained for the Heli and Cart molecules are the sum of individual energies in Eq. (3-5). In other words, in this model, first, each interaction is calculated separately, then the sum of these individual energies is considered as the total energy (see section 3.1.2.1). Also, it was stated that this assumption is correct for only up-to-first-order energies, but in higher-order energies, the cross terms (repulsion-dispersion, repulsion-electrostatic, etc.) also appear in the calculation of the total energy, and hence, this may not be a viable assumption. To simplify the formulation, in Model II, it was assumed these cross terms do not contribute to the total energy (the 𝑉෠௦௘௣ case). However, since the simulations in Chapter 6 and Chapter 7 were done based on the 𝑉෠௦௘௣ case of Model II and since the purpose of this chapter is to compare the results of models II and III, first, the impact of the aforementioned assumption in the 𝑉෠௦௘௣ case on the total energies of the Heli and Cart molecules in Model II must be investigated. This comparison can also give us an idea about the impact of these cross terms in the calculation of the energies in Model III. Therefore, all these terms were included in the total energy (the 𝑉෠௧௢௧ case) and the results are presented in Figure 8-2. Once again, it should be mentioned that the energies shown in this figure are all up-to-fourth-order energies for the following two cases: 1) the case when the energies are first calculated separately and then summed up (the 𝑉௦௘௣ case of Model II), and 2) the case when all the cross terms are also included in the calculation of each correction energy in the PT series (the 𝑉௧௢௧ case). As seen in the figure, at all points above the surface, these two energies are the same. Also, the energy data along the saddle-anion-saddle channels (Figure D-2 in Appendix D) show similar trends to the energies presented in Figure 8-2. These results suggest that the cross-term interactions are negligible compared to the other terms in the total energy and hence, the 299 assumption that was made in the 𝑉௦௘௣ case of Model II is a reasonable assumption. This also means that, the simulation results presented and discussed in Chapter 7 are also valid. (a) (b) Figure 8-2 The comparison between total energies of the Heli and Cart molecules in the 𝑽𝒔𝒆𝒑 and 𝑽𝒕𝒐𝒕 cases of Model II for the MgO substrate and along the channels of a. anion-cation-anion, and b. saddle-cation-saddle. All the energies are up-to-fourth-order (4th) PT energies. -0.65-0.45-0.25Energy [kcal/mol]Position of the moleculeHeli (Vsep)Cart (Vsep)Heli (Vtot)Cart (Vtot)Anion AnionCation-0.65-0.40-0.15Energy [kcal/mol]Position of the moleculeHeli (Vsep)Cart (Vsep)Heli (Vtot)Cart (Vtot)SaddleCationSaddle300 In the next step, the results of Model III are compared with the 𝑉௦௘௣ case of Model II. Figure 8-3 illustrates the minimum energies of the Heli and Cart molecules in the 𝑉௦௘௣ case of Model II as well as the minimum energies of the three mixed states (MS) of the ortho-H2 molecule obtained in Model III. The data in this figure are only for the positions of the ortho-H2 molecule along the anion-cation-anion channel. Since in Model III the energies are calculated only up to first order, for consistency, the data for Model II are also up-to-first-order energies. In fact, based on the data in Figure 8-1, it is assumed that, in Model II, the first-order energies are approximately the same as the fourth-order energies. It should be mentioned that, in Figure 8-3, the three MS states are, in fact, mixtures of the H2 rotational states in the isolated phase and are named solely based on their energies: MS1 being the lowest energy, MS2 the next lowest and MS3 the highest energy. 301 Figure 8-3 Comparison of the minimum energies of the two states of the ortho-H2 molecule in the 𝑽𝒔𝒆𝒑 case of Model II and the three states of this molecule (MS1, MS2, and MS3) in Model III over the MgO substrate. The energies presented in this figure are only up to first order and at positions along the anion-cation-anion channel. The results in Figure 8-3 show that along the anion-cation-anion channel, the MS2 state always has the same energy as the Heli case in Model II and implies that the MS2 state in Model III is in fact the Heli state in Model II. Another interesting result in Figure 8-3 is that, over the cation site, two of the three MS states of Model III have the same energy (MS1 and MS2), which is equal to the energy of the Heli state in Model II. As seen, these MS states are also the lowest energy states over the cation sites. The third state over this site has the highest energy (MS3), which is equal to the energy of a Cart molecule in Model II. Thus, over cations, the three states of the ortho-H2 molecule of model III split into the Heli and Cart states of Model II. The results -0.65-0.45-0.25Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)MS1 (1st)MS2 (1st)MS3 (1st)CationAnion Anion302 over the anion site are similar to the energies discussed over the cation site. Above this site, Model III gives two similar mixed states (MS1 and MS2), with energies similar to the energy of the Heli molecule in Model II. As expected, these two states have the highest energy over the anion site. The lowest-energy state over this site is MS3, which has the same energy as the Cart molecule in Model II. Therefore, over anions, the three mixed states of Model III also split into the Heli and Cart states of Model II, but here the Cart molecule has the lowest energy among all. Moreover, the results of Figure 8-3 show that, only over the cation and anion sites, are the results of models II and III the same. As the ortho-H2 molecule gets further away from these sites, three distinguishable mixed states are obtained for this molecule in model III. It is interesting to note that, although one of the MS states of Model III always has the same energy as the Heli state of Model II, none of these MS states are equal in energy to the Cart state of this model. Also, at intermediate points between the cation and anion sites, neither Heli nor Cart states of Model II is the minimum energy state among all the states of models II and III. In fact, the MS1 state of Model III is the lowest energy state along this channel. Figure 8-4 illustrates the minimum energies of the three mixed states of Model III as well as the Heli and Cart states of Model II for positions of these molecules along the saddle-cation-saddle channels. The results are similar to the data presented in Figure 8-4 in that the MS2 energy always shows a similar energy to the Heli state of Model II. Also, over the saddle site, even though the Cart state is slightly lower in energy than the Heli state, neither of these states is the lowest-energy state. 303 Figure 8-4 Comparison of the minimum energies of the two states of the ortho-H2 molecule in Model II (the 𝑽𝒔𝒆𝒑 case) and the three states of this molecule (MS1, MS2, and MS3) in Model III. The energies presented in this figure are only up to first order and at positions along the saddle-cation-saddle channel of MgO. In Figure 8-5, the minimum energies of the different states of models II and III are illustrated along the saddle-anion-saddle channel. Similar to the previous cases presented in Figure 8-3 and Figure 8-4, the MS2 state has the same energy as the Heli molecule of Model II at all points along the saddle-anion-saddle channel. Although, MS3 and MS2 states show similar energies near the anion site, their energies become different as the molecule gets closer to the saddle site. -0.70-0.50-0.30Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)MS1 (1st)MS2 (1st)MS3 (1st)CationSaddle Saddle304 Figure 8-5 Comparison of the minimum energies of the two states of the ortho-H2 molecule in Model II (the 𝑽𝒔𝒆𝒑 case) and the three states of this molecule (MS1, MS2, and MS3) in Model III. The energies presented in this figure are only up to first order and at positions along the saddle-anion-saddle channel of MgO. 8.1.2 The NaCl substrate In this section, the same step-by-step analysis as section 8.1.1 is used for the NaCl substrate. As expected, the results are for the ortho-H2 molecules on NaCl are very similar to those discussed for the MgO substrate in section 8.1.1. For example, for the NaCl substrate and in Model II, if the up-to-first-order energies are compared with the up-to-fourth-order energies (similar to Figure 8-1), the largest deviations are seen for the Heli/Na+ (7.6%) and Cart/Na+ (6.8%) systems. These are the maximum differences seen in the energies of these molecules (see Figure D-3 and Figure D-4 in Appendix D). In addition, similar to the Figure 8-2 of the MgO -0.65-0.45-0.25Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)MS1 (1st)MS2 (1st)MS3 (1st)SaddleAnionSaddle305 substrate, the 𝑉௦௘௣ and 𝑉௧௢௧ energies are the same for both the Heli and Cart molecules at all positions over the NaCl substrate (see Figure D-5 and Figure D-6 in Appendix D). In order to compare the results of models II and III for ortho-H2/NaCl, in Figure 8-6, the minimum energies of this molecule along the anion-cation-anion as well as saddle-cation-saddle channels are presented. As seen in the figure, over either the anion or the cation site, the ortho-H2 molecule of Model III splits up into three states. On both these sites, two of these states are close in energy to the Heli molecule of Model II and the third one is the same as the Cart molecule of this model. Similar to the MgO case (Figure 8-3, Figure 8-4, and Figure 8-5) and over the channels shown in subfigures (a) and (b) of Figure 8-6, the energy of the MS2 state of Model III is close to the energy of the Heli molecule of Model II. The same trends are seen for the energies of the MS2 and Heli states over the saddle-anion-saddle channel (see Figure D-7 in Appendix D). Also, as seen in Figure 8-6, the energy of the Cart molecule is only equal to the energy of one of the states of Model III when this molecule sits over an anion or cation site or midway between these two ions. In addition, at all positions over the surface, always MS1 is the lowest-energy state among all the ortho-H2 states of models II and III. Thus, these results show that, if the PT calculations are done properly (DPT formulation instead of NDPT), the only state of an ortho-H2 molecule that can be seen in the experiments, is the MS1 state and that pure Cart and Heli molecules are never observed in the experiments except when they sit right on top of a cation or an anion site. 306 (a) (b) Figure 8-6 Comparison of the minimum energies of the two states of the ortho-H2 molecule in Model II (the 𝑽𝒔𝒆𝒑 case) and the three states of this molecule (MS1, MS2, and MS3) in Model III over the NaCl substrate. The energies presented in this figure are only up to first order and at positions along the a. anion-cation-anion channel, and b. saddle-cation-saddle channel. -0.80-0.60-0.40-0.20Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)MS1 (1st)MS2 (1st)MS3 (1st) CationAnion Anion-0.80-0.60-0.40-0.20Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)MS1 (1st)MS2 (1st)MS3 (1st) CationSaddleSaddle307 8.2 H2 adsorbed layers In this section, it is discussed how the interaction of a single ortho-H2 molecule with the MgO and NaCl substrates in Model III affects the structures and stabilities of the adsorbed H2 layers on these substrates. It is also discussed how using the total potential operator (𝑉෠௧௢௧) in Model II may affect these layers compared to the case of the separate potential operator (𝑉෠௦௘௣) used in Chapter 7. Since over NaCl, only one structure is observed, the discussion will be presented first for the NaCl substrate. 8.2.1 The NaCl substrate 8.2.1.1 A monolayer of Para molecules For the Para molecules, it was explained that Model II is enough for calculating the interactions between these molecules with the surface. However, the operator that was used in this model was the 𝑉෠௦௘௣ case. To understand how using the total potential operator (𝑉෠௧௢௧) affects the energy of the layer, first the impact of using this operator on the energy of a single Para molecule over the NaCl surface must be investigated. Figure 8-7 shows how these energies in the 𝑉௦௘௣ and 𝑉௧௢௧ cases are compared along the saddle-cation-saddle channel. Based on the energies in Figure 8-7, along the saddle-cation-saddle channel, the energies are the same except for the positions close to the cation site. The maximum deviation from the energy in the 𝑉௧௢௧ case is about 6.9% right on top of the cation site. It should be added that the difference between the energies is only seen over the cation site. Over anion, the energies of these two versions of Model II are the same. 308 Figure 8-7 The up-to-fourth-order minimum energies of a Para molecule over NaCl along the saddle-cation-saddle channel when Model II is used in the calculation of the energy in the 𝑽𝒔𝒆𝒑 and 𝑽𝒕𝒐𝒕 cases. The results of Figure 8-7 show that if 𝑉෠௧௢௧ is used as the operator in the calculation of the energy of a Para molecule with the NaCl surface, it will cause two important changes in the elements that affect the structure and stability of the Para layers over NaCl. As discussed in Chapter 6 and Chapter 7, these elements are the binding energy of the Para to the Na+ site, the Para-Para interaction energy, as well as the MiEB of this molecule over NaCl. As seen in Figure 8-7, using the 𝑉௧௢௧ case, the binding energy of the Para molecule to the Na+ site increases by 6.9%. But, based on the results shown in Chapter 5 (subfigure (b) of Figure 5-6), it is not expected to see any significant changes in the Para-Para interactions. Since over the saddle site, the 𝑉௦௘௣ and 𝑉௧௢௧ energies are the same, the stronger binding energy to the cation site in the 𝑉௧௢௧ -0.80-0.55-0.30Energy [kcal/mol]Position of the moleculePara (Vsep)Para (Vtot)SaddleCationSaddle309 case will result in an increase in the MiEB of the molecule which is about 6.9%. Because all the Para molecules sit over Na+ sites in a monolayer of Para molecules over NaCl (see section 7.1.2), the stronger binding energy over cation as well as the larger MiEB of the Para molecule result in an increase in the stability of the layer. Therefore, using 𝑉௧௢௧ energies (the more complete version of Model II), it is expected to see a more stable p(1×1) structure for a monolayer of Para molecules over the NaCl surface. 8.2.1.2 A monolayer of n-H2 molecules In section 6.1.1, it was discussed that a monolayer of n-H2 molecules over NaCl is formed by only the Heli molecules (which is an ortho-H2 molecule) sitting over the Na+ sites in a p(1×1) structure. In section 8.1.2, it was discussed that in Model II using the total potential operator (𝑉෠௧௢௧) instead of the separate operator (𝑉෠௦௘௣), do not affect the energies. This means that the structure and energy of the p(1×1) layer will not change even if the 𝑉෠௧௢௧ case of Model II is used in the simulations. It was also discussed that, by using Model III, different results from Model II are expected. In fact, in Model III the MS1 state (which is neither Heli nor Cart) is always the lowest energy state over the NaCl surface. This means that at all positions over the surface, the MS1 state is the state of an ortho-H2 molecule that must be seen over the surface. Over the cations, however, this state is equal in energy to the Heli state of Model II. Therefore, as long as all the Heli molecules sit over the cation sites, the results of models II and III must be the same. Therefore, a p(1×1) structure will still be seen in the MC-NVT simulations if Model III is used in the simulations. But, using Model III can affect the stability of the layer. To discuss this, the data presented in subfigure (b) of Figure 8-6 can be used. As seen, over cation, the binding energy is the same for 310 the MS1 state of Model III and the Heli state of Model II. Over the saddle site, however, the MS1 state has a lower energy. This difference in fact affects the MiEB energy from 0.38442 kcal/mol for the Heli molecule to 0.31227 kcal/mol for the MS1 state (a 19% lower MiEB for the MS1 state). A lower MiEB means that the monolayer of n-H2 molecules in Model III are stable up to lower temperatures compared to Model II. In order to see how the thermal stability of the n-H2 layer changes, the energies of an MS1 state with the Para molecule in Model II are compared. This comparison is done because the MiEB barrier of the MS1 state in Model III is almost equal to the MiEB of a Para molecule in Model II (see Table 4-26). In section 7.1.2, it was seen that a monolayer of Para molecules in Model II forms a p(1×1) structure. Since the MS1 state has a stronger binding energy and a comparable MiEB to the Para molecule of Model II, therefore it is expected that, using Model III, the simulated monolayers of n-H2 molecules over NaCl are stable up to temperatures higher than what was seen for a monolayer of Para molecules using Model II (see section 7.1.2). Once again, it should be added that, based on the calculations of Chapter 5 (subfigure (a) of Figure 5-6), the MS1-MS1 interaction energies in Model III should be the same as the Para-Para interaction energies in Model II. 8.2.2 The MgO substrate (the UBE case) In section 7.2.3, it was discussed that the UBE case is probably a better case for the binding energy of an H2 molecule to the MgO surface. Therefore, in this section, only the results of this case are discussed. 311 8.2.2.1 Layers of Para molecules Figure 8-8 illustrates the minimum up-to-fourth-order energies of a Para molecule over MgO and along the saddle-cation-saddle channel using Model II and in the 𝑉௦௘௣ and 𝑉௧௢௧ cases. The results show that there is no difference between these energies along this channel. Although not shown here, our calculations show that these energies along the saddle-cation-saddle and saddle-anion-saddle channels are also the same. This suggests that, even when the more complete version of Model II (the 𝑉௧௢௧ cases) is used in the simulations, no change will be seen in the simulation results of layers of Para molecules over MgO; either at 0.5 coverage (section 7.2.1.1.2) or 0.75 coverage (section 7.2.2.4), or at 0.83 coverage (section 7.2.3). Figure 8-8 The up-to-fourth order minimum energies of a Para molecule over MgO along the saddle-cation-saddle channel when Model II is used in the calculation of the energy in 𝑽𝒔𝒆𝒑 and 𝑽𝒕𝒐𝒕 cases. -0.55-0.45-0.35Energy [kcal/mol]Position of the moleculePara (Vsep)Para (Vtot)SaddleCationSaddle312 8.2.2.2 Layers of n-H2 molecules 8.2.2.2.1 Half (0.5) coverage In section 7.2.1.1.1, it was discussed that at this coverage the c(2×2) structure is formed by only Heli molecules. In section 8.1.1, it was shown that using the 𝑉෠௧௢௧ case of Model II will result in the same energies as the 𝑉෠௦௘௣ case for the Heli molecules (see Figure 8-2). Therefore, the structure and energy of the c(2×2) structure in section 7.2.1.1.1 will not change if the 𝑉෠௧௢௧ case is used in the simulations. On the other hand, in section 8.1.1, it was shown that the Heli state of Model II can only be seen over the cation site. In fact, over this site the Heli molecule is the lowest-energy state of the ortho-H2 molecule. However, at other positions over the surface, the lowest-energy state of the ortho-H2 molecule over the surface is the MS1 state. Here, the question is how this new state changes the simulation results that were obtained at 0.5 coverage using Model II (in the UBE case). To investigate this, first this assumption was used that in models II and III the up-to-first-order energies are the same as the up-to-fourth-order energies (similar to Figure 8-1 in section 8.1.1). Then the data of Figure 8-4 are used. According to this figure, over cation, the binding energy of the MS1 state in Model III is equal to the Heli state in Model II. The MiEB of the MS1 state, however, is about 0.17869 kcal/mol, which is 27.3% smaller than the MiEB of the Heli molecule (0.24603 kcal/mol, see Table 4-16). It should be added that this amount is based on the assumption that first- and fourth-order energies are the same in Model III, but based on the results of Model II this difference in MiEB could go down to 22%. Since the MS1-MS1 interactions are expected to be the same as the Heli-Heli interactions, the similar binding energies over cation but lower MiEB for MS1 results in one of the following three cases: 1) a 313 less stable c(2×2) structure, which is stable up to a temperature lower than 26.7 K (based on the statistical model of section 7.2.1.1.1); or 2) a metastable c(2×2) structure like the c(2×2) Para in section 7.2.1.2.2; or 3) an unstable c(2×2) structure like the layer of Para molecules in section 7.2.1.1.2. In order to discuss which one of these cases is more likely, the three energies in forming the c(2×2) structure must be used. These energies can be compared for the MS1 molecule in Model III with the corresponding ones for the Para molecule in the LBE case of Model II (see Table 7-13). This former case is used because in section 7.2.1.2.2 (the LBE case of Model II) it was seen that the Para molecules form a metastable c(2×2) structure. In order to do this comparison, first, it is assumed that the MS1-MS1 and Para-Para interaction energies are the same (this assumption seems reasonable based on the energies in Figure 5-5). Also, using the assumption that, in Model III, the up-to-first- and up-to-fourth-order energies are the same, the binding energy of an MS1 molecule to the Mg2+ site (~ -0.60055) is 7.4% stronger than the Para molecule in the LBE case of Model II. Besides, the MiEB of MS1 is about 11.1% larger than the Para molecule in the LBE case of Model II. The stronger binding energy of the MS1 molecule to the cation site as well as its larger MiEB compared to the Para molecule in the LBE case of Model II suggests that a stable (or at least metastable which means island-like) c(2×2) structure may form in a layer of n-H2 (only ortho-H2, see section 6.2.1.1.1) molecules when Model III is used in the simulations. The MC-μVT simulations using the complete version of Model III can shed further light on the structure and stability of a layer of n-H2 molecules over MgO at half coverage. 314 8.2.2.2.2 Coverage of 0.75 and 0.83 In section 7.2.3, it was seen that at this coverage, an aggregated structure that is formed by a layer of Heli molecules only (the H layer) is the most stable structure (Table 7-15) for a layer of n-H2 molecules over MgO. In addition, the RDF data of this layer (subfigure (b) of Figure 7-25) shows that the Heli molecules in this layer sit either right on top of the cations or over sites near the cation sites. It was explained that, this formation of the Heli molecules is because of their strong binding energy to the cation site as well as their large MiEB over the surface. In fact, the repulsive interaction between the Heli molecules cannot overcome this barrier and the Heli molecules sit close to each other on the neighboring cation sites. In Model III, it was shown that the binding energy of the MS1 molecule to the cation site is the same as the Heli molecule in Model II. The MiEB of the MS1 molecule, however, is 27.3% smaller. This smaller MiEB may cause the repulsive interaction between the MS1 molecules to overcome the MiEB of this molecule over the surface and push the molecules further away from over the cations and these MS1 molecules may now be able to form a c(4×2) structure. But no accurate prediction can be made unless the complete version of Model III is used in MC-μVT simulations. On the other hand, Table 7-15 shows that the second lowest-energy c(4×2) structure is formed by the HP layer (Figure 7-23). In this layer the Para molecules are mobile and move between saddle and cation sites. According to Table 4-21, the energy of the Para molecule over this site is -0.38206 kcal/mol (which is slightly stronger than the Heli binding energies over this site, i.e., -0.37682). Based on Figure 8-4, this binding energy for the MS1 molecule is -0.43308 kcal/mol. Compared to the H layer, this lower energy may cause the MS1 molecules to stay more localized around the saddle sites in a structure similar to the HP layer shown in Figure 7-23. Therefore, it is possible that the c(4×2) structure (or the hcp structure) is formed by the MS1 molecules. Based 315 on these data, however, it cannot be predicted with certainty that the structure that is formed by the n-H2 molecules in Model III. MC-μVT simulations using the complete version of Model III must be used to examine the structure and stability of a layer of n-H2 molecules over MgO and at 0.75 coverage. Same is true for the coverage of 0.83. 8.3 Summary 8.3.1 The interaction of H2 with NaCl and MgO Since Model III was only done up to first-order corrections, the assumption that the 4th and 1st order energies of the ortho-H2 molecules in the 𝑉௦௘௣ case of Model II are the same was examined. Only the UBE case of Model II was used, because in Chapter 7 it was explained that this case might be a more realistic case for the binding energy of an H2 molecule to the surface. It was shown that, the aforementioned assumption is reasonable for ortho-H2 molecules at all points over the surface. The largest deviation between these energies was reported on the cation site, which was about 5.6% in the MgO case and 7.6% in the NaCl case. Then the assumption that the 𝑉௦௘௣ case of Model II give reasonably close 4th energies to the 𝑉௧௢௧ case (the complete version) of this model was also investigated. The results of the ortho-H2 molecule showed that there is no difference between the 4th energies obtained in these two cases over both the MgO and NaCl substrates. In the next step, the first-order PT energy of a single ortho-H2 molecule with the surface in Model III was compared with the corresponding energy of the Heli and Cart states in the 𝑉௦௘௣ case of Model II for both MgO and NaCl surfaces. As discussed in Chapter 3, Model III gives three different energy states (eigenstates) over both the MgO and NaCl surfaces, which are called 316 MS1, MS2, and MS3. The results show that at all points over the surface, the MS1 case of Model III always has the lowest energy over the surface among all the eigenstates of models II and III. Also, over the cation and anion sites, two of these three eigenstates are equal in energy to the Heli molecule of model II, whereas the last eigenstate has the same energy as the Cart molecule of Model II. Therefore, only over either the cation or the anion site, the Cart or Heli eigenstates of Model II can be seen over the surface when Model III is used in the calculation of the energy. It was also shown that at all points over the surface, the energy of the MS2 state of Model III is equal to the energy of the Heli molecule of Model II. 8.3.2 The H2 layers over NaCl and MgO As discussed, using the 𝑉௧௢௧ case of Model II (instead of the 𝑉௦௘௣ case of this model) will result in more stable p(1×1) monolayers of Para molecules over NaCl, which is because of the 6.9% stronger binding energy of Para over cation as well as its 6.9% larger MiEB over the surface in the 𝑉௧௢௧ case of Model II. Over MgO, however, there will not be any changes in the simulation results of Para molecules when the 𝑉௧௢௧ case of Model II is used instead of the 𝑉௦௘௣ case, since the binding energy of Para over Na+ and the MiEB of this molecule remain unchanged. In addition, for monolayers of n-H2 over NaCl, Model III will result in less stable p(1×1) layers compared to Model II, which is mostly due to the 19% smaller MiEB of the MS1 molecules compared to the Heli molecules of Model II. Also, over MgO, at 0.5 coverage, Model III should result in a less stable c(2×2) structure due to the 27.3% smaller MiEB of the MS1 state compared to the Heli state of Model II over this surface. But, at higher coverages, there are indications that c(n×2) structures will be stabilized but no accurate predictions can be made 317 about these structures. Ideally, MC-μVT simulations using the complete version of Model III must be used to examine the properties of these layers. 318 Concluding remarks and future work 9.1 Conclusions Based on previous experimental and theoretical studies on the adsorption of quantum H2 molecules over ionic surfaces, it was hypothesized that the structures and stabilities of the adsorbed films depend on the rotational states of the H2 molecules. In order to examine this hypothesis, a potential model (Model I) was proposed that incorporate the rotational state of the molecule into the equations for the H2-surface as well as H2-H2 interactions. Our model is the first model of this type that depends on both the rotational state of the molecule as well as its position over the surface. This also makes it suitable for large-scale molecular simulations. To develop this model the NDPT method was used employing the well-known quantum mechanical rigid rotor model for H2 as the reference system. This PT method was only used up to first order and only in the calculation of the electrostatic interactions (Model I). Then the interaction potential equations of Model I were tested on the H2 layers on both NaCl and MgO using MC-NVT molecular simulations. For NaCl, the results are in great agreement with the experimental data, but for an MgO substrate with sub-monolayers of para-H2 at 0.5 coverage, discrepancies are seen between our simulation results and the experimental findings. The differences between the results of Model I and the experiments led us to the development of Model II, an improved version of Model I. This new model tries to fix the drawbacks in Model I by removing some its assumptions. The main difference between models II and I is in Model II’s use of higher-order PT correction terms. Since Model II uses a set of corrections to the energy and the wavefunctions, the convergence of the PT series had to be analyzed. It was shown that except for the interaction of the quadrupole moment of H2 with the ionic surface 319 (𝐸ொ௘௟௦), the other types of interactions give well-converged energies up to second order. For 𝐸ொ௘௟௦, going up to fourth-order energy gives reasonably converged energies, provided that the molecule does not sit any closer than 2.50 Å to the surface (on both NaCl and MgO substrates). In the next step, the proposed model (Model II) was used in the MC-NVT simulations to see how adsorbed layers of H2 over NaCl and MgO compare with the data obtained in the preceding experimental and theoretical studies. For NaCl, as expected, the results are still in great agreement with the experiments for both n-H2 and para-H2 layers. Also, for MgO, the results of n-H2 at half coverage still show great agreement with the experiment. Interestingly enough, now the simulated layers of para-H2 also show agreement with the experimental data. This is in contrast with the results of Model I. In addition, at higher coverages of 0.75 and 0.83, the simulation results for the Para molecules are in good agreement with the experimental findings. Also, the external vibrations were in good agreement with the experiments. Problems arise, however, when layers of n-H2 molecules over MgO are examined at 0.75 coverage. The discrepancies between the simulation results and the experimental data for these layers led us to the development of Model III, which is based on the DPT method. This method is used only to modify the interactions related to the ortho-H2 molecules. For this model, only H2-surface calculations were done and only up to first-order corrections in the energy. Also, the cross interactions between different components of the energy were assumed to be negligible. The results show that the lowest-energy state over the surface is neither a Heli nor a Cart molecule, but a mixed state of these H2 species over the surface (the MS1 state). Over cations, this mixed state in Model III is equal in energy to the Heli molecule in Model II, and over the anion site, its energy is equal to the energy of the Cart molecule in Model II. In fact, the main success of 320 Model III was to show that the Heli and Cart molecules can only be seen over the surface if they sit over either an anion or a cation. This is the first time in the field that such results are reported. 9.2 Future Work In this three-step study the models used in the simulations were sequentially refined, but there are still some parts left that need to be revisited and modified to develop a more complete model for the simulations of H2 films on ionic surfaces. These improvements have two main directions; one is to remove the assumptions made in the previous models and the other one is based on the methods used in the simulations. 9.2.1 The assumptions 9.2.1.1 The accurate binding energy One of the most important assumptions was to use binding energies that are not very well known. In this study, it was shown that even slight changes in the energies could lead us to the results that do not necessarily reflect the properties of the real system. To calculate the accurate binding energies between H2 and the ionic surface, one way is to perform more accurate electronic structure quantum mechanical (ab initio) calculations. Also, updated experimental energy measurements may be required for the benchmarking of the calculated energies for comparison with ab initio calculations and simulation results. These experiments can be done based on our simulation results, for example at 0.5 coverage over MgO, the energy of the Heli H2 molecules right on top of a cation can be measured. This experimental energy can then be used for the following purposes: 1) as a reference energy for the ab initio quantum mechanical 321 calculations, and 2) as a reference energy to find a better set of dispersion and repulsion parameters in our semi empirical model and avoid problems similar to the one in our study for both MgO and NaCl substrates (see section 4.3.2.2). In addition, one of the main problems about using LiF as a substrate is the lack of data available for the binding energy of the ortho- and para-H2 molecules to the surface from either experiment or theory. Therefore, both experiment and ab initio quantum mechanical calculations should be done on these surfaces to find the binding energies of these H2 molecules over the surface. Then our models can be used in the molecular simulation studies to calculate the properties of the H2/LiF systems and answer questions97 that have remained in the field for more than 15 years. 9.2.1.2 The zero-point energy Since the temperature of the system is very low, the inclusion of the zero-point energy is crucial for accurate calculation of the properties of the system. In order to properly include this energy in the calculations, accurate quantum mechanical potentials of the H2 molecules over the ionic surfaces are required. Then by using these potentials and following Briquez26, one can find the Hamiltonian for the zero-point motion of the H2 molecule and include it explicitly in the model. It should however be mentioned that, in the formulas of the Briquez, the zero-point-motion Hamiltonian is not quite compatible with the formula proposed in this study, which is an integration of three spherical harmonic functions (see Eq. (A1-1) in section A.1). Thus, the first step should be to find a way to convert Briquez’ Hamiltonian to a one that can be utilized in our model. 322 9.2.2 The methods 9.2.2.1 The NDPT and DPT methods As was explained in Chapter 4, for the H2-surface interactions and for all the H2 species, fifth-order PT correction terms were estimated. But, in order to get better-converged energies in the NDPT model (Model II), fifth-order PT terms can be calculated and added to the calculations. Also, even though it was shown that the effect of using the 𝑉௦௘௣ case instead of the 𝑉௧௢௧ case in our simulations is negligible, it is better to use the 𝑉௧௢௧ formulation in all of the PT correction terms higher than the first order. Also, for the ortho-H2 molecules, the complete version of the DPT model (Model III) must be developed, which means the PT equations for the first-, second-, third-, fourth-, and fifth-order correction terms in energy. Also, the equations for the wavefunctions must also be developed to see how the distribution of the ortho-H2 molecule varies when the molecule moves over the surface. On the other hand, as was shown in Chapter 5, a more complete version of the H2-H2 interactions should also be developed, which is when higher-order interactions are included in the potential. 9.2.2.2 MC simulations As was explained in Chapter 6, the best ensemble for the adsorption process is the μVT ensemble (GCMC simulations). For better and easier comparisons with the experimental data, MC-NVT simulations were used to validate the proposed models. But in the future, first experimental studies must be done on the H2 adsorbed layers to find the accurate pressure of the 323 systems. In fact, the MC-μVT simulation is a better tool to study these layers if a specific coverage or structure is not studied. 324 Bibliography 1. Fukutani, K. & Sugimoto, T. 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(Rensselaer Polytechnic Institute, 1962). 334 Appendix A The integral of the product of three spherical harmonics In order to evaluate integrals of the product of three spherical harmonic functions, the following formula is used98,99: න න ϒ௃భ,௠಻భ (𝜃, 𝜙) ϒ௃మ,௠಻మ (𝜃, 𝜙) ϒ௃య,௠಻య (𝜃, 𝜙) sin 𝜃 (𝑑𝜃)𝑑𝜙 గ଴ଶగ଴= ඨ(2𝐽ଵ + 1)(2𝐽ଶ + 1)(2𝐽ଷ + 1)4𝜋 . ൬𝐽ଵ 𝐽ଶ 𝐽ଷ𝑚௃భ 𝑚௃మ 𝑚௃య൰ . ቀ𝐽ଵ 𝐽ଶ 𝐽ଷ0 0 0ቁ (A1-1) where the matrices are the Wigner 3-j symbols and are calculated using the following equation98: ൬𝐾ଵ 𝐾ଶ 𝐾ଷ𝑚௄భ 𝑚௄మ 𝑚௄య൰= (−1)௄భି௄మି௠಼య . 1ඥ(2𝐾ଷ + 1) . 𝐶(𝐾ଵ, 𝑚௄భ , 𝐾ଶ, 𝑚௄మ , 𝐾ଷ, −𝑚௄య) (A1-2) where 𝐶(𝐾ଵ, 𝑚௄భ , 𝐾ଶ, 𝑚௄మ , 𝐾ଷ, 𝑚௄య) is the Clebsch-Gordan coefficient. These coefficients are calculated using the following formula52,100: 𝐶(𝐾ଵ, 𝑚௄భ , 𝐾ଶ, 𝑚௄మ , 𝐾ଷ, 𝑚௄య)= 𝛿௠಼భା௠಼మ ,௠಼య ඥ(2𝐾ଷ + 1)/(𝐾ଵ + 𝐾ଶ + 𝐾ଷ + 1)! × ඥ(𝐾ଵ + 𝐾ଶ − 𝐾ଷ)! (𝐾ଶ + 𝐾ଷ − 𝐾ଵ)! (𝐾ଷ + 𝐾ଵ − 𝐾ଶ)! × ට൫𝐾ଵ + 𝑚௄భ൯! ൫𝐾ଵ − 𝑚௄భ൯! ൫𝐾ଶ + 𝑚௄మ൯! ൫𝐾ଶ − 𝑚௄మ൯! ൫𝐾ଷ + 𝑚௄య൯! ൫𝐾ଷ − 𝑚௄య൯! × ෍{(−1)௔(𝐾ଵ + 𝐾ଶ − 𝐾ଷ − 𝑎)! (𝐾ଷ − 𝐾ଵ − 𝑚௄మ + 𝑎)! ௔ ×1൫𝐾ଷ − 𝐾ଶ + 𝑚௄భ + 𝑎൯! ൫𝐾ଵ − 𝑚௄భ − 𝑎൯! ൫𝐾ଶ + 𝑚௄మ − 𝑎൯! 𝑎!} (A1-3) 335 where 𝛿௜,௝ is the Kronecker delta, and the index of summation, 𝑎, takes on values in the following range: max (0, 𝐾ଶ − 𝐾ଷ − 𝑚௄భ , 𝐾ଵ − 𝐾ଷ + 𝑚௄మ) ≤ 𝑎 ≤ min (𝐾ଵ + 𝐾ଶ − 𝐾ଷ, 𝐾ଵ − 𝑚௄భ , 𝐾ଶ + 𝑚௄మ) On the one hand, the Clebsch-Gordan coefficient is defined only if 𝐾ଵ, 𝑚௄భ , 𝐾ଶ, 𝑚௄మ , 𝐾ଷ, 𝑎𝑛𝑑 𝑚௄య are either integer or half-integer. Also, 𝐾ଵ, 𝐾ଶ, 𝑎𝑛𝑑 𝐾ଷ must be non-negative52. Besides, these conditions must be satisfied as well; ห𝑚௄భห ≤ 𝐾ଵ, ห𝑚௄మห ≤𝐾ଶ, 𝑎𝑛𝑑 ห𝑚௄యห ≤ 𝐾ଷ. Moreover, 𝐾ଵ + 𝑚௄భ , 𝐾ଶ + 𝑚௄మ , 𝐾ଷ + 𝑚௄య , 𝑎𝑛𝑑 𝐾ଵ + 𝐾ଶ + 𝐾ଷ must be either integer or half-integer non-negative numbers52. On the other hand, the Clebsch-Gordan coefficient is zero if 𝑚௄భ + 𝑚௄మ − 𝑚௄య ≠ 0 or 𝐾ଷ <|𝐾ଵ − 𝐾ଶ| or when 𝐾ଷ > 𝐾ଵ + 𝐾ଶ 52. 336 An example for the derivation of the 𝑿𝒊 equations in Table 3-1 The starting point is the potential equation proposed by Hill53,54,89: 𝑉෠ுమି௦௨௥ொ 𝑜𝑟 𝑉෠ொ௘௟௦= −Θටπ30 ෍{ 𝐴௡௠ exp[𝑖𝐺(𝑛𝑥 + 𝑚𝑦)] × exp (−𝛾௡௠ 𝑧)[ exp (𝑖 2 𝜉) ϒଶଶ(𝜃, 𝜙)ᇱ௡௠+ exp (𝑖 2 𝜉) ϒଶଶ(𝜃, 𝜙)) + exp (𝑖 2 𝜉) ϒଶଶ(𝜃, 𝜙)) + exp (𝑖 2 𝜉) ϒଶଶ(𝜃, 𝜙))+ exp (𝑖 2 𝜉) ϒଶଶ(𝜃, 𝜙)] } The prime in the summation means that the {0,0} vector energy vanishes to zero. Now set 𝑀௘௟௦ = −Θටగଷ଴ , where Θ is the quadrupole moment of the H2 molecule78,79. Only show the derivation for the “𝑎𝑙𝑙 |2|, |3|” set is shown. This set means all the {+2,+3}, {+2,–3}, {–2,+3}, and {–2,–3} Fourier terms (or vectors) are used in the calculation of the energy. Therefore, for this set Hill’s equation is written as: 𝑉෠௘௟௦ = 𝑀 × { +𝐴ଶଷ. exp൫𝑖𝐺(2𝑥 + 3𝑦)൯ . exp(−𝛾ଶଷ 𝑧) . ൥൬2 − 3𝑖√13൰ଶΥଶଶ + ቆ√132 − 3𝑖ቇଶΥଶଶഥ − 2𝑖 ൬2 − 3𝑖√13൰ Υଶଵ + 2𝑖 ቆ√132 − 3𝑖ቇ Υଶଵഥ − √6 Υଶ଴൩ +𝐴ଶଷഥ. exp (𝑖𝐺(2𝑥 − 3𝑦). 𝑒𝑥𝑝(−𝛾ଶଷഥ 𝑧) . ൥൬2 + 3𝑖√13൰ଶΥଶଶ + ቆ√132 + 3𝑖ቇଶΥଶଶഥ − 2𝑖 ൬2 + 3𝑖√13൰ Υଶଵ + 2𝑖 ቆ√132 + 3𝑖ቇ Υଶଵഥ − √6 Υଶ଴൩ +𝐴ଶഥ ଷ. exp൫𝑖𝐺(−2𝑥 + 3𝑦)൯. exp(−𝛾ଶഥ ଷ 𝑧) 337 ൥൬−2 − 3𝑖√13൰ଶΥଶଶ + ቆ√13−2 − 3𝑖ቇଶΥଶଶഥ − 2𝑖 ൬−2 − 3𝑖√13൰ Υଶଵ + 2𝑖 ቆ√13−2 − 3𝑖ቇ Υଶଵഥ − √6 Υଶ଴}൩ +𝐴ଶഥ ଷഥ. exp൫𝑖𝐺(−2𝑥 − 3𝑦)൯. exp(−𝛾ଶഥ ଷഥ 𝑧) ൥൬−2 + 3𝑖√13൰ଶΥଶଶ + ቆ√13−2 + 3𝑖ቇଶΥଶଶഥ − 2𝑖 ൬−2 + 3𝑖√13൰ Υଶଵ + 2𝑖 ቆ√13−2 + 3𝑖ቇ Υଶଵഥ − √6 Υଶ଴}൩ } According to Kroes and Mowrey53,54,89, in Hill’s equation, 𝛾ଶଷ = 𝛾ଶଷഥ = 𝛾ଶഥଷ = 𝛾ଶഥ ଷഥ and 𝐴ଶଷ = 𝐴ଶଷഥ = 𝐴ଶഥଷ = 𝐴ଶഥ ଷഥ, therefore: 𝑉෠௘௟௦ = 𝑀. 𝐴ଶଷ. exp(−𝛾ଶଷ 𝑧) × { + exp൫𝑖𝐺(2𝑥 + 3𝑦)൯ . ቈ൬4 − 12𝑖 − 913൰ Υଶଶ + ൬134 − 12𝑖 − 9൰ Υଶଶഥ − 2𝑖 ൬2 − 3𝑖√13൰ Υଶଵ + 2𝑖 ቆ√132 − 3𝑖ቇ Υଶଵഥ − √6 Υଶ଴቉ + exp൫𝑖𝐺(2𝑥 − 3𝑦)൯ ቈ൬4 + 12𝑖 − 913൰ Υଶଶ + ൬134 + 12𝑖 − 9൰ Υଶଶഥ − 2𝑖 ൬2 + 3𝑖√13൰ Υଶଵ + 2𝑖 ቆ√132 + 3𝑖ቇ Υଶଵഥ − √6 Υଶ଴቉ + exp൫−𝑖𝐺(2𝑥 − 3𝑦)൯ ቈ൬4 + 12𝑖 − 913൰ Υଶଶ + ൬134 + 12𝑖 − 9൰ Υଶଶഥ + 2𝑖 ൬2 + 3𝑖√13൰ Υଶଵ − 2𝑖 ቆ√132 + 3𝑖ቇ Υଶଵഥ − √6 Υଶ଴቉ + exp൫−𝑖𝐺(2𝑥 + 3𝑦)൯ ቈ൬4 − 12𝑖 − 913൰ Υଶଶ + ൬134 − 12𝑖 − 9൰ Υଶଶഥ + 2𝑖 ൬2 − 3𝑖√13൰ Υଶଵ − 2𝑖 ቆ√132 − 3𝑖ቇ Υଶଵഥ − √6 Υଶ଴቉ } 338 𝑉෠௘௟௦ = 𝑀. 𝐴ଶଷ. exp (−𝛾ଶଷ 𝑧) × { + ൬−5 − 12𝑖13൰ Υଶଶ൫𝑒௜ீ(ଶ௫ାଷ௬) + 𝑒ି௜ (ଶ௫ାଷ௬)൯ + ൬13−5 − 12𝑖൰ Υଶଶഥ൫𝑒௜ீ(ଶ௫ାଷ௬) + 𝑒ି௜ (ଶ௫ାଷ௬)൯ −2𝑖 ൬2 − 3𝑖√13൰ Υଶଵ൫𝑒௜ீ(ଶ௫ାଷ௬) − 𝑒ି௜ீ(ଶ௫ାଷ௬)൯ + 2𝑖 ቆ√132 − 3𝑖ቇ Υଶଵഥ൫𝑒௜ீ(ଶ௫ାଷ௬) − 𝑒ି௜ீ(ଶ௫ାଷ௬)൯ −√6 Υଶ଴൫𝑒௜ீ(ଶ௫ାଷ௬) + 𝑒ି௜ீ(ଶ௫ାଷ௬)൯ + ൬−5 + 12𝑖13൰ Υଶଶ൫𝑒௜ீ(ଶ௫ିଷ௬) − 𝑒ି௜ (ଶ௫ିଷ௬)൯ + ൬13−5 + 12𝑖൰ Υଶଶഥ൫𝑒௜ீ(ଶ௫ିଷ௬) + 𝑒ି௜ீ(ଶ௫ିଷ௬)൯ −2𝑖 ൬2 + 3𝑖√13൰ Υଶଵ൫𝑒௜ீ(ଶ௫ିଷ௬) − 𝑒ି௜ீ(ଶ௫ିଷ௬)൯ + 2𝑖 ቆ√132 + 3𝑖ቇ Υଶଵഥ൫𝑒௜ீ(ଶ௫ିଷ௬) − 𝑒ି௜ (ଶ௫ିଷ௬)൯ −√6 Υଶ଴൫𝑒௜ீ(ଶ௫ିଷ௬) + 𝑒ି௜ (ଶ௫ିଷ௬)൯ } But ▪ −1312𝑖 + 5×12𝑖 − 512𝑖 − 5= −1312𝑖 − 5−144 − 25=12𝑖 − 513 ▪ √132 − 3𝑖×2 + 3𝑖2 + 3𝑖= √13.2 + 3𝑖4 + 9=3𝑖 + 2√13 ▪+1312𝑖 − 5×12𝑖 + 512𝑖 + 5= 13 ×12𝑖 + 5−144 − 25= −12𝑖 + 513 ▪√133𝑖 + 2×3𝑖 − 23𝑖 − 2= √13.3𝑖 − 2−9 − 4= −3𝑖 − 2√13 Therefore, it can be written: 𝑉෠௘௟௦ = 𝑀. 𝐴ଶଷ. exp (−𝛾ଶଷ 𝑧) × { −12𝑖 + 513. Υଶଶ. 2 cos൫𝐺(2𝑥 + 3𝑦)൯ +12𝑖 − 513Υଶଶഥ. 2 cos൫𝐺(2𝑥 + 3𝑦)൯ 339 +2𝑖 ൬3𝑖 − 2√13൰ Υଶଵ. 2𝑖. sin൫𝐺(2𝑥 + 3𝑦)൯ + 2𝑖3𝑖 + 2√13. Υଶଵഥ. 2𝑖. sin ൫𝐺(2𝑥 + 3𝑦)൯ −√6 Υଶ଴. 2. cos൫𝐺(2𝑥 + 3𝑦)൯ +12𝑖 − 513. Υଶଶ. 2 cos൫𝐺(2𝑥 − 3𝑦)൯ −12𝑖 + 513Υଶଶഥ. 2 cos൫𝐺(2𝑥 − 3𝑦)൯ −2𝑖 ൬3𝑖 + 2√13൰ Υଶଵ. 2𝑖. sin൫𝐺(2𝑥 − 3𝑦)൯ − 2𝑖3𝑖 − 2√13. Υଶଵഥ. 2𝑖. sin ൫𝐺(2𝑥 − 3𝑦)൯ −√6 Υଶ଴. 2. cos൫𝐺(2𝑥 − 3𝑦)൯ } 𝑉෠௘௟௦ = 𝑀. 𝐴ଶଷ. exp (−𝛾ଶଷ 𝑧) × { −513. Υଶଶ. 2. cos൫𝐺(2𝑥 + 3𝑦)൯ −12𝑖13. Υଶଶ. 2 cos൫𝐺(2𝑥 + 3𝑦)൯ +12𝑖13. Υଶଶഥ. 2 cos൫𝐺(2𝑥 + 3𝑦)൯ −513Υଶଶഥ. 2 cos൫𝐺(2𝑥 + 3𝑦)൯ −6𝑖√13. Υଶଵ. 2sin൫𝐺(2𝑥 + 3𝑦)൯ +413Υଶଵ. 2sin൫𝐺(2𝑥 + 3𝑦)൯ −6𝑖√13Υଶଵഥ. 2sin൫𝐺(2𝑥 + 3𝑦)൯ −413Υଶଵഥ. 2sin൫𝐺(2𝑥 + 3𝑦)൯ −√6 Υଶ଴. 2 cos൫𝐺(2𝑥 + 3𝑦)൯ +12𝑖13. Υଶଶ. 2 cos൫𝐺(2𝑥 − 3𝑦)൯ −513. Υଶଶ. 2 cos൫𝐺(2𝑥 − 3𝑦)൯ −12𝑖13Υଶଶഥ. 2 cos൫𝐺(2𝑥 − 3𝑦)൯ −513. Υଶଶഥ. 2 cos൫𝐺(2𝑥 − 3𝑦)൯ +6𝑖√13. Υଶଵ. 2sin൫𝐺(2𝑥 − 3𝑦)൯ +4√13Υଶଵ. 2sin൫𝐺(2𝑥 − 3𝑦)൯ +6𝑖√13. Υଶଵഥ. 2sin൫𝐺(2𝑥 − 3𝑦)൯ −4√13Υଶଵഥ. 2sin൫𝐺(2𝑥 − 3𝑦)൯ 340 −√6 Υଶ଴. 2 cos൫𝐺(2𝑥 − 3𝑦)൯ } 𝑉෠௘௟௦ = 𝑀. 𝐴ଶଷ. exp (−𝛾ଶଷ 𝑧) × { −1013Υଶଶൣcos൫𝐺(2𝑥 + 3𝑦)൯ + cos൫𝐺(2𝑥 − 3𝑦)൯൧ −24𝑖13Υଶଶൣcos൫𝐺(2𝑥 + 3𝑦)൯ − cos൫𝐺(2𝑥 − 3𝑦)൯൧ +24𝑖13Υଶଶഥൣcos൫𝐺(2𝑥 + 3𝑦)൯ − cos൫𝐺(2𝑥 − 3𝑦)൯൧ −1013Υଶଶഥൣcos൫𝐺(2𝑥 + 3𝑦)൯ + cos൫𝐺(2𝑥 − 3𝑦)൯൧ −12𝑖√13Υଶଵൣsin൫𝐺(2𝑥 + 3𝑦)൯ − sin൫𝐺(2𝑥 − 3𝑦)൯൧ +8√13Υଶଵ. ൣsin൫𝐺(2𝑥 + 3𝑦)൯ + sin൫𝐺(2𝑥 − 3𝑦)൯൧ −12𝑖√13Υଶଵഥൣsin൫𝐺(2𝑥 + 3𝑦)൯ − sin൫𝐺(2𝑥 − 3𝑦)൯൧ −8√13Υଶଵഥ. ൣsin൫𝐺(2𝑥 + 3𝑦)൯ + sin൫𝐺(2𝑥 − 3𝑦)൯൧ −2√6 Υଶ଴ൣcos൫𝐺(2𝑥 + 3𝑦)൯ + cos൫𝐺(2𝑥 − 3𝑦)൯൧ } ⸫ 𝑉෠௘௟௦ = 𝑀. 𝐴ଶଷ. exp (−𝛾ଶଷ 𝑧) × { −1013ൣcos൫𝐺(2𝑥 + 3𝑦)൯ + cos൫𝐺(2𝑥 − 3𝑦)൯൧. (Υଶଶഥ + Υଶଶ) +24𝑖13ൣcos൫𝐺(2𝑥 + 3𝑦)൯ − cos൫𝐺(2𝑥 − 3𝑦)൯൧. (Υଶଶഥ − Υଶଶ) −12𝑖√13ൣsin൫𝐺(2𝑥 + 3𝑦)൯ − sin൫𝐺(2𝑥 − 3𝑦)൯൧. (Υଶଵഥ + Υଶଵ) 341 −8√13ൣsin൫𝐺(2𝑥 + 3𝑦)൯ + sin൫𝐺(2𝑥 − 3𝑦)൯൧. (Υଶଵഥ − Υଶଵ) −2√6 Υଶ଴ൣcos൫𝐺(2𝑥 + 3𝑦)൯ + cos൫𝐺(2𝑥 − 3𝑦)൯൧Υଶ଴ } 342 Derivation of the first-order 𝑬𝑸𝒆𝒍𝒔 energy Here it is shown how Eq. (3-15) is developed. First, it should be noted that, the reference wavefunction, ϒ௄,௠಼, for a Cart molecule is the ϒଵ,଴ function. To calculate the first-order energy, 𝐸ொ௘௟௦(ଵ) , Eq. (3-12) for the operator 𝑉෠ொ௘௟௦ is plugged into Eq. (3-9): 𝐸ொ௘௟௦಴ೌೝ೟(ଵ) =< Υଵ଴ห𝑉෠ொ௘௟௦หΥଵ଴ >= < Υଵ଴|{𝑋଴ (Υଶଶഥ + Υଶଶ) + 𝑖𝑋ଵ (Υଶଶഥ − Υଶଶ) + 𝑋ଶ (Υଶଵഥ − Υଶଵ) + 𝑖𝑋ଷ (Υଶଵഥ + Υଶଵ) + 𝑋ସ Υଶ଴}|Υଵ଴ > = 𝑋଴ < Υଵ଴|Υଶଶഥ|Υଵ଴ > +𝑋଴ < Υଵ଴|Υଶଶ|Υଵ଴ > +. . . +𝑋ସ < Υଵ଴|Υଶ଴|Υଵ଴ > As seen, to evaluate the first-order energy, the integrals of the products of three spherical harmonics that appear in the equation of energy must be evaluated. But all these integrals vanish to zero except the ϒଶ,଴ term. This is because for the rest of the terms, 𝑚௄భ + 𝑚௄మ − 𝑚௄య ≠ 0, which based on the Clebsch-Gordan coefficient properties in section A.1, it results in 𝐶 = 0. Therefore, the first-order energy for the Cart molecule can be written as: 𝐸ொ௘௟௦಴ೌೝ೟(ଵ) = 𝑋ସ < Υଵ଴|Υଶ଴|Υଵ଴ > Using Eq. (A1-1) in section A.1, < ϒଵ,଴|ϒଶ,଴|ϒଵ,଴ > = 0.252. Thus: 𝐸ொ௘௟௦಴ೌೝ೟(ଵ) = 𝑋ସ × 0.252 which is Eq. (3-15). 343 The 𝑫(𝒊)𝒈 functions used in section 3.1.2.1.1 In order to develop Eq. (3-23) from Briquez et al.’s 26 equation, this equation must be simplified more. This was done by grouping different sets of variables and constants in their equation into functions that are called 𝐷(𝑖)௚ functions and are listed in the following table. In these functions, 𝐶଴௦ (ఈ) is either the 𝐶଺ or 𝐶ଵଶ coefficient in the LJ interactions with 𝛼 equal to either 6 or 12, and 𝑠 is either the cation or the anion in the crystal. 𝛾଴ is the anisotropy factor and is equal to 0.105 a.u. for H2. 𝑔 =ଶగ௖ √ଶ, and 𝑧଴௣ = 𝑧଴– 𝑝 (𝑐/√2), where 𝑧଴ is the vertical distance of the H2 molecule from the top layer of ions in the crystal (surface plane), 𝑝 is the number of the ionic layer in the crystal, and 𝐾௡ is the modified Bessel function of the second kind. Table A4-1 The 𝑫(𝒊)𝒈 functions used in Table 3-3 of section 3.1.2.1.1. 𝐷(0) =2 𝜋ଷ ଶൗ𝑐ଶ , 𝐷(1) = (−1)ఈ ଶൗ . 𝐶଴௦ (ఈ) 𝐷(2) =2 + 𝛾଴(𝛼 − 2)𝑧଴௣ఈିଶ , 𝐷(5)௚ =2 . (1 + 𝛾଴)ቀ𝛼2 − 1ቁ ! . ቆ𝑔2. 𝑍଴௣ቇ . 𝐾ఈଶିଵ 𝐷(6)௚ =ଷ ఊబቀഀమିଵቁ! . ൬ ௚ଶ ௓బ೛൰ఈଶൗ , 𝐷(7)௚ =ଶ√ହ. 𝑧଴௣ ቆ𝑧଴௣. 𝐾ഀమ(𝑔. 𝑧଴௣) −ఈଷ௚. 𝐾ഀమିଵ൫𝑔. 𝑧଴௣൯ቇ 𝐷(8)௚ =𝑧଴௣ଶ𝑔. ඨ215 . 𝐾ఈଶିଵ (𝑔. 𝑧଴௣) , 𝐷(9)௚ =𝑧଴௣ଶ𝑔ଶ. ඨ215. 𝐾ఈଶିଵ(𝑔. 𝑧଴௣) 344 Derivation of the second-order dispersion and repulsion energy for Para Following Eq. (2-8), the second-order dispersion and repulsion energy in the NDPT procedure for the Para molecule can be written in the following form: 𝐸஽ோುೌೝೌ(ଶ) =𝑉଴ଶ𝑉ଶ଴𝐸଴ − 𝐸ଶ+𝑉଴ସ𝑉ସ଴𝐸଴ − 𝐸ସ+. .. (A5-1) Using the potential operator of Eq. (3-23), for 𝐽 = 4,6,8, . .., all the 𝑉଴௃ terms in Eq. (A5-1) vanish to zero. This happens because by following the Clebsch-Gordan coefficient properties in section A.1, 𝐾ଷ > 𝐾ଵ + 𝐾ଶ in the integrals of the form < Υ଴଴|𝑒𝑖𝑡ℎ𝑒𝑟 Υଶ௠಻𝑜𝑟 Υ଴଴|Υସ௠಻ >, which results in 𝐶 = 0. Thus, Eq. (A5-1) becomes: 𝐸஽ோುೌೝೌ(ଶ) =𝑉଴ଶ𝑉ଶ଴𝐸଴ − 𝐸ଶ (A5-2) Also, for simplicity 𝐷(0) . ∑ ∑ 𝐷(1) ∑௣ୀ଴.ିଵ,... ௦ఈୀ଺,ଵଶ is replaced by 𝑓஽ோ in the following steps. Therefore, Eq. (A5-2) is written as: 𝐸஽ோುೌೝೌ(ଶ)=< Υଶଶഥ| 𝑓஽ோ [𝐾஽ோ(2) + 𝑖 𝐾஽ோ(3)]Υଶଶഥ |Υ଴଴ > < Υ଴଴| 𝑓஽ோ [𝐾஽ோ(2) − 𝑖 𝐾஽ோ(3)]Υଶଶ |Υଶଶഥ >𝐸଴଴ − 𝐸ଶଶഥ+< Υଶଵഥ| 𝑓஽ோ [𝐾஽ோ(5) + 𝑖 𝐾஽ோ(4)]Υଶଵഥ |Υ଴଴ > < Υ଴଴| 𝑓஽ோ [−𝐾஽ோ(5) + 𝑖 𝐾஽ோ(4)]Υଶଵ |Υଶଵഥ >𝐸଴଴ − 𝐸ଶଵഥ+< Υଶ଴| 𝑓஽ோ 𝐾஽ோ(1)Υଶ଴|Υ଴଴ > < Υ଴଴| 𝑓஽ோ 𝐾஽ோ(1)Υଶ଴ |Υଶ଴ >𝐸଴଴ − 𝐸ଶ଴ + < Υଶଵ| 𝑓஽ோ [−𝐾஽ோ(5) + 𝑖 𝐾஽ோ(4)]Υଶଵ |Υ଴଴ > < Υ଴଴| 𝑓஽ோ [𝐾஽ோ(5) + 𝑖 𝐾஽ோ(4)]Υଶଵഥ |Υଶଵ >𝐸଴଴ − 𝐸ଶଵ + < Υଶଶ| 𝑓஽ோ [𝐾஽ோ(2) − 𝑖 𝐾஽ோ(3)]Υଶଶ |Υ଴଴ > < Υ଴଴| 𝑓஽ோ [𝐾஽ோ(2) + 𝑖 𝐾஽ோ(3)]Υଶଶഥ |Υଶଶ >𝐸଴଴ − 𝐸ଶଶ 345 where in each integral the spherical harmonic function between the bars only that function survives in the operator. For example, in the integral < Υଶଶഥ| 𝑓஽ோ . [𝐾஽ோ(2) −𝑖 𝐾஽ோ(3)]Υଶଶഥ |Υ଴଴ >, only the Υଶଶഥ term in the operator 𝑉෠஽ோ survives (based on the properties of the Clebsch-Gordan coefficients as well as Eq. (A1-1) and Eq. (A1-2) given in section A.1). Also, note that, all the 𝐸ଶ௠಻ energies are the same (because they only depend on 𝐽) and then can all be shown simply by 𝐸ଶ. Thus: 𝐸஽ோುೌೝೌ(ଶ) = +0.282 × 𝑓஽ோ [𝐾஽ோ(2) + 𝑖 𝐾஽ோ(3)] × 0.282 × 𝑓஽ோ [𝐾஽ோ(2) − 𝑖 𝐾஽ோ(3)]𝐸଴ − 𝐸ଶ +0.282 × 𝑓஽ோ[𝐾஽ோ(5) + 𝑖 𝐾஽ோ(4)] × (−0.282) × 𝑓஽ோ[−𝐾஽ோ(5) + 𝑖 𝐾஽ோ(4)]𝐸଴ − 𝐸ଶ + 0.282 × 𝑓஽ோ 𝐾஽ோ(1) × 0.282 × 𝑓஽ோ 𝐾஽ோ(1)𝐸଴ − 𝐸ଶ + 0.282 × 𝑓஽ோ [−𝐾஽ோ(5) + 𝑖 𝐾஽ோ(4)] × (−0.282) × 𝑓஽ோ [𝐾஽ோ(5) + 𝑖 𝐾஽ோ(4)]𝐸଴ − 𝐸ଶ +0.282 × 𝑓஽ோ [𝐾஽ோ(2) − 𝑖 𝐾஽ோ(3)] × 0.282 × 𝑓஽ோ [𝐾஽ோ(2) + 𝑖 𝐾஽ோ(3)]𝐸଴ − 𝐸ଶ 𝐸஽ோುೌೝೌ(ଶ) = +0.282ଶ𝐸଴ − 𝐸ଶ× {[𝑓஽ோ 𝐾஽ோ(2)]ଶ+[𝑓஽ோ 𝐾஽ோ(3)]ଶ} +−0.282ଶ𝐸଴ − 𝐸ଶ× {−[𝑓஽ோ 𝐾஽ோ(5)]ଶ−[𝑓஽ோ 𝐾஽ோ(4)]ଶ} 346 +0.282ଶ𝐸଴ − 𝐸ଶ× {[𝑓஽ோ 𝐾஽ோ(1)]ଶ} +−0.282ଶ𝐸଴ − 𝐸ଶ× {−[𝑓஽ோ 𝐾஽ோ(5)]ଶ−[𝑓஽ோ 𝐾஽ோ(4)]ଶ} +0.282ଶ𝐸଴ − 𝐸ଶ× {[𝑓஽ோ 𝐾஽ோ(2)]ଶ+[𝑓஽ோ 𝐾஽ோ(3)]ଶ} And the sum of these terms results in Eq. (3-27) in section 3.1.2.1.1. 347 Derivation of the 𝑽𝟎𝟐𝑽𝟐𝟐𝑽𝟐𝟎 term in (3-45) for the Para molecule For simplicity, (𝐾𝑚௄ห2𝑚௃ห𝐿𝑚௅) is used to show each one of the 𝑉௄௃ terms in the 𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴ equation. The bolded term between the bars show the Υଶ௠಻ term that survives in the 𝑉෠ொ௘௟௦ operator (Eq. (3-12)). For example: (00|𝟐𝟐|22ത) = 0.282 × (𝑋଴ + 𝑖𝑋ଵ), where 0.282 is in fact the integral of 〈Υ଴଴|Υଶଶ|Υଶଶഥ〉 and (𝑋଴ + 𝑖𝑋ଵ) is the associated spatial-dependent Υଶଶ term in the equation of the 𝑉෠ொ௘௟௦ operator. Also, for simplicity, all the numbers obtained from the integrals are only shown down to the third decimal place, but in the code all the actual digits after the decimal point have been used in the calculations. 𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴ = + (00|𝟐𝟐|22ത) × [(22ത|𝟐𝟎|22ത). (22ത|𝟐𝟐ഥ|00) + (22ത|𝟐𝟏ഥ|21ത)(21ത|𝟐𝟏ഥ|00)+ (22ത|𝟐𝟐ഥ|20). (20|𝟐𝟎|00)] + (00|𝟐𝟏|21ത) × [(21ത|𝟐𝟏|22ത). (22ത|𝟐𝟐ഥ|00) + (21ത|𝟐𝟎|21ത)(21ത|𝟐𝟏ഥ|00)+ (21ത|𝟐𝟏ഥ|20). (20|𝟐𝟎|00) + (21ത|𝟐𝟐ഥ|21)(21|𝟐𝟏|00)] +(00|𝟐𝟎|20) × [(20|22|22ത). (22ത|𝟐𝟐ഥ|00) + (20|𝟐𝟏|21ത)(21ത|𝟐𝟏ഥ|00)+ (20|𝟐𝟎|00). (20|𝟐𝟎|00) + (20|𝟐𝟏ഥ|21)(21|𝟐𝟏|00)+ (20|𝟐𝟐ഥ|22)(22|𝟐𝟐|00)] +(00|𝟐𝟏ഥ|21) × [(21|𝟐𝟐|21ത). (21ത|𝟐𝟏ഥ|00) + (21|𝟐𝟏|20)(20|𝟐𝟎|00)+ (21|𝟐𝟎|21). (21|𝟐𝟏|00) + (21|𝟐𝟏ഥ|22)(22|𝟐𝟐|00)] +(00|𝟐𝟐ഥ|22) × [(22|𝟐𝟐|20). (20|𝟐𝟎|00) + (22|𝟐𝟏|21)(21|𝟐𝟏|00)+ (22|𝟐𝟎|22)(22|𝟐𝟐|00)] 348 𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴ = +0.282. (𝑋଴ + 𝑖𝑋ଵ) × (0.180). 𝑋ସ × 0.282. (𝑋଴ + 𝑖𝑋ଵ) +0.282. (𝑋଴ + 𝑖𝑋ଵ) × 0.220.× (𝑋ଶ + 𝑖𝑋ଷ)0.282. (𝑋ଶ + 𝑖𝑋ଷ) +0.282. (𝑋଴ + 𝑖𝑋ଵ) × (−180). (𝑋଴ + 𝑖𝑋ଵ) × 0.282. 𝑋ସ −0.282. (−𝑋ଶ + 𝑖𝑋ଷ) × (−0.220). (−𝑋ଶ + 𝑖𝑋ଷ) × 0.282. (𝑋଴ − 𝑖𝑋ଵ) −0.282. (−𝑋ଶ + 𝑖𝑋ଷ) × (9.01 × 10ିଶ). (−𝑋ଶ + 𝑖𝑋ଷ) × 0.282. (𝑋଴ − 𝑖𝑋ଵ) −0.282. (−𝑋ଶ + 𝑖𝑋ଷ) × (9.01 × 10ିଶ). (−𝑋ଶ + 𝑖𝑋ଷ) × 0.282. 𝑋ସ −0.282. (−𝑋ଶ + 𝑖𝑋ଷ) × (−0.226). (−𝑋଴ − 𝑖𝑋ଵ) × 0.282. (−𝑋ଶ + 𝑖𝑋ଷ) +0.282. 𝑋ସ × (𝑋ସ) × (−0.180). (𝑋଴ + 𝑖𝑋ଵ) × 0.282(𝑋଴ − 𝑖𝑋ଵ) +0.282. 𝑋ସ × (−9.01 × 10ିଶ). (−𝑋ଶ + 𝑖𝑋ଷ) × 0.282. (𝑋ଶ − 𝑖𝑋ଷ) +0.282. 𝑋ସ × 0.180. 𝑋ସ × 0.282. 𝑋ସ +0.282. 𝑋ସ × (−9.01 × 10ିଶ) . (𝑋ଶ + 𝑖𝑋ଷ).× 0.282(−𝑋ଶ + 𝑖𝑋ଷ) +0.282. 𝑋ସ × (−0.180) . (𝑋଴ + 𝑖𝑋ଵ).× 0.282(𝑋଴ + 𝑖𝑋ଵ) −0.282. (𝑋ଶ + 𝑖𝑋ଷ) × (−0.220). (𝑋଴ + 𝑖𝑋ଵ) × 0.282. (𝑋ଶ + 𝑖𝑋ଷ) −0.282. (𝑋ଶ + 𝑖𝑋ଷ) × (9.01 × 10ିଶ). (−𝑋ଶ + 𝑖𝑋ଷ) × 0.282. 𝑋ସ −0.282. (𝑋ଶ + 𝑖𝑋ଷ) × (9.01 × 10ିଶ). 𝑋ସ × 0.282. (−𝑋ଶ + 𝑖𝑋ଷ) −0.282. (𝑋ଶ + 𝑖𝑋ଷ) × (−0.220). (𝑋ଶ + 𝑖𝑋ଷ ) × 0.282. (𝑋଴ + 𝑖𝑋ଵ) +0.282. (𝑋଴ − 𝑖𝑋ଵ) × (−0.180). (𝑋଴ + 𝑖𝑋ଵ ) × 0.282. 𝑋ସ +0.282. (𝑋଴ + 𝑖𝑋ଵ) × 0.220 . (−𝑋ଶ + 𝑖𝑋ଷ ) × 0.282. (−𝑋ଶ + 𝑖𝑋ଷ) 𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴ = 349 −0.282ଶ × 0.180 . (𝑋଴ଶ + 𝑋ଵଶ). 𝑋ସ −0.282ଶ × 0.180 . (𝑋଴ଶ + 𝑋ଵଶ). 𝑋ସ +0.282ଶ × 9.01 × 10ିଶ . (𝑋ଶଶ + 𝑋ଷଶ). 𝑋ସ +0.282ଶ × 9.01 × 10ିଶ . (𝑋ଶଶ + 𝑋ଷଶ). 𝑋ସ −0.282ଶ × 0.180 . (𝑋଴ଶ + 𝑋ଵଶ). 𝑋ସ +0.282ଶ × 9.011 × 10ିଶ . (𝑋ଶଶ + 𝑋ଷଶ). 𝑋ସ +0.282ଶ × 0.180 . 𝑋ସଷ +0.282ଶ × 9.011 × 10ିଶ . (𝑋ଶଶ + 𝑋ଷଶ). 𝑋ସ −0.282ଶ × 0.180 . (𝑋଴ଶ + 𝑋ଵଶ). 𝑋ସ +0.282ଶ × 9.011 × 10ିଶ . (𝑋ଶଶ + 𝑋ଷଶ). 𝑋ସ +0.282ଶ × 9.011 × 10ିଶ . (𝑋ଶଶ + 𝑋ଷଶ). 𝑋ସ −0.282ଶ × 0.180 . (𝑋଴ଶ + 𝑋ଵଶ). 𝑋ସ −0.282ଶ × 0.180 . (𝑋଴ଶ + 𝑋ଵଶ). 𝑋ସ +3 × 0.282ଶ × 0.220 × 2 × [𝑋଴(𝑋ଶଶ − 𝑋ଷଶ) − 2𝑋ଵ𝑋ଶ𝑋ଷ] 𝑉଴ଶ . 𝑉ଶଶ . 𝑉ଶ଴ = +( 𝑋଴ଶ + 𝑋ଵଶ). 𝑋ସ. [− 0.282ଶ × 0.180 − 0.282ଶ × 0.180 − 0.282ଶ × 0.180 − 0.282ଶ × 0.180 − 0.282ଶ × 0.180 − 0.282ଶ × 0.180] +( 𝑋ଶଶ + 𝑋ଷଶ). 𝑋ସ. [+0.282ଶ × 9.011 × 10ିଶ + 0.282ଶ × 9.011 × 10ିଶ+ 0.282ଶ × 9.01 × 10ିଶ + 0.282ଶ × 9.01 × 10ିଶ] + 𝑋ସଷ . 0.282ଶ × 0.180 + [𝑋଴ (𝑋ଶଶ − 𝑋ଷଶ) − 2𝑋ଵ𝑋ଶ𝑋ଷ] .0.282ଶ × 0.220 × 2 × 3 350 ⸫ 𝑉଴ଶ . 𝑉ଶଶ. 𝑉ଶ଴ = 0.282ଶ × {−(𝑋଴ଶ + 𝑋ଵଶ). 𝑋ସ × 6 × 0.180 + (𝑋ଶଶ + 𝑋ଷଶ). 𝑋ସ × 4 × 9.011 × 10ିଶ + 𝑋ସଷ × 0.180+ [𝑋଴(𝑋ଶଶ − 𝑋ଷଶ) − 2. 𝑋ଵ𝑋ଶ𝑋ଷ] × 2 × 0.220 × 3} Substituting this last equation into Eq. (3-35) results in Eq. (3-36). 351 The (𝑽𝟏𝟑. 𝑽𝟑𝟑. 𝑽𝟑𝟏) terms used in Eq. (3-38) for the Heli and Cart molecules For Heli: 𝑉ଵଷ. 𝑉ଷଷ. 𝑉ଷଵ =12× { +𝑋ସ. (𝑋଴ଶ + 𝑋ଵଶ). [2 × (8.25 × 10ିଶ)ଶ × 0.126 − 2 × 8.25 × 10ିଶ × 0.206 × 0.202− 2 × 0.202 × 206 × 8.25 × 10ିଶ − 4 × 0.202 × 0.132 × 0.319− 2 × 0.319ଶ × 0.210] −𝑋ସ. (𝑋ଶଶ + 𝑋ଷଶ). [−2 × 0.168 + 4 ×× 0.143 × 5.94 × 10ିଶ − 4 × 0.202− 4 × 0.202 × 0.162 × 0.261] +𝑋ସଷ. [2 × 0.202ଶ × 0.126] +[2. 𝑋଴(𝑋ଶଶ + 𝑋ଷଶ) − 4. 𝑋ଵ𝑋ଶ𝑋ଷ]. [2 × 8.25 × 10ିଶ × 5.94 × 10ିଶ × 0.143+ 2 × 0.143 × 0.188 × 0.261 + 2 × 0.261 × 319] } For Cart: 𝑉ଵଷ𝑉ଷଷ𝑉ଷଵ = { +𝑋ସ. (𝑋଴ଶ + 𝑋ଵଶ). [−4 × 0.184 × 0.188] +𝑋ସ. (𝑋ଶଶ + 𝑋ଷଶ). [+2 × 0.233ଶ × 0.126 − 4 × 0.233 × 5.94 × 10ିଶ × 0.247] +𝑋ସଷ. [0.247ଶ × 0.168] +[2𝑋଴(𝑋ଶଶ − 𝑋ଷଶ) − 4𝑋ଵ𝑋ଶ𝑋ଷ] [−2 × 0.184 × 0.162 × 0.233 + 0.233ଶ × 0.206] } 352 The 𝑽𝟎𝟐𝑽𝟐𝟐𝑽𝟐𝟐𝑽𝟐𝟎 and 𝑽𝟎𝟐𝑽𝟐𝟒𝑽𝟒𝟐𝑽𝟐𝟎 terms used in Eq. (3-40) for the Para molecules 𝑉଴ଶ . 𝑉ଶଶ . 𝑉ଶଶ . 𝑉ଶ଴ = 0.282ଶ × { +𝑋ସଶ . (𝑋଴ଶ + 𝑋ଵଶ). [4 × 0.180ଶ] +𝑋ସ. [𝑋଴(𝑋ଶଶ − 𝑋ଷଶ) − 2𝑋ଵ𝑋ଶ𝑋ଷ] × 2 × [ −4 × 0.180 × 0.220 + 8 × 0.220 × 9.01 × 10ିଶ] +(𝑋ଶଶ + 𝑋ଷଶ)ଶ. [2 × 0.220ଶ + 4 × (9.01 × 10ିଶ)ଶ] +(𝑋ଶଶ + 𝑋ଷଶ) . (𝑋଴ଶ + 𝑋ଵଶ). [+8 × 0.220ଶ − 8 × 9.01 × 10ିଶ × 0.180] +𝑋ସଶ. (𝑋ଶଶ + 𝑋ଷଶ). [8 × (9.01 × 10ିଶ)ଶ + 4 × 9.01 × 10ିଶ × 0.180] +(𝑋଴ଶ + 𝑋ଵଶ)ଶ. [4 × 0.180ଶ] +𝑋ସସ. 0.180ଶ } -------------------------------------------------------------------- 𝑉଴ଶ. 𝑉ଶସ. 𝑉ସଶ. 𝑉ଶ଴ = 0.282ଶ × { +(𝑋଴ଶ + 𝑋ଵଶ)ଶ. [2 × 0.337ଶ + 4 × (4.02 × 10ିଶ)ଶ] +(𝑋଴ଶ + 𝑋ଵଶ). (𝑋ଶଶ + 𝑋ଷଶ). [ 8 × 0.238ଶ + 8 × (9.01 × 10ିଶ)ଶ + 8 × 4.02 × 10ିଶ × 0.161] +𝑋ସଶ. (𝑋଴ଶ + 𝑋ଵଶ). [8 × 0.156ଶ + 4 × 4.02 × 10ିଶ × 0.241] +(𝑋ଶଶ + 𝑋ଷଶ)ଶ. [2 × 0.254ଶ + 4 × 0.161ଶ] +𝑋ସଶ. (𝑋ଶଶ + 𝑋ଷଶ). [+8 × 0.220ଶ − 4 × 0.161 × 0.241] +𝑋ସସ. 0.241ଶ +𝑋ସ. [ 𝑋଴(𝑋ଶଶ − 𝑋ଷଶ) − 2𝑋ଵ𝑋ଶ𝑋ଶ] × 2 × [4 × 0.156 × 0.254 − 8 × 9.01 × 10ିଶ × 0.220] } 353 The 𝑽𝟏𝟑𝑽𝟑𝟑𝑽𝟑𝟑𝑽𝟑𝟏 and 𝑽𝟏𝟑𝑽𝟑𝟓𝑽𝟓𝟑𝑽𝟑𝟏 terms used in Eq. (3-42) for the Heli molecules 𝑉ଵଷ. 𝑉ଷଷ. 𝑉ଷଷ. 𝑉ଷଵ =12 × { +(𝑋଴ଶ + 𝑋ଵଶ)ଶ. [+2 × [(8.25 × 10ିଶ)ଶ × 0.132ଶ + (8.25 × 10ିଶ)ଶ × 0.206ଶ + 0.319ଶ × 0.132ଶ ]+ 4 × 8.25 × 10ିଶ × 0.206 × 0.132 × 0.319 ] − (𝑋଴ଶ + 𝑋ଵଶ). (𝑋ଶଶ + 𝑋ଷଶ). ൣ[+2 × [−(8.25 × 10ିଶ)ଶ × 0.162ଶ−(8.25 × 10ିଶ)ଶ × (5.9 × 10ିଶ)ଶ − 0.261ଶ × 0.188ଶ− 0.319ଶ × 0.210ଶ ]+ 4× [ 8.25 × 10ିଶ × 0.162 × 0.188 × 0.143 + 0.143ଶ × 0.188ଶ− 8.2 × 10ିଶ × 5.9 × 10ିଶ × 0.188 × 0.261− 0.143 × 5.9 × 10ିଶ × 0.206 × 8.25 × 10ିଶ+ 0.261 × 0.162 × 0.206 × 8.2 × 10ିଶ − 0.319 × 0.132 × 5.9 × 10ିଶ × 0.143+ 0.319 × 0.132 × 0.162 × 0.261 − 0.319 × 0.210 × 0.188 × 0.143]൧ +(𝑋ସ)ଶ. (𝑋଴ଶ + 𝑋ଵଶ). ൣ+2 × [ (8.25 × 10ିଶ)ଶ × 0.126ଶ + 0.202ଶ × 0.206ଶ + 0.202ଶ × 0.132ଶ + 0.319ଶ × 0.210ଶ ]+ 4 × [ −0.319 × 0.132 × 0.126 × 0.202 + 0.319 × 0.210 × 0.132 × 0.202]+ 8 × [−8.25 × 10ିଶ × 0.126 × 0.206 × 0.202]൧ +𝑋ସ. [𝑋଴(𝑋ଶଶ − 𝑋ଷଶ) − 2𝑋ଵ𝑋ଶ𝑋ଷ]. 354 2 × ൣ2 × [+ 8.25 × 10ିଶ × 0.126 × 5.9 × 10ିଶ × 0.143− 0.202 × 0.206 × 5.9 × 10ିଶ × 0.143+ 8.25 × 10ିଶ × 5.9 × 10ିଶ × 0.168 × 0.143− 8.2 × 10ିଶ × (5.9 × 10ିଶ)ଶ × 0.202 + 0.266 × 0.188 × 0.168 × 0.143− 0.261 × 0.188 × 5.9 × 10ିଶ × 0.202 + 0.202 × 0.162 × 0.188 × 0.143+ 0.202 × 0.162 × 0.210 × 0.319 − 0.202 × 0.132 × 0.210 × 0.261− 0.319 × 0.210ଶ × 0.261 ]൧ +(𝑋ଶଶ + 𝑋ଷଶ)ଶ. [4 × [−0.143ଶ × (5.9 × 10ିଶ)ଶ − 0.143 × 5.9 × 10ିଶ × 0.162 × 0.206]+ 2 × [+0.261ଶ × 0.162ଶ + 0.261ଶ × 0.210ଶ]] −(𝑋ସ)ଶ. (𝑋ଶଶ + 𝑋ଷଶ). [2 × [−0.143ଶ × 0.168ଶ − 0.202ଶ × (5.9 × 10ିଶ)ଶ − 0.202ଶ × 0.162ଶ]+ 4 × [+0.143 × 0.168 × 5.9 × 10ିଶ × 0.202+ 0.143 × 5.9 × 10ିଶ × 0.126 × 0.202 − 0.202 × 0.126 × 0.162 × 0.261]] +(𝑋ସ)ସ. [2 × (0.202)ଶ × (0.126)ଶ] } -------------------------------------------------------------------- 𝑉ଵଷ.𝑉ଷହ. 𝑉ହଷ. 𝑉ଷଵ =12 × { +(𝑋଴ଶ × 𝑋ଵଶ)ଶ. [+2 × [+( 8.25 × 10ିଶ)ଶ. 0.200ଶ + (8.25 × 10ିଶ)ଶ. (9.2 × 10ିଶ)ଶ + 8.25 × 10ିଶ × 9.2 × 10ିଶ × 0.319 × 2.3 × 10ିଶ 355 + 0.319 × 2.3 × 10ିଶ × 8.25 × 10ିଶ × 9.2 × 10ିଶ +0.319ଶ × (2.3 × 10ିଶ)ଶ + 0.319ଶ × 0.347ଶ] ] − (𝑋଴ଶ + 𝑋ଵଶ). (𝑋ଶଶ + 𝑋ଷଶ). [+2 × [−(8.25 × 10ିଶ)ଶ × 0.245ଶ − (8.2 × 10ିଶ)ଶ × 0.169ଶ − 0.261ଶ × (5.3 × 10ିଶ)ଶ] +4 × [−(8.25 × 10ିଶ)ଶ × 0.245 × 0.143 × 0.141− 8.25 × 10ିଶ × 0.169 × 0.261 × 5.3 × 10ିଶ+ 8.25 × 10ିଶ × 9.2 × 10ିଶ × 0.143 × 0.214+ 8.25 × 10ିଶ × 9.2 × 10ିଶ × 0.261 × 0.117 + 0.143ଶ × 0.141ଶ+ 0.143 × 0.214 × 0.319 × 2.3 × 10ିଶ − 0.143 × 0.141 × 0.319 × 6.3 × 10ିଶ+ 0.261 × 0.117 × 0.319 × 2.3 × 10ିଶ − 0.261 × 0.268 × 0.319 × 0.219] ] − (𝑋ସଶ)ଶ. (𝑋଴ଶ + 𝑋ଵଶ). [+2 × ൣ+(8.25 × 10ିଶ)ଶ × 0.227ଶ + 0.202ଶ × (9.2 × 10ିଶ)ଶ + 0.202ଶ × 0.200ଶ+ 0.319ଶ × 0.126ଶ] + 8 × [8.25 × 10ିଶ × 0.227 × 0.202 × 9.2 × 10ିଶ]൧ +4 × [+0.202 × 0.227 × 0.319 × 2.3 × 10ିଶ + 0.202 × 0.200 × 0.319 × 0.126] ] + 𝑋ସ. [𝑋଴. (𝑋ଶଶ − 𝑋ଷଶ) − 2𝑋ଵ 𝑋ଶ 𝑋ଷ]. 2 × [ 356 +2 × [+8.25 × 10ିଶ × 0.227 × 0.143 × 0.214 − 8.25 × 10ିଶ × 0.169 × 0.143 × 0.239− 8.25 × 10ିଶ × 0.169ଶ × 0.202 + 0.2028.2 × 10ିଶ × 0.143 × 0.214− 0.202 × 0.245 × 0.143 × 0.141 − 0.202 × 0.245 × 0.319 × 6.3 × 10ିଶ− 0.261 × 5.3 × 10ିଶ × 0.143 × 0.239 − 0.261 × 5.3 × 10ିଶ × 0.202 × 0.169− 0.261 × 0.190 × 0.143 × 0.141 − 0.261 × 0.190 × 0.319 × 6.3 × 10ିଶ+ 0.261 × 0.253 × 0.202 × 0.200] +1 × [+0.261 × 0.253 × 0.319 × 0.126] ] + (𝑋ଶଶ + 𝑋ଷଶ)ଶ. [+4 × [+0.143ଶ × 0.214ଶ + 0.143 × 0.214 × 0.261 × 0.117] +2 × [+0.261ଶ × 0. 117ଶ + 0.261ଶ × 0. 253ଶ] ] [+4 × [+0.143ଶ × 0.214ଶ − 0.143 × 0.239 × 0.202 × 0.169] +2 × [+0.143 × 0.214 × 0.202 × 0.227 − 0.202ଶ × 0.169ଶ] ] −(𝑋ସ)ଶ. (𝑋ଶଶ + 𝑋ଷଶ). [+2 × [−0.143ଶ × 0.239ଶ − 0.202ଶ × 0.169ଶ − 0.202ଶ × 0.245ଶ − 0.261ଶ × 0.190ଶ] +4 × [−0.143 × 0.239 × 0.202 × 0.169 + 0.143 × 0.214 × 0.202 × 0.227+ 0.202 × 0.227 × 0.261 × 0.117 − 0.202 × 0.227 × 0.261 × 0.117] ] +(𝑋ସ)ସ. [2 × (0.202)ଶ × (0.227)ଶ] } 357 The 𝑽𝟏𝟑𝑽𝟑𝟑𝑽𝟑𝟑𝑽𝟑𝟏 and 𝑽𝟏𝟑𝑽𝟑𝟓𝑽𝟓𝟑𝑽𝟑𝟏 terms used in Eq. (3-42) for the Cart molecules (the 𝑽෡𝒕𝒐𝒕 case) 𝑉ଵଷ. 𝑉ଷଷ. 𝑉ଷଷ. 𝑉ଷଵ = −(𝑋଴ଶ + 𝑋ଵଶ). (𝑋ଶଶ + 𝑋ଷଶ). [+2 × [− 0.184ଶ × 0.210ଶ + 0.233ଶ × 0.132ଶ + 0.184ଶ × 0.162ଶ − 0.233ଶ × 0.206ଶ] +4 × [−0.184 × 0.210 × 0.132 × 0.233 + 0.184 × 0.162 × 0.206 × 0.233] +8 × [0.184 × 0.188 × 5.9 × 10ିଶ × 0.233] ] +(𝑋ଶଶ + 𝑋ଷଶ)ଶ. [+2 × 0.233ଶ × 0.162ଶ + 4 × 0.233ଶ × (5.9 × 10ିଶ)ଶ] +𝑋ସ. [𝑋଴(𝑋ଶଶ + 𝑋ଷଶ) − 2𝑋ଵ𝑋ଶ𝑋ଷ]. 2 × [ +2 × [+0.247 × 0.188 × 0.162 × 0.233 − 0.184 × 0.162 × 0.126 × 0.233+ 0.233ଶ × 0.206 × 0.126 + 0.184 × 0.162 × 5.9 × 10ିଶ × 0.274− 0.233 × 0.206 × 5.9 × 10ିଶ × 0.247] ] +(𝑋ସ)ଶ. (𝑋଴ଶ + 𝑋ଵଶ). ൣ+2 × [0.247 × 0.188 × 0.162 × 0.233] + 4 × [−0.184 × 0.162 × 0.126 × 0.233]൧ −(𝑋ସ)ଶ. (𝑋ଶଶ + 𝑋ଷଶ). [+2 × [−0.233ଶ × 0.126ଶ + 3] +4 × [0.233 × 0.126 × 5.9 × 10ିଶ × 0.247 + 0.233 × 5.9 × 10ିଶ × 0.168 × 0.247] ] +(𝑋଴ଶ + 𝑋ଵଶ)ଶ. ൣ4 × [0.184ଶ × 0.188ଶ]൧ 358 +(𝑋ସ)ସ. [0.247ଶ × 0.168ଶ] -------------------------------------------------------------------- 𝑉ଵଷ. 𝑉ଷହ. 𝑉ହଷ. 𝑉ଷଵ = +(𝑋଴ଶ + 𝑋ଵଶ)ଶ. ൣ+2 × [+ 0.184ଶ × 0.268ଶ] + 4 × [0.184ଶ × (5.3 × 10ିଶ)ଶ]൧ −(𝑋଴ଶ + 𝑋ଵଶ). (𝑋ଶଶ + 𝑋ଷଶ). [+2 × [−0.184ଶ × 0.253ଶ − 0.184ଶ × 0.117ଶ − 0.233ଶ × 0.200ଶ − 0.233ଶ × (9.2 × 10ିଶ)ଶ] +4 × [+0.184 × 0.253 × 0.233 × 0.200 + 0.184 × 0.117 × 0.233 × 9.2 × 10ିଶ− 0.184 × 5.3 × 10ିଶ × 0.233 × 0.169] ] +(𝑋ସ)ଶ. (𝑋଴ଶ + 𝑋ଵଶ). [+2 × [+0.184ଶ × 0.190ଶ + 0.247ଶ × 0.141ଶ] +4 × [+0.184 × 0.190 × 0.247 × 0.141 + 0.184 × 5.3 × 10ିଶ × 0.247 × 0.239] ] +𝑋ସ. [𝑋଴. (𝑋ଶଶ + 𝑋ଷଶ) − 2 𝑋ଵ. 𝑋ଶ. 𝑋ଷ]. 2 × [ +2 × [−0.184 × 0.190 × 0.233 × 0.245 + 0.184 × 0.117 × 0.233 × 0.245− 0.247 × 0.214 × 0.184 × 0.117 − 0.247 × 0.117 × 0.247 × 0.214− 0.247 × 0.141 × 0.233 × 0.245 + 0.247 × 0.214 × 0.233 × 9.2 × 10ିଶ− 0.233ଶ × 9.2 × 10ିଶ × 0.227] ] +(𝑋ଶଶ + 𝑋ଷଶ)ଶ. ൣ+2 × [0.233ଶ × 0.245ଶ] + 4 × [0.233ଶ × 0.169ଶ]൧ +(𝑋ସ)ଶ. (𝑋ଶଶ + 𝑋ଷଶ) . [+2 × [−0.233ଶ × 0.227ଶ − 0.233 × 0.169 × 0.247 × 0.239 + 0.247ଶ × 0.214ଶ] 359 +4 × [+0.233 × 0.169 × 0.247 × 0.239 − 0.247 × 0.239 × 0.233 × 0.169] ] +(𝑋ସ)ସ. [0.247ଶ × 0.239ଶ] 360 The 𝑽𝟎𝟐𝑽𝟐𝟐𝑽𝟐𝟎 term in the third-order energy for the Para molecules in the 𝑽෡𝒕𝒐𝒕 case 𝑉଴ଶ𝑉ଶଶ𝑉ଶ଴ = 0.282ଶ × { − ቂ൫𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൯ଶቃ +2 × ൣ൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ × 6 × 0.180 − 𝑊(0) × 2൧ + ቂ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ቃ × ൣ൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ × 4 × 9.011 × 10ିଶ + 𝑊(0) × 2൧ +൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ଷ× 0.180 +൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ଶ× 𝑊(0) +[ ൫𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൯ × ൣ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ − ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯൧ −2 × ൫𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൯. ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ . ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ ] × 2 × 0.220 × 3 } 361 The 𝑽𝟏𝟑𝑽𝟑𝟑𝑽𝟑𝟏 terms in the third-order energy for the Heli and Cart molecules in the 𝑽෡𝒕𝒐𝒕 case For Heli: 𝑉ଵଷ𝑉ଷଷ𝑉ଷଵ =12× { ቂ൫𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൯ଶ. ቃ { 𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ [+2 × (8.25 × 10ିଶ) × 0.126− 2× (8.25 × 10ିଶ) × 0.206 × 0.202 − 2 × 0.202 × 0.206 × 8.25 ×× 10ିଶ− 4 × 0.202 × 0.132 × 0.319 − 2 × 0. 319ଶ × 0.210)] +𝑊(0). [2 × (8.25 × 10ିଶ)ଶ + 2 × 0. 319ଶ] } − ቂ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ଶቃ × {+൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ . (−2 × 0. 143ଶ × 0.168 + 4 × 0.143 × 5.94 × 10ିଶ × 0.202 − 4 × 0.202 × 0.162 × 0.261) +𝑊(0) × (−2 × 143ଶ) } +൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ଷ. (0 × 0.202ଶ × 0.126) +൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ଶ. 𝑊(0). (2 × 0.202ଶ) +{2 × ൫𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൯. ቂ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ଶ− ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ଶቃ 362 −4 × ൫𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൯. ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ . ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ ൯ } [2 × 8.25 × 10ିଶ × 5.94 × 10ିଶ × 0.143 + 2 × 0.143 × 0.188 × 0.261+ 2 × 0.261 × 0.210 × 0.319] } For Cart: 𝑉ଵଷ. 𝑉ଷଷ. 𝑉ଷଵ = +൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯. ቂ+൫𝑓஽ோ 𝐾஽ோ(2) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൯ଶቃ × (−4 × 0.184 × 0.188) + ቂ+൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ଶ+ ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ଶቃ. {+൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ × (+2 × 0.233ଶ × 0.126 − 4 × 0.233 × 5.94 × 10ିଶ × 0.247) +𝑊(0) × 2 × 0.233ଶ } +൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ଷ× 0.247ଶ × 0.168 +൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ଶ× 0.247ଶ × 𝑊(0) +{+2 × ൫𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൯. ቂ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯ଶ− ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ଶቃ −4 × ൫𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൯. ൫𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൯. ൫𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൯ } × (−2 × 0.184 × 0.162 × 0.233 + 0.233ଶ × 0.206) 363 The cubic equation in Model III to calculate the first-order dispersion and repulsion energy + 𝐸஽ோ௣ ଷ − [+2 × (−0.216 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0)] . 𝐸஽ோ௣ ଶ + [+ (−0.126 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0))ଶ +2 × ൫−0.126 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0)൯ × (+0.252 𝑓஽ோ 𝐾஽ோ(1) +0.282 𝑓஽ோ 𝐾஽ோ(0) ) −2 × 0.218ଶ × ([𝑓஽ோ 𝐾஽ோ(5)]ଶ + [𝑓஽ோ 𝐾஽ோ(4)]ଶ) − 0.309ଶ. ([𝑓஽ோ 𝐾஽ோ(5)]ଶ + [𝑓஽ோ 𝐾஽ோ(4)]ଶ)]. 𝐸஽ோ௣ −[+ ൫−0.126 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0)൯ଶ. ൫+0.252 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0)൯ −2 × ൫−0.126 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0)൯ × 0.218ଶ . ([𝑓஽ோ 𝐾஽ோ(5)]ଶ + [𝑓஽ோ 𝐾஽ோ(4)]ଶ) − ൫+0.252 𝑓஽ோ 𝐾஽ோ(1) + 0.282 𝑓஽ோ 𝐾஽ோ(0)൯ × 0.309ଶ. ([𝑓஽ோ 𝐾஽ோ(2)]ଶ + [𝑓஽ோ 𝐾஽ோ(3)]ଶ) + 0.309 × 0.218ଶ × {+ 2 × 𝑓஽ோ 𝐾஽ோ(2) . [𝑓஽ோ 𝐾஽ோ(5)]ଶ − 2 × 𝑓஽ோ 𝐾஽ோ(2) . [𝑓஽ோ 𝐾஽ோ(4)]ଶ+ 4 × 𝑓஽ோ 𝐾஽ோ(3) × 𝑓஽ோ 𝐾஽ோ(5) × 𝑓஽ோ 𝐾஽ோ(4)} ] = 0 364 The cubic equation in Model III to calculate the first-order H2 quadrupole energy + 𝐸ொ௘௟௦௣ ଷ − [−2 × 0.216 + 0.252]. 𝑋ସ . 𝐸ொ௘௟௦௣ ଶ + [( +0.126ଶ − 2 × 0.126 × 0.252 ). 𝑋ସଶ − 2 × 0. 218ଶ . (𝑋ଶଶ + 𝑋ଷଶ)− 0.309ଶ. (𝑋଴ଶ + 𝑋ଵଶ)]. 𝐸ொ௘௟௦௣ −ൣ+0.216ଶ × 0.256. 𝑋ସଷ + 2 × 0.126 × 0.218ଶ. 𝑋ସ. (𝑋ଶଶ + 𝑋ଷଶ)− 0.309ଶ × 0.252 . 𝑋ସ . (𝑋଴ଶ + 𝑋ଵଶ)+ 0.309 × 0.218ଶ. [2𝑋଴ (𝑋ଶଶ − 𝑋ଷଶ) + 4𝑋ଵ𝑋ଶ𝑋ଷ]൧ = 0 365 The cubic equation in Model III to calculate the first-order H2 induced dipole energy +𝐸௜௡ௗଶ௣ ଷ − (−2 × 0.126 + 0.252). 𝐶௜௡ௗబ . 𝐶௜௡ௗభ . 𝐸௜௡ௗଶ௣ ଶ +𝐶௜௡ௗబଶ . ൣ(+0.126ଶ − 2 × 0.126 × 0.252). 𝐶௜௡ௗభଶ − 2 × 0.218ଶ൧. 𝐸௜௡ௗଶ௣ +𝐶௜௡ௗబଷ . ቂ+0.126ଶ × 0.252 . 𝐶௜௡ௗభଷ + 2 × 0.126 × 0.218ଶ. 𝐶௜௡ௗభ . ൫𝐶௜௡ௗఱଶ . 𝐶௜௡ௗరଶ ൯− −0.309ଶ × 0.252. 𝐶௜௡ௗభ . ൫𝐶௜௡ௗమଶ + 𝐶௜௡ௗయଶ ൯− 0.309ଶ × 0.218ଶ × ൣ2 × 𝐶௜௡ௗమ . ൫𝐶௜௡ௗఱଶ . 𝐶௜௡ௗరଶ ൯ + 4 × 𝐶௜௡ௗయ𝐶௜௡ௗర𝐶௜௡ௗఱ൧ቃ = 0 366 Appendix B The development of the different terms of the H2-H2 interaction energy in Eq. (3-54) Based on Eq. (3-55), the H2-H2 interaction energy has four terms. Here, each term is derived based on the wavefunctions of molecules 1 (𝛹௄,௠಼) and 2 (𝛹௅,௠ಽᇱ) as follows: 𝐸଴଴଴ = < 𝛹௄,௠಼ · 𝛹௅,௠ಽᇱห𝑉෠଴଴଴ห𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ > Using the equation for 𝑉෠଴଴଴ in Table 3-8, this energy is written as: 𝐸଴଴଴ = < 𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ|𝑉଴଴଴|𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ > = < 𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ|𝑉଴଴଴|𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ > = < 𝛹௄,௠಼|𝑉଴଴଴|𝛹௄,௠಼ >·< 𝛹௅,௠ಽᇱ|𝛹௅,௠ಽᇱ > = 𝑉଴଴଴ ·< 𝛹௄,௠಼|𝛹௄,௠಼ >·< 𝛹௅,௠ಽᇱ|𝛹௅,௠ಽᇱ > = 𝑉଴଴଴ × 𝐼଴଴଴(1) × 𝐼଴଴଴(2) Therefore: 𝐸଴଴଴ = 𝑉଴଴଴ × 𝐼଴଴଴(1) × 𝐼଴଴଴(2) (B1-1) where 𝐼଴଴଴(1) = < 𝛹௄,௠಼|𝛹௄,௠಼ > (B1-2) and 𝐼଴଴଴(2) = < 𝛹௅,௠ಽᇱ|𝛹௅,௠ಽᇱ > (B1-3) Following Eq. (3-55), the equation for 𝐸ଶ଴ଶ is: 𝐸ଶ଴ଶ = < 𝛹௄,௠಼ · 𝛹௅,௠ಽᇱห𝑉෠ଶ଴ଶห𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ > 367 Using the equation for 𝑉෠ଶ଴ଶ in Table 3-8, this energy can be written as: 𝐸ଶ଴ଶ = < 𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ ቮ𝑉ଶ଴ଶ · √5𝜋 · ቌඨ32(Υଶଶഥ + Υଶଶ) − Υଶ଴ቍቮ 𝛹௄,௠಼ · 𝛹௅,௠ಽᇱ > 𝐸ଶ଴ଶ = < 𝛹௄,௠಼ ቮ𝑉ଶ଴ଶ · √5𝜋 · ቌඨ32(Υଶଶഥ + Υଶଶ) − Υଶ଴ቍቮ 𝛹௄,௠಼ > × < 𝛹௅,௠ಽᇱ|𝛹௅,௠ಽᇱ > = {< 𝛹௄,௠಼ ቮ𝑉ଶ଴ଶ · √5𝜋 · ඨ32· Υଶଶഥቮ 𝛹௄,௠಼ > + < 𝛹௄,௠಼ ቮ𝑉ଶ଴ଶ · √5𝜋 · ඨ32· Υଶଶቮ 𝛹௄,௠಼ > − < 𝛹௄,௠಼ห𝑉ଶ଴ଶ · √5𝜋 · Υଶ଴ห𝛹௄,௠಼ >ൟ × < 𝛹௅,௠ಽᇱ|𝛹௅,௠ಽᇱ > = 𝑉ଶ଴ଶ × ቐ< 𝛹௄,௠಼ ቮ√5𝜋 · ඨ32· Υଶଶഥቮ 𝛹௄,௠಼ > + < 𝛹௄,௠಼ ቮ√5𝜋 · ඨ32· Υଶଶቮ 𝛹௄,௠಼ > −< 𝛹௄,௠಼ห√5𝜋 · Υଶ଴ห𝛹௄,௠಼ >ቑ × < 𝛹௅,௠ಽᇱ|𝛹௅,௠ಽᇱ > Therefore: 𝐸ଶ଴ଶ = 𝑉ଶ଴ଶ × ቐඨ15𝜋2· 𝑆ଶଶഥ(1) + ඨ15𝜋2· 𝑆ଶଶ(1) − √5𝜋 · 𝑆ଶ଴(1)ቑ× 𝐼ଶ଴ଶ(2) (B1-4) where 368 𝑆ଶଶഥ(1) = < 𝛹௄,௠಼|Υଶଶഥ|𝛹௄,௠಼ > (B1-5) 𝑆ଶଶ(1) = < 𝛹௄,௠಼|Υଶଶ|𝛹௄,௠಼ > (B1-6) 𝑆ଶ଴(1) = < 𝛹௄,௠಼|Υଶ଴|𝛹௄,௠಼ > (B1-7) and 𝐼ଶ଴ଶ(2) = < 𝛹௅,௠ಽᇱ|𝛹௅,௠ಽᇱ > (B1-8) Similarly, 𝐸଴ଶଶ has the following form: 𝐸଴ଶଶ = 𝑉଴ଶଶ × 𝐼଴ଶଶ(1)× ቐඨ15𝜋2· 𝑆ଶଶഥ(2) + ඨ15𝜋2· 𝑆ଶଶ(2) − √5𝜋 · 𝑆ଶ଴(2)ቑ (B1-9) where 𝑆ଶଶഥ(2), 𝑆ଶଶ(2), and 𝑆ଶ଴(2) are evaluated using Eq. (B1-5), (B1-6), (B1-7), and (B1-8) respectively, except that in all these equations, the 𝛹௄,௠಼ functions are replaced by the 𝛹௅,௠ಽᇱ functions. Also, 𝐼଴ଶଶ(1) has the same form as Eq. (B1-8), with 𝛹௅,௠ಽᇱ being replaced by 𝛹௄,௠಼. Similarly, 𝐸ଶଶସ has the following form: 369 𝐸ଶଶସ = 𝑉ଶଶସ × ൬454√70൰ ൬2𝜋15൰ . ቐ39 × [𝑆ଶଶഥ(1) 𝑆ଶଶഥ(2) + 𝑆ଶଶ(1) 𝑆ଶଶ(2)]− 1 × [𝑆ଶଶഥ(1) · 𝑆ଶଶ(2) + 𝑆ଶଶ(1) · 𝑆ଶଶഥ(2)]− 28 × [𝑆ଶଵഥ(1) · 𝑆ଶଵഥ(2) + 𝑆ଶଵ(1) · 𝑆ଶଵ(2)]+ 4 × [𝑆ଶଵഥ(1) · 𝑆ଶଵ(2) + 𝑆ଶଵ(1) · 𝑆ଶଵഥ(2)]− 10ඨ32× ൣ𝑆ଶ଴(2) · [𝑆ଶଶഥ(1) + 𝑆ଶଶ(1)] + 𝑆ଶ଴(1)· [𝑆ଶଶഥ(2) + 𝑆ଶଶ(2)]൧ + 18 × 𝑆ଶ଴(1) · 𝑆ଶ଴(2)ቑ (B1-10) where 𝑆ଶଶഥ(𝑖), 𝑆ଶଶ(𝑖), and 𝑆ଶ଴(𝑖) (for 𝑖 = 1 𝑎𝑛𝑑 2) have already been defined in Eq. (B1-5), (B1-6), and (B1-7). It is emphasized again that for 𝑖 = 2, all the 𝛹௄,௠಼ functions in these equations are replaced by the 𝛹௅,௠ಽᇱ functions. Also, one may define: 𝑆ଶଵഥ(1) =< 𝛹௄,௠಼|Υଶଵഥ|𝛹௄,௠಼ > (B1-11) and 𝑆ଶଵ(1) =< 𝛹௄,௠಼|Υଶଵ|𝛹௄,௠಼ > (B1-12) Same equations are used for the 𝑆ଶଵഥ(2) as well as 𝑆ଶଵ(2) functions, with the 𝛹௄,௠಼ functions replaced by the 𝛹௅,௠ಽᇱ functions in these equations. Therefore, to calculate the H2-H2 interaction energy for each pair of H2 molecules, first the 𝐸௜௝௞ terms (𝑖, 𝑗, 𝑎𝑛𝑑 𝑘 indices are the same as 𝑉௜௝௞(𝑟) in Table 3-8) must be calculated. To calculate these terms, all that needs to be done is to 370 calculate 𝐼଴଴଴(1), 𝐼଴଴଴(2), 𝐼ଶ଴ଶ(2), 𝐼଴ଶଶ(1), as well as the 𝑆ଶ௠಻(𝑖) integrals (𝑓𝑜𝑟 𝑚௃ =−2, −1,0,1,2 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2). 371 The development of the 𝑰𝟎𝟎𝟎(𝟏), 𝑰𝟎𝟎𝟎(𝟐), 𝑰𝟐𝟎𝟐(𝟐), 𝑰𝟎𝟐𝟐(𝟏), and the 𝑺𝟐𝒎𝑱(𝒊) terms (𝒊 = 𝟏 𝒂𝒏𝒅 𝟐) for Para, Heli, and Cart in Model I First, these integrals are calculated for the Para molecule in Model I: 𝐼଴଴଴(1) =< ϒ଴଴|ϒ଴଴ > The result of this integral is the Kronecker delta function, which is equal to 1 when the spherical harmonic functions in the integral are the same. Therefore, for molecule 1: 𝐼଴଴଴(1) = 1 Similarly, 𝐼଴଴଴(2) = 1. Also, for 𝑚௃ = −2, −1,0,1,2 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2: 𝑆ଶ௠಻(𝑖) =< ϒ଴଴ቚΥଶ௠಻ቚϒ଴଴ >= 0 This integral is zero because, according to section A.1, the Clebsch-Gordan coefficient is zero if 𝑚௄భ + 𝑚௄మ − 𝑚௄య ≠ 0 or 𝐾ଷ < |𝐾ଵ − 𝐾ଶ|. Also, similar to 𝐼଴଴଴(𝑖) shown above, 𝐼ଶ଴ଶ(2) =𝐼଴ଶଶ(1) = 1. It should be noted again that in the equation above for 𝑆ଶ௠಻(𝑖), when 𝑖 =𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 2, all the spherical harmonic functions in the integral are replaced by the primed harmonic functions (because they are the wavefunctions of molecule 2). For the Heli molecule: 𝐼଴଴଴(1) =< ϒଵଵ|ϒଵଵ >= 1 Similarly: 𝐼଴଴଴(2) = 𝐼ଶ଴ଶ(2) = 𝐼଴ଶଶ(1) = 1 Also, for 𝑚௃ = −2, −1,1,2 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2: 𝑆ଶ௠಻(𝑖) =< ϒଵଵቚΥଶ௠಻ቚϒଵଵ >= 0 But for 𝑚௃ = 0 and 𝑖 = 1,2, the following result is obtained: 372 𝑆ଶ଴(𝑖) =< ϒଵଵ|Υଶ଴|ϒଵଵ >= −0.126 Similarly, for the Cart molecule: 𝐼଴଴଴(1) = 𝐼଴଴଴(2) = 𝐼ଶ଴ଶ(2) = 𝐼଴ଶଶ(1) = 1 Also, for 𝑚௃ = −2, −1,1,2 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2: 𝑆ଶ௠಻(𝑖) =< ϒଵଵቚΥଶ௠಻ቚϒଵଵ >= 0 But for 𝑚௃ = 0 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2, the following result is obtained: 𝑆ଶ଴(𝑖) =< ϒଵ଴|Υଶ଴|ϒଵ଴ >= +0.252 373 Derivation of the 𝝈𝒊𝒋 coefficients as well as the normalization factor in Eq. (3-61) For the Para molecule in Model II, the reference state is Υ଴଴, thus: 𝜎଴଴ = 1 The rest of the coefficients are then calculated using Eq. (3-60) and based on this wavefunction as follows: 𝜎ଶଶഥ =1𝐸଴ − 𝐸ଶ × < Υଶଶഥ|𝑉෠ுమି௦௨௥௙|Υ଴଴ > Based on the properties of the Clebsch-Gordan coefficients in section A.1, only Υଶଶഥ terms in the 𝑉෠ுమି௦௨௥௙ operator survive: 𝜎ଶଶഥ =1𝐸଴ − 𝐸ଶ × < Υଶଶഥ| ൫ൣ𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൧ + 𝑖. ൣ𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൧൯Υଶଶഥ |Υ଴଴ > =1𝐸଴ − 𝐸ଶ × < Υଶଶഥ|Υଶଶഥ|Υ଴଴ > × ൫ൣ𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൧ + 𝑖. ൣ𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൧൯ Therefore: 𝜎ଶଶഥ =1𝐸଴ − 𝐸ଶ × 0.282 × ൫ൣ𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൧ + 𝑖. ൣ𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൧൯ Similarly, the rest of the coefficients can be calculated: 374 𝜎ଶଵഥ =1𝐸଴ − 𝐸ଶ × 0.282 × ൫ൣ𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൧ + 𝑖. ൣ𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൧൯ 𝜎ଶ଴ =1𝐸଴ − 𝐸ଶ × 0.282 × ൫𝑓஽ோ 𝐾஽ோ(1) + 𝑋ସ + 𝐶௜௡ௗబ𝐶௜௡ௗభ൯ 𝜎ଶଵ =1𝐸଴ − 𝐸ଶ × 0.282 × ൫−ൣ𝑓஽ோ 𝐾஽ோ(5) + 𝑋ଶ + 𝐶௜௡ௗబ𝐶௜௡ௗఱ൧ + 𝑖. ൣ𝑓஽ோ 𝐾஽ோ(4) + 𝑋ଷ + 𝐶௜௡ௗబ𝐶௜௡ௗర൧൯ 𝜎ଶଶ =1𝐸଴ − 𝐸ଶ × 0.282 × ൫ൣ𝑓஽ோ 𝐾஽ோ(2) + 𝑋଴ + 𝐶௜௡ௗబ𝐶௜௡ௗమ൧ − 𝑖. ൣ𝑓஽ோ 𝐾஽ோ(3) + 𝑋ଵ + 𝐶௜௡ௗబ𝐶௜௡ௗయ൧൯ 𝑁୔ୟ୰ୟ =1ට𝜎଴଴ଶ + 𝜎ଶଶഥଶ + 𝜎ଶଵഥଶ + 𝜎ଶ଴ଶ + 𝜎ଶଵଶ + 𝜎ଶଶଶ 375 Derivation of the Para-Para interaction energy in Model II As explained in section B.1, first the integrals 𝐼଴଴଴(1), 𝐼଴଴଴(2), 𝐼ଶ଴ଶ(2), 𝐼଴ଶଶ(1), and 𝑆ଶ௠಻(𝑖) (for 𝑚௃ = −2, −1, 0, 1, 2 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2) must be calculated. In order to calculate these terms Eq. (B1-1), (B1-3), (B1-5), (B1-6), (B1-7), (B1-8), (B1-11), and (B1-12) must be used. 𝐼଴଴଴(1) = 𝑁୔ୟ୰ୟଶ . < (𝜎଴଴Υ଴଴ + 𝜎ଶଶഥΥଶଶഥ+. . . +𝜎ଶଶΥଶଶ)|(𝜎଴଴Υ଴଴ + 𝜎ଶଶഥΥଶଶഥ+. . . +𝜎ଶଶΥଶଶ) > According to the properties of the spherical harmonic functions explained in section A.1, only the like Υ௜௝ terms in the integral survive, i.e.: 𝐼଴଴଴(1) = 𝑁୔ୟ୰ୟଶ . {< 𝜎଴଴Υ଴଴|𝜎଴଴Υ଴଴ > +< 𝜎ଶଶഥΥଶଶഥ|𝜎ଶଶഥΥଶଶഥ > +. . . +< 𝜎ଶଶΥଶଶ|𝜎ଶଶΥଶଶ >} = 𝑁୔ୟ୰ୟଶ . ൛𝜎଴଴ଶ . < Υ଴଴|Υ଴଴ > +𝜎ଶଶഥଶ . < Υଶଶഥ|Υଶଶഥ > +. . . +𝜎ଶଶଶ . < Υଶଶ|Υଶଶ >ൟ ⸫ 𝐼଴଴଴(1) = 𝑁୔ୟ୰ୟଶ . ൫𝜎଴଴ଶ + 𝜎ଶଶഥଶ +. . . +𝜎ଶଶଶ ൯ Similarly, the 𝐼଴଴଴(2) for molecule 2 is: 𝐼଴଴଴(2) = 𝑁௉௔௥௔ᇱమ . < (𝜎଴଴ᇱ Υ଴଴ᇱ + 𝜎ଶଶഥᇱ Υଶଶഥᇱ+. . . +𝜎ଶଶᇱ Υଶଶᇱ)|(𝜎଴଴ᇱ Υ଴଴ᇱ + 𝜎ଶଶഥᇱ Υଶଶഥᇱ+. . . +𝜎ଶଶᇱ Υଶଶᇱ) > Similar to 𝐼଴଴଴(1), only the like Υ௜௝ᇱ terms in the integral survive, which is because of the properties of the spherical harmonic functions explained in section A.1. Thus, 𝐼଴଴଴(2) is: 376 𝐼଴଴଴(2) = 𝑁௉௔௥௔ᇱమ . ቄ𝜎଴଴ᇱమ . < Υ଴଴ᇱ|Υ଴଴ᇱ > +𝜎ଶଶഥᇱమ . < Υଶଶഥᇱ|Υଶଶഥᇱ > +. . . +𝜎ଶଶᇱమ . < Υଶଶᇱ|Υଶଶᇱ >ቅ ⸫ 𝐼଴଴଴(2) = 𝑁௉௔௥௔ᇱమ . ቀ𝜎଴଴ᇱమ + 𝜎ଶଶഥᇱమ +. . . +𝜎ଶଶᇱమ ቁ Also, 𝑆ଶଶഥ(1) = 𝑁୔ୟ୰ୟଶ . < (𝜎଴଴Υ଴଴ + 𝜎ଶଶഥΥଶଶഥ+. . . +𝜎ଶଶΥଶଶ)|Υଶଶഥ|(𝜎଴଴Υ଴଴ + 𝜎ଶଶഥΥଶଶഥ+. . . +𝜎ଶଶΥଶଶ) > But, based on the properties of the Clebsch-Gordan coefficients, only the following terms survive: 𝑆ଶଶഥ(1) = 𝑁୔ୟ୰ୟଶ . [< 𝜎଴଴ϒ଴଴|Υଶଶഥ|𝜎ଶଶϒଶଶ > +< 𝜎ଶଶഥΥଶଶഥ|Υଶଶഥ|𝜎଴଴ϒ଴଴ > +< 𝜎ଶଶഥΥଶଶഥ|Υଶଶഥ|𝜎ଶ଴ϒଶ଴ > +< 𝜎ଶଵഥΥଶଵഥ|Υଶଶഥ|𝜎ଶଵϒଶଵ > +< 𝜎ଶ଴ϒଶ଴|Υଶଶഥ|𝜎ଶଶϒଶଶ >] = 𝑁୔ୟ୰ୟଶ . [𝜎଴଴𝜎ଶଶ < ϒ଴଴|Υଶଶഥ|ϒଶଶ > +𝜎ଶଶഥ𝜎଴଴ < Υଶଶഥ|Υଶଶഥ|ϒ଴଴ > +𝜎ଶଶഥ𝜎ଶ଴ < Υଶଶഥ|Υଶଶഥ|ϒଶ଴ > +𝜎ଶଵഥ𝜎ଶଵ < Υଶଵഥ|Υଶଶഥ|ϒଶଵ > +𝜎ଶ଴𝜎ଶଶ < ϒଶ଴|Υଶଶഥ|ϒଶଶ >] ⸫ 𝑆ଶଶഥ(1) = 𝑁୔ୟ୰ୟଶ . [+0.282 𝜎଴଴. (𝜎ଶଶ + 𝜎ଶଶഥ) − 0.180 𝜎ଶ଴. (𝜎ଶଶ + 𝜎ଶଶഥ) − 0.220 𝜎ଶଵഥ. 𝜎ଶଵ] Similarly: 𝑆ଶଶ(1) = 𝑁୔ୟ୰ୟଶ . [+0.282 𝜎଴଴. (𝜎ଶଶ + 𝜎ଶଶഥ) − 0.180 𝜎ଶ଴. (𝜎ଶଶ + 𝜎ଶଶഥ) − 0.220 𝜎ଶଵഥ. 𝜎ଶଵ] 𝑆ଶ଴(1) = 𝑁୔ୟ୰ୟଶ . ൣ+2 × 0.282 𝜎଴଴. 𝜎ଶ଴ − 0.180 ൫𝜎ଶଶഥଶ + 𝜎ଶଶଶ ൯ + 9.01 × 10ିଶ ൫𝜎ଶଵഥଶ + 𝜎ଶଵଶ ൯+ 0.180 𝜎ଶ଴ଶ ൧ 𝐼ଶ଴ଶ(2) = 𝑁௉௔௥௔ᇱమ . ቀ𝜎଴଴ᇱమ + 𝜎ଶଶഥᇱమ +. . . +𝜎ଶଶᇱమ ቁ 377 𝐼଴ଶଶ(1) = 𝑁୔ୟ୰ୟଶ . ൫𝜎଴଴ଶ + 𝜎ଶଶഥଶ +. . . +𝜎ଶଶଶ ൯ 𝑆ଶଵഥ(1) = 𝑁୔ୟ୰ୟଶ . [+0.282 𝜎଴଴. (𝜎ଶଵഥ − 𝜎ଶଵ) + 9.01 × 10ିଶ 𝜎ଶ଴. (𝜎ଶଵഥ − 𝜎ଶଵ) +0.220 (𝜎ଶଶഥ𝜎ଶଵഥ − 𝜎ଶଵ𝜎ଶଶ)] and 𝑆ଶଵ(1) = 𝑁୔ୟ୰ୟଶ . [−0.282 𝜎଴଴. (𝜎ଶଵഥ − 𝜎ଶଵ) − 9.01 × 10ିଶ 𝜎ଶ଴. (𝜎ଶଵഥ − 𝜎ଶଵ) −0.220 (𝜎ଶଵഥ𝜎ଶଶഥ − 𝜎ଶଵ𝜎ଶଶ)] All the 𝑆ଶ௠಻(2) (for 𝑚௃ = −2, −1, 0, 1, 2 and 𝑖 = 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 1 𝑜𝑟 2) integrals are the same as the 𝑆ଶ௠಻(1) terms shown above, but with the 𝜎௜௝ coefficients (for molecule 1) replaced by the 𝜎௜௝ᇱ coefficients (for molecule 2). 378 Appendix C The interaction parameters used in models I and II In this section, the initial and final interaction parameters used in Model I (Eq. (3-10)) and Model II (Eq. (3-23)) of this study are presented. Table C1-1 The initial interaction parameters used in the Tang-Toennies potential (Eq. (3-10)) of Model I before fitting to the desired energies. These parameters were obtained from Refs. 78,79 for H2-H2 interactions and Ref.25 for ion-ion interactions. For the H2-ion interactions some mixing rules of Ref. 25 were used. Aij [kcal/mol] βij [Å-1] C6 ij [(kcal/mol). Å6] C8 ij [(kcal/mol) . Å8] C10 ij [(kcal/mol) . Å10] Mg2+ ‒ H2 176589.1090 5.4677 25.6204 81.0806 314.3288 O2- ‒ H2 2251.0181 2.6254 375.0232 2153.3657 15146.5439 Na+ ‒ H2 79450.6719 4.6241 49.6285 166.7864 686.6354 Cl- ‒ H2 21399.7812 3.0156 649.02124 4732.8355 42278.5469 379 Table C1-2 The final interaction parameters used in the Tang-Toennies potential (Eq. (3-10)) of Model I after fitting the energies to the desired values. For MgO, two separate adjustments were done, one for the UBE case and one for the LBE case. The difference in the parameters listed in this table with Table C1-1 (before the adjustment) is only in the Aij parameter. The energies presented in section 4.3 are calculated using the parameters listed in this table. Aij [kcal/mol] βij [Å-1] C6 ij [(kcal/mol). Å6] C8 ij [(kcal/mol) . Å8] C10 ij [(kcal/mol) . Å10] Mg2+ ‒ H2 (UBE) 208834.2810 5.4677 25.6204 81.0806 314.3288 O2- ‒ H2 (UBE) 2662.0540 2.6254 375.0232 2153.3657 15146.5439 Mg2+ ‒ H2 (LBE) 201594.1268 5.4677 25.6204 81.0806 314.3288 O2- ‒ H2 (LBE) 2569.7622 2.6254 375.0232 2153.3657 15146.5439 Na+ ‒ H2 106463.9060 4.6241 49.6285 166.7864 686.6354 Cl- ‒ H2 24609.7480 3.0156 649.02124 4732.8355 42278.5469 380 Table C1-3 The initial interaction parameters used in Eq. (3-23) (the LJ potential of Briquez et al.26 used in Model II, see also section A.4) before fitting the energies to the desired values. These parameters were obtained from fitting the LJ potential to the energy obtained from the Tang-Toennies potential in Model I. C12 ij [(kcal/mol). Å12] C6 ij [(kcal/mol). Å6] Mg2+ ‒ H2 11883.1794 45.1043 O2- ‒ H2 739015.4948 387.9801 Na+ ‒ H2 47362.1521 81.4291 Cl- ‒ H2 1734063.5736 896.2244 Table C1-4 The final interaction parameters used in Eq. (3-23) (the LJ potential of Briquez et al.26 used in Model II, see also section A.4) after fitting the energies to the desired values. The energies presented in section 4.3.2 are calculated using the parameters listed in this table. C12 ij [(kcal/mol). Å12] C6 ij [(kcal/mol). Å6] (UBE) Mg2+ ‒ H2 12952.6656 40.5938 O2- ‒ H2 805526.8893 349.1821 (LBE) Mg2+ ‒ H2 12037.6607 40.5938 O2- ‒ H2 748622.6962 349.1821 Na+ ‒ H2 65122.9591 69.2147 Cl- ‒ H2 2384337.4130 761.7907 381 The Fourier series convergence (Chapter 4) In this section, the supplementary data related to section 4.2.1.2.1 are presented. Figure C2-1 The LJ energy for the Para/Mg2+ system in the UBE case after adding each Fourier term into the potential. (𝒌,𝒋) is defined in section 4.2 of the text. -0.6-0.4-0.20.00.20.40.60.82.0 4.0 6.0 8.0Energy [kcal/mol]Height [Å](0,0)(0,1)(1,1)(0,2)(1,2)(2,2)(0,3)382 Figure C2-2 The interaction of the quadrupole moment of Para H2 with the MgO surface for the Para/Mg2+ system and in the UBE case after adding each Fourier term into this potential. The (𝒌, 𝒋) vector set is defined in section 4.2 of the text. Figure C2-3 The interaction of the quadrupole moment of Cart H2 with the MgO surface for the Cart/Mg2+ system and in the UBE case after adding each Fourier term into this potential. The (𝒌, 𝒋) vector set is defined in section 4.2 of the text. -0.20-0.15-0.10-0.050.002.6 3.1 3.6Energy [kcal/mol]Height [Å](0,1)(1,2)(0,3)(2,3)0.00.10.20.30.40.52.5 3.0 3.5 4.0Energy [kcal/mol]Height [Å](0,1)(1,2)(0,3)(2,3)383 The PT series convergence for H2/MgO systems in the UBE case (section 4.2.2.1) In this section, the PT series convergence in the remaining H2/MgO systems in the UBE case is discussed. In fact, this section presents supplementary data for section 4.2.2.1. C.3.1. Heli/Saddle The next case that is considered here is a Heli molecule sitting above a saddle site (Heli/Saddle). This site is considered as the second most attractive site for the Heli H2 species on the ionic surface. The results for this system are shown in Table C3-1 (this section). The minimum of the potential occurs at about 3.45 Å, thus the RI in our data is from 3.15 to 3.45 Å. Same procedure as in section 4.2.2.1.1 is used in the convergence analysis and only the final results are presented in Table C3-2 (this section). As seen in this table, the 4th/Etot ratios are all less than 1.00% and hence, even going up to third-order PT terms in EQels is enough to get a well-converged energy in the RI. But, since the third-order energies in EQels are all zero, therefore, in this case going up to second order is suggested. The reason is that in this system, the minimum energy occurs at a height far from the surface. Nevertheless, all PT terms up to the fourth-order correction terms have been included in EQels. Also, in Table C3-2 (this section), it is shown that including up to second-order PT energies in both EDR and Eind2 results in a well-converged energy in the RI. C.3.2. Other systems As was mentioned in section 4.2.2.1, since the results of systems Cart/Mg2+, Para/Saddle, Cart/Saddle, Heli/O2-, Para/O2-, and Cart/O2- are very similar to the cases that have just been discussed, only the data for some of these systems are presented in Figure C3-1, and in Table 384 C3-3, Table C3-4, and Table C3-5 at the end of this section of Appendix C. It should be mentioned that, in all these systems, wherever 4th/Etot ratios are smaller than 1%, even going up to third-order PT energies is enough in the calculations. 385 Table C3-1 The energy results for Heli/Saddle system in the UBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 23.72011 29.10464 -0.39615 -4.35064 -0.00221 0.00000 -0.80654 0.00000 0.11781 0.00000 0.00000 0.00000 0.00000 2.05 19.28802 24.08670 -0.25402 -4.00505 -0.00189 0.00000 -0.65488 0.00000 0.07767 0.00000 0.00000 0.00000 0.00000 2.10 15.66000 19.96819 -0.16476 -3.69101 -0.00163 0.00000 -0.53154 0.00000 0.05117 0.00000 0.00000 0.00000 0.00000 2.15 12.69117 16.58134 -0.10805 -3.40546 -0.00141 0.00000 -0.43128 0.00000 0.03368 0.00000 0.00000 0.00000 0.00000 2.20 10.26237 13.79146 -0.07160 -3.14560 -0.00122 0.00000 -0.34983 0.00000 0.02216 0.00000 0.00000 0.00000 0.00000 2.25 8.27574 11.48975 -0.04792 -2.90895 -0.00106 0.00000 -0.28369 0.00000 0.01457 0.00000 0.00000 0.00000 0.00000 2.30 6.65108 9.58799 -0.03236 -2.69324 -0.00092 0.00000 -0.23000 0.00000 0.00958 0.00000 0.00000 0.00000 0.00000 2.35 5.32269 8.01435 -0.02205 -2.49644 -0.00080 0.00000 -0.18643 0.00000 0.00629 0.00000 0.00000 0.00000 0.00000 2.40 4.23686 6.71034 -0.01514 -2.31674 -0.00070 0.00000 -0.15109 0.00000 0.00413 0.00000 0.00000 0.00000 0.00000 2.45 3.34961 5.62814 -0.01048 -2.15247 -0.00061 0.00000 -0.12242 0.00000 0.00271 0.00000 0.00000 0.00000 0.00000 2.50 2.62500 4.72869 -0.00730 -2.00217 -0.00054 0.00000 -0.09918 0.00000 0.00178 0.00000 0.00000 0.00000 0.00000 2.55 2.03363 3.97998 -0.00512 -1.86451 -0.00047 0.00000 -0.08035 0.00000 0.00117 0.00000 0.00000 0.00000 0.00000 2.60 1.55145 3.35577 -0.00361 -1.73829 -0.00041 0.00000 -0.06508 0.00000 0.00077 0.00000 0.00000 0.00000 0.00000 2.65 1.15879 2.83453 -0.00256 -1.62244 -0.00036 0.00000 -0.05271 0.00000 0.00050 0.00000 0.00000 0.00000 0.00000 2.70 0.83952 2.39856 -0.00183 -1.51600 -0.00032 0.00000 -0.04268 0.00000 0.00033 0.00000 0.00000 0.00000 0.00000 2.75 0.58046 2.03333 -0.00131 -1.41809 -0.00028 0.00000 -0.03456 0.00000 0.00022 0.00000 0.00000 0.00000 0.00000 2.80 0.37080 1.72683 -0.00095 -1.32793 -0.00025 0.00000 -0.02799 0.00000 0.00014 0.00000 0.00000 0.00000 0.00000 2.85 0.20167 1.46921 -0.00069 -1.24482 -0.00022 0.00000 -0.02266 0.00000 0.00009 0.00000 0.00000 0.00000 0.00000 386 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.90 0.06579 1.25229 -0.00050 -1.16813 -0.00020 0.00000 -0.01835 0.00000 0.00006 0.00000 0.00000 0.00000 0.00000 2.95 -0.04281 1.06933 -0.00036 -1.09728 -0.00017 0.00000 -0.01485 0.00000 0.00004 0.00000 0.00000 0.00000 0.00000 3.00 -0.12903 0.91475 -0.00027 -1.03175 -0.00016 0.00000 -0.01202 0.00000 0.00003 0.00000 0.00000 0.00000 0.00000 3.05 -0.19690 0.78393 -0.00020 -0.97109 -0.00014 0.00000 -0.00973 0.00000 0.00002 0.00000 0.00000 0.00000 0.00000 3.10 -0.24974 0.67302 -0.00015 -0.91488 -0.00012 0.00000 -0.00788 0.00000 0.00001 0.00000 0.00000 0.00000 0.00000 3.15 -0.29027 0.57882 -0.00011 -0.86272 -0.00011 0.00000 -0.00638 0.00000 0.00001 0.00000 0.00000 0.00000 0.00000 3.20 -0.32076 0.49869 -0.00008 -0.81429 -0.00010 0.00000 -0.00516 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.25 -0.34303 0.43041 -0.00006 -0.76926 -0.00009 0.00000 -0.00418 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.30 -0.35863 0.37212 -0.00005 -0.72736 -0.00008 0.00000 -0.00338 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.35 -0.36880 0.32227 -0.00003 -0.68833 -0.00007 0.00000 -0.00274 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.40 -0.37458 0.27958 -0.00003 -0.65193 -0.00006 0.00000 -0.00222 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.45 -0.37682 0.24295 -0.00002 -0.61797 -0.00006 0.00000 -0.00179 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.50 -0.37623 0.21147 -0.00001 -0.58624 -0.00005 0.00000 -0.00145 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.55 -0.37339 0.18437 -0.00001 -0.55657 -0.00005 0.00000 -0.00118 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.60 -0.36876 0.16100 -0.00001 -0.52880 -0.00004 0.00000 -0.00095 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.65 -0.36275 0.14082 -0.00001 -0.50279 -0.00004 0.00000 -0.00077 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.70 -0.35567 0.12337 -0.00001 -0.47840 -0.00003 0.00000 -0.00062 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.80 -0.33933 0.09512 0.00000 -0.43402 -0.00003 0.00000 -0.00041 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.90 -0.32132 0.07378 0.00000 -0.39481 -0.00002 0.00000 -0.00027 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4.00 -0.30269 0.05756 0.00000 -0.36006 -0.00002 0.00000 -0.00018 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 387 Table C3-2 The energies and PEPR quantities for a system of Heli/Saddle (MgO) in the UBE case after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 3.05 up to 3.45 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table C3-1). 4th in the ratios refers to the corresponding EQels PT correction term in its PT series presented in Table C3-1. Heights are in Å, energies are in kcal/mol, and the ratios are in %. Height Etot EQels(tot) 4th/EQels(tot) 4th/Etot EDR(2)/Etot Eind2(2)/Etot 2.00 23.72011 -0.68874 17.10 0.50 1.46 0.00 2.05 19.28802 -0.57721 13.46 0.40 1.12 0.00 2.10 15.66000 -0.48037 10.65 0.33 0.87 0.00 2.15 12.69117 -0.39760 8.47 0.27 0.69 0.00 2.20 10.26237 -0.32767 6.76 0.22 0.54 0.00 2.25 8.27574 -0.26911 5.42 0.18 0.43 0.00 2.30 6.65108 -0.22042 4.35 0.14 0.35 0.00 2.35 5.32269 -0.18013 3.49 0.12 0.28 0.00 2.40 4.23686 -0.14695 2.81 0.10 0.23 0.00 2.45 3.34961 -0.11971 2.27 0.08 0.19 0.00 2.50 2.62500 -0.09740 1.83 0.07 0.16 0.00 2.55 2.03363 -0.07918 1.48 0.06 0.13 0.00 2.60 1.55145 -0.06431 1.19 0.05 0.11 0.00 2.65 1.15879 -0.05220 0.96 0.04 0.09 0.00 2.70 0.83952 -0.04235 0.78 0.04 0.08 0.00 2.75 0.58046 -0.03435 0.63 0.04 0.07 0.00 2.80 0.37080 -0.02784 0.51 0.04 0.07 0.00 2.85 0.20167 -0.02257 0.41 0.05 0.08 0.00 2.90 0.06579 -0.01829 0.33 0.09 0.14 0.00 2.95 -0.04281 -0.01481 0.27 0.09 0.12 0.00 3.00 -0.12903 -0.01200 0.22 0.02 0.02 0.00 3.05 -0.19690 -0.00972 0.18 0.01 0.01 0.00 3.10 -0.24974 -0.00787 0.14 0.00 0.00 0.00 3.15 -0.29027 -0.00637 0.12 0.00 0.00 0.00 3.20 -0.32076 -0.00516 0.09 0.00 0.00 0.00 3.25 -0.34303 -0.00418 0.08 0.00 0.00 0.00 388 Height Etot EQels(tot) 4th/EQels(tot) 4th/Etot EDR(2)/Etot Eind2(2)/Etot 3.30 -0.35863 -0.00338 0.06 0.00 0.00 0.00 3.35 -0.36880 -0.00274 0.05 0.00 0.00 0.00 3.40 -0.37458 -0.00222 0.04 0.00 0.00 0.00 3.45 -0.37682 -0.00179 0.03 0.00 0.00 0.00 3.50 -0.37623 -0.00145 0.03 0.00 0.00 0.00 3.55 -0.37339 -0.00117 0.03 0.00 0.00 0.00 3.60 -0.36876 -0.00095 0.02 0.00 0.00 0.00 3.65 -0.36275 -0.00077 0.01 0.00 0.00 0.00 3.70 -0.35567 -0.00062 0.02 0.00 0.00 0.00 3.75 -0.34779 -0.00050 0.00 0.00 0.00 0.00 3.80 -0.33933 -0.00041 0.00 0.00 0.00 0.00 3.85 -0.33045 -0.00033 0.00 0.00 0.00 0.00 3.90 -0.32132 -0.00027 0.00 0.00 0.00 0.00 3.95 -0.31203 -0.00022 0.00 0.00 0.00 0.00 4.00 -0.30269 -0.00018 0.00 0.00 0.00 0.00 389 (a) (b) (c) Figure C3-1 The 2nd/1st PEPR for a system of Cart/Mg2+ in the UBE case for a. dispersion, repulsion and the sum of these interactions, b. anisotropic H2 induced dipole-surface (induction), and c. H2 quadrupole-surface interaction energy. 0.000.020.040.062.4 2.8 3.2 3.6 4.02nd/1st PEPR (%)Height [Å]VRVDVDR0.01.02.03.04.02.4 2.6 2.8 3.0 3.2 3.42nd/1st PEPR (%)Height [Å]Anisotropic Induction02040602.4 2.6 2.8 3.0 3.2 3.42nd/1st PEPR (%)Height [Å]H₂ quadrupole-surface390 Table C3-3 The energy results for Cart/Mg2+ in the UBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic H2 induced dipole-surface (induction), and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 3.50429 10.46419 -0.02019 -3.77582 -0.00148 2.27315 -2.93969 -1.31417 2.80799 -3.56778 0.00000 -0.35937 -0.07348 2.05 2.80206 8.98871 -0.01449 -3.49597 -0.00129 2.03875 -2.36468 -0.94811 1.81693 -2.88746 0.00000 -0.29085 -0.04813 2.10 2.29874 7.73123 -0.01051 -3.24223 -0.00113 1.82921 -1.90358 -0.68479 1.17743 -2.33687 0.00000 -0.23539 -0.03152 2.15 1.91868 6.65845 -0.00770 -3.01146 -0.00099 1.64174 -1.53341 -0.49509 0.76402 -1.89126 0.00000 -0.19050 -0.02065 2.20 1.61649 5.74186 -0.00571 -2.80102 -0.00088 1.47392 -1.23593 -0.35825 0.49634 -1.53063 0.00000 -0.15418 -0.01352 2.25 1.36532 4.95749 -0.00427 -2.60864 -0.00078 1.32359 -0.99667 -0.25943 0.32277 -1.23876 0.00000 -0.12478 -0.00886 2.30 1.14939 4.28518 -0.00322 -2.43239 -0.00070 1.18885 -0.80409 -0.18800 0.21009 -1.00255 0.00000 -0.10098 -0.00580 2.35 0.95957 3.70808 -0.00244 -2.27062 -0.00063 1.06804 -0.64897 -0.13631 0.13685 -0.81138 0.00000 -0.08173 -0.00380 2.40 0.79049 3.21200 -0.00187 -2.12187 -0.00056 0.95967 -0.52395 -0.09889 0.08920 -0.65666 0.00000 -0.06614 -0.00249 2.45 0.63897 2.78500 -0.00143 -1.98487 -0.00051 0.86242 -0.42314 -0.07177 0.05818 -0.53145 0.00000 -0.05353 -0.00163 2.50 0.50305 2.41704 -0.00111 -1.85852 -0.00046 0.77513 -0.34182 -0.05211 0.03797 -0.43011 0.00000 -0.04332 -0.00107 2.55 0.38134 2.09961 -0.00086 -1.74184 -0.00041 0.69676 -0.27619 -0.03785 0.02479 -0.34809 0.00000 -0.03506 -0.00070 2.60 0.27279 1.82548 -0.00067 -1.63395 -0.00037 0.62637 -0.22321 -0.02750 0.01619 -0.28172 0.00000 -0.02838 -0.00046 2.65 0.17648 1.58853 -0.00052 -1.53408 -0.00034 0.56314 -0.18042 -0.01998 0.01058 -0.22800 0.00000 -0.02297 -0.00030 2.70 0.09155 1.38353 -0.00041 -1.44154 -0.00030 0.50634 -0.14586 -0.01452 0.00691 -0.18452 0.00000 -0.01859 -0.00020 2.75 0.01714 1.20602 -0.00032 -1.35570 -0.00027 0.45529 -0.11793 -0.01056 0.00452 -0.14934 0.00000 -0.01504 -0.00013 2.80 -0.04759 1.05219 -0.00025 -1.27599 -0.00025 0.40942 -0.09536 -0.00768 0.00295 -0.12086 0.00000 -0.01217 -0.00008 2.85 -0.10346 0.91877 -0.00020 -1.20192 -0.00023 0.36818 -0.07712 -0.00558 0.00193 -0.09782 0.00000 -0.00985 -0.00006 391 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.90 -0.15129 0.80297 -0.00015 -1.13302 -0.00020 0.33112 -0.06237 -0.00406 0.00126 -0.07916 0.00000 -0.00797 -0.00004 2.95 -0.19186 0.70237 -0.00012 -1.06889 -0.00019 0.29779 -0.05045 -0.00295 0.00083 -0.06407 0.00000 -0.00645 -0.00002 3.00 -0.22591 0.61491 -0.00010 -1.00913 -0.00017 0.26784 -0.04081 -0.00215 0.00054 -0.05185 0.00000 -0.00522 -0.00002 3.05 -0.25415 0.53881 -0.00008 -0.95342 -0.00015 0.24090 -0.03301 -0.00156 0.00035 -0.04196 0.00000 -0.00423 -0.00001 3.10 -0.27722 0.47256 -0.00006 -0.90143 -0.00014 0.21667 -0.02671 -0.00114 0.00023 -0.03396 0.00000 -0.00342 -0.00001 3.15 -0.29573 0.41482 -0.00005 -0.85288 -0.00013 0.19489 -0.02161 -0.00083 0.00015 -0.02749 0.00000 -0.00277 0.00000 3.20 -0.31023 0.36447 -0.00004 -0.80751 -0.00012 0.17530 -0.01748 -0.00060 0.00010 -0.02225 0.00000 -0.00224 0.00000 3.25 -0.32123 0.32052 -0.00003 -0.76507 -0.00010 0.15769 -0.01415 -0.00044 0.00007 -0.01800 0.00000 -0.00181 0.00000 3.30 -0.32918 0.28213 -0.00002 -0.72536 -0.00010 0.14184 -0.01145 -0.00032 0.00004 -0.01457 0.00000 -0.00147 0.00000 3.35 -0.33449 0.24857 -0.00002 -0.68817 -0.00009 0.12759 -0.00926 -0.00023 0.00003 -0.01179 0.00000 -0.00119 0.00000 3.40 -0.33752 0.21920 -0.00001 -0.65332 -0.00008 0.11477 -0.00749 -0.00017 0.00002 -0.00954 0.00000 -0.00096 0.00000 3.45 -0.33861 0.19348 -0.00001 -0.62063 -0.00007 0.10324 -0.00606 -0.00012 0.00001 -0.00772 0.00000 -0.00078 0.00000 3.50 -0.33804 0.17094 -0.00001 -0.58996 -0.00007 0.09287 -0.00491 -0.00009 0.00001 -0.00625 0.00000 -0.00063 0.00000 3.55 -0.33606 0.15117 -0.00001 -0.56115 -0.00006 0.08355 -0.00397 -0.00007 0.00001 -0.00506 0.00000 -0.00051 0.00000 3.60 -0.33290 0.13381 -0.00001 -0.53408 -0.00006 0.07516 -0.00321 -0.00005 0.00000 -0.00409 0.00000 -0.00041 0.00000 3.65 -0.32876 0.11856 0.00000 -0.50862 -0.00005 0.06761 -0.00260 -0.00003 0.00000 -0.00331 0.00000 -0.00033 0.00000 3.70 -0.32381 0.10514 0.00000 -0.48467 -0.00005 0.06082 -0.00210 -0.00003 0.00000 -0.00268 0.00000 -0.00027 0.00000 3.80 -0.31207 0.08293 0.00000 -0.44087 -0.00004 0.04922 -0.00138 -0.00001 0.00000 -0.00176 0.00000 -0.00018 0.00000 3.90 -0.29865 0.06566 0.00000 -0.40195 -0.00003 0.03983 -0.00090 -0.00001 0.00000 -0.00115 0.00000 -0.00012 0.00000 4.00 -0.28429 0.05218 0.00000 -0.36727 -0.00003 0.03224 -0.00059 0.00000 0.00000 -0.00075 0.00000 -0.00008 0.00000 392 Table C3-4 The energy results for Cart/Saddle in the UBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic H2 induced dipole-surface (induction), and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 24.65995 29.98779 -0.36263 -4.45075 -0.00286 0.00000 -0.50409 0.00000 -0.06685 0.00000 0.00000 0.00000 0.00000 2.05 20.08141 24.83217 -0.24073 -4.09964 -0.00250 0.00000 -0.40930 0.00000 -0.04407 0.00000 0.00000 0.00000 0.00000 2.10 16.32867 20.59895 -0.16166 -3.78028 -0.00219 0.00000 -0.33221 0.00000 -0.02903 0.00000 0.00000 0.00000 0.00000 2.15 13.25331 17.11597 -0.10969 -3.48961 -0.00193 0.00000 -0.26955 0.00000 -0.01911 0.00000 0.00000 0.00000 0.00000 2.20 10.73355 14.24522 -0.07512 -3.22487 -0.00169 0.00000 -0.21864 0.00000 -0.01258 0.00000 0.00000 0.00000 0.00000 2.25 8.66942 11.87529 -0.05187 -2.98357 -0.00148 0.00000 -0.17730 0.00000 -0.00827 0.00000 0.00000 0.00000 0.00000 2.30 6.97891 9.91586 -0.03608 -2.76344 -0.00130 0.00000 -0.14375 0.00000 -0.00544 0.00000 0.00000 0.00000 0.00000 2.35 5.59474 8.29343 -0.02527 -2.56247 -0.00115 0.00000 -0.11652 0.00000 -0.00357 0.00000 0.00000 0.00000 0.00000 2.40 4.46180 6.94807 -0.01780 -2.37883 -0.00101 0.00000 -0.09443 0.00000 -0.00235 0.00000 0.00000 0.00000 0.00000 2.45 3.53488 5.83083 -0.01261 -2.21085 -0.00089 0.00000 -0.07651 0.00000 -0.00154 0.00000 0.00000 0.00000 0.00000 2.50 2.77695 4.90165 -0.00898 -2.05707 -0.00078 0.00000 -0.06199 0.00000 -0.00101 0.00000 0.00000 0.00000 0.00000 2.55 2.15766 4.12770 -0.00642 -1.91614 -0.00069 0.00000 -0.05022 0.00000 -0.00066 0.00000 0.00000 0.00000 0.00000 2.60 1.65213 3.48206 -0.00461 -1.78685 -0.00061 0.00000 -0.04067 0.00000 -0.00044 0.00000 0.00000 0.00000 0.00000 2.65 1.23998 2.94259 -0.00333 -1.66813 -0.00054 0.00000 -0.03294 0.00000 -0.00029 0.00000 0.00000 0.00000 0.00000 2.70 0.90448 2.49113 -0.00241 -1.55900 -0.00047 0.00000 -0.02668 0.00000 -0.00019 0.00000 0.00000 0.00000 0.00000 2.75 0.63192 2.11271 -0.00175 -1.45858 -0.00042 0.00000 -0.02160 0.00000 -0.00012 0.00000 0.00000 0.00000 0.00000 2.80 0.41105 1.79499 -0.00128 -1.36607 -0.00037 0.00000 -0.01749 0.00000 -0.00008 0.00000 0.00000 0.00000 0.00000 2.85 0.23263 1.52778 -0.00093 -1.28076 -0.00033 0.00000 -0.01416 0.00000 -0.00005 0.00000 0.00000 0.00000 0.00000 393 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.90 0.08908 1.30269 -0.00069 -1.20202 -0.00029 0.00000 -0.01147 0.00000 -0.00003 0.00000 0.00000 0.00000 0.00000 2.95 -0.02585 1.11275 -0.00051 -1.12925 -0.00026 0.00000 -0.00928 0.00000 -0.00002 0.00000 0.00000 0.00000 0.00000 3.00 -0.11728 0.95221 -0.00037 -1.06193 -0.00023 0.00000 -0.00752 0.00000 -0.00001 0.00000 0.00000 0.00000 0.00000 3.05 -0.18943 0.81628 -0.00028 -0.99960 -0.00021 0.00000 -0.00608 0.00000 -0.00001 0.00000 0.00000 0.00000 0.00000 3.10 -0.24576 0.70099 -0.00021 -0.94182 -0.00018 0.00000 -0.00492 0.00000 -0.00001 0.00000 0.00000 0.00000 0.00000 3.15 -0.28914 0.60305 -0.00015 -0.88820 -0.00016 0.00000 -0.00399 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.20 -0.32193 0.51970 -0.00012 -0.83839 -0.00015 0.00000 -0.00323 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.25 -0.34606 0.44864 -0.00009 -0.79208 -0.00013 0.00000 -0.00261 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.30 -0.36314 0.38797 -0.00007 -0.74898 -0.00012 0.00000 -0.00211 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.35 -0.37448 0.33607 -0.00005 -0.70883 -0.00011 0.00000 -0.00171 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.40 -0.38117 0.29161 -0.00004 -0.67138 -0.00010 0.00000 -0.00139 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.45 -0.38412 0.25345 -0.00003 -0.63643 -0.00009 0.00000 -0.00112 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.50 -0.38405 0.22065 -0.00002 -0.60378 -0.00008 0.00000 -0.00091 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.55 -0.38159 0.19240 -0.00002 -0.57324 -0.00007 0.00000 -0.00073 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.60 -0.37723 0.16804 -0.00001 -0.54465 -0.00006 0.00000 -0.00059 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.65 -0.37138 0.14700 -0.00001 -0.51788 -0.00006 0.00000 -0.00048 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.70 -0.36438 0.12879 -0.00001 -0.49277 -0.00005 0.00000 -0.00039 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.80 -0.34803 0.09932 0.00000 -0.44707 -0.00004 0.00000 -0.00026 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 3.90 -0.32983 0.07706 0.00000 -0.40670 -0.00004 0.00000 -0.00017 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 4.00 -0.31091 0.06013 0.00000 -0.37091 -0.00003 0.00000 -0.00011 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 394 Table C3-5 The energy results for the Heli/O2- system in the UBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic H2 induced dipole-surface (induction), and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 169.04317 210.61067 -25.64748 -7.29668 -0.02092 1.13657 -1.95980 -3.01077 -2.79626 -3.56778 0.00000 0.17969 -0.04898 2.05 129.69173 156.70255 -14.19841 -6.42920 -0.01573 1.01937 -1.57646 -2.17212 -1.80934 -2.88746 0.00000 0.14542 -0.03208 2.10 99.04666 117.44106 -7.97261 -5.69063 -0.01193 0.91460 -1.26906 -1.56885 -1.17251 -2.33687 0.00000 0.11769 -0.02101 2.15 75.52428 88.62939 -4.53785 -5.05876 -0.00912 0.82087 -1.02228 -1.13426 -0.76083 -1.89126 0.00000 0.09525 -0.01376 2.20 57.59981 67.33236 -2.61657 -4.51563 -0.00703 0.73696 -0.82396 -0.82076 -0.49427 -1.53063 0.00000 0.07709 -0.00902 2.25 43.98299 51.48077 -1.52760 -4.04666 -0.00546 0.66179 -0.66445 -0.59436 -0.32142 -1.23876 0.00000 0.06239 -0.00591 2.30 33.64339 39.60344 -0.90254 -3.63996 -0.00427 0.59443 -0.53606 -0.43070 -0.20921 -1.00255 0.00000 0.05049 -0.00387 2.35 25.78344 30.64695 -0.53937 -3.28577 -0.00337 0.53402 -0.43265 -0.31229 -0.13628 -0.81138 0.00000 0.04086 -0.00253 2.40 19.79577 23.85148 -0.32590 -2.97607 -0.00267 0.47984 -0.34930 -0.22655 -0.08883 -0.65666 0.00000 0.03307 -0.00166 2.45 15.22194 18.66507 -0.19901 -2.70421 -0.00214 0.43121 -0.28210 -0.16442 -0.05794 -0.53145 0.00000 0.02677 -0.00109 2.50 11.71740 14.68417 -0.12277 -2.46468 -0.00172 0.38757 -0.22788 -0.11938 -0.03781 -0.43011 0.00000 0.02166 -0.00071 2.55 9.02349 11.61175 -0.07649 -2.25285 -0.00139 0.34838 -0.18413 -0.08670 -0.02468 -0.34809 0.00000 0.01753 -0.00047 2.60 6.94598 9.22790 -0.04811 -2.06489 -0.00113 0.31318 -0.14881 -0.06299 -0.01612 -0.28172 0.00000 0.01419 -0.00031 2.65 5.33873 7.36880 -0.03054 -1.89753 -0.00093 0.28157 -0.12028 -0.04578 -0.01053 -0.22800 0.00000 0.01148 -0.00020 2.70 4.09153 5.91174 -0.01956 -1.74804 -0.00077 0.25317 -0.09724 -0.03327 -0.00688 -0.18452 0.00000 0.00929 -0.00013 2.75 3.12102 4.76429 -0.01263 -1.61410 -0.00064 0.22765 -0.07862 -0.02419 -0.00450 -0.14934 0.00000 0.00752 -0.00009 2.80 2.36390 3.85645 -0.00823 -1.49374 -0.00053 0.20471 -0.06358 -0.01759 -0.00294 -0.12086 0.00000 0.00609 -0.00006 2.85 1.77196 3.13494 -0.00540 -1.38526 -0.00044 0.18409 -0.05141 -0.01279 -0.00192 -0.09782 0.00000 0.00493 -0.00004 395 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.90 1.30832 2.55899 -0.00358 -1.28724 -0.00037 0.16556 -0.04158 -0.00931 -0.00126 -0.07916 0.00000 0.00399 -0.00002 2.95 0.94471 2.09728 -0.00239 -1.19842 -0.00032 0.14890 -0.03364 -0.00677 -0.00082 -0.06407 0.00000 0.00323 -0.00002 3.00 0.65931 1.72562 -0.00160 -1.11774 -0.00027 0.13392 -0.02721 -0.00492 -0.00054 -0.05185 0.00000 0.00261 -0.00001 3.05 0.43528 1.42524 -0.00109 -1.04428 -0.00023 0.12045 -0.02201 -0.00358 -0.00035 -0.04196 0.00000 0.00211 -0.00001 3.10 0.25953 1.18151 -0.00074 -0.97724 -0.00020 0.10834 -0.01781 -0.00261 -0.00023 -0.03396 0.00000 0.00171 0.00000 3.15 0.12187 0.98300 -0.00051 -0.91592 -0.00017 0.09745 -0.01441 -0.00190 -0.00015 -0.02749 0.00000 0.00138 0.00000 3.20 0.01434 0.82071 -0.00035 -0.85972 -0.00015 0.08765 -0.01166 -0.00138 -0.00010 -0.02225 0.00000 0.00112 0.00000 3.25 -0.06931 0.68756 -0.00024 -0.80809 -0.00013 0.07884 -0.00943 -0.00101 -0.00006 -0.01800 0.00000 0.00091 0.00000 3.30 -0.13399 0.57792 -0.00017 -0.76059 -0.00011 0.07092 -0.00763 -0.00073 -0.00004 -0.01457 0.00000 0.00073 0.00000 3.35 -0.18358 0.48734 -0.00012 -0.71679 -0.00010 0.06380 -0.00617 -0.00053 -0.00003 -0.01179 0.00000 0.00059 0.00000 3.40 -0.22116 0.41225 -0.00009 -0.67633 -0.00009 0.05739 -0.00500 -0.00039 -0.00002 -0.00954 0.00000 0.00048 0.00000 3.45 -0.24915 0.34980 -0.00006 -0.63890 -0.00008 0.05162 -0.00404 -0.00028 -0.00001 -0.00772 0.00000 0.00039 0.00000 3.50 -0.26951 0.29769 -0.00004 -0.60421 -0.00007 0.04644 -0.00327 -0.00021 -0.00001 -0.00625 0.00000 0.00031 0.00000 3.55 -0.28378 0.25408 -0.00003 -0.57202 -0.00006 0.04177 -0.00265 -0.00015 -0.00001 -0.00506 0.00000 0.00025 0.00000 3.60 -0.29320 0.21746 -0.00002 -0.54209 -0.00005 0.03758 -0.00214 -0.00011 0.00000 -0.00409 0.00000 0.00021 0.00000 3.65 -0.29876 0.18664 -0.00002 -0.51423 -0.00005 0.03380 -0.00173 -0.00008 0.00000 -0.00331 0.00000 0.00017 0.00000 3.70 -0.30126 0.16060 -0.00001 -0.48826 -0.00004 0.03041 -0.00140 -0.00006 0.00000 -0.00268 0.00000 0.00014 0.00000 3.80 -0.29953 0.11984 -0.00001 -0.44135 -0.00003 0.02461 -0.00092 -0.00003 0.00000 -0.00176 0.00000 0.00009 0.00000 3.90 -0.29179 0.09030 0.00000 -0.40028 -0.00003 0.01992 -0.00060 -0.00002 0.00000 -0.00115 0.00000 0.00006 0.00000 4.00 -0.28049 0.06867 0.00000 -0.36415 -0.00002 0.01612 -0.00039 -0.00001 0.00000 -0.00075 0.00000 0.00004 0.00000 396 The PT series convergence for H2/MgO systems in the LBE case (section 4.2.2.1) As stated in section 4.1.2, the only difference between the UBE and LBE cases in Model II for the MgO substrate is in the repulsion (EDR) energy. This means that, in the LBE case, all parts of the energy are the same as those in the UBE case, except ER (or EDR). Also, because of a stronger energy in this case, the optimum heights shift to lower values compared to the UBE case. Thus, the molecule sits closer to the surface, which may result in energies in RI that are not well-converged. This effect is analyzed in detail in this section. Like the UBE case, only systems for which the optimum height occurs at distances closer than 3.10 Å to the surface are discussed. C.4.1. Heli/Mg2+ Table C4-1 shows the data for this system. The only difference with the energies of the same system (Heli/Mg2+) in the UBE case (Table 4-7) is in ER, which is smaller for the LBE case. Also, since the EQels energies are the same, similar to the UBE case, the PT series in EQels become divergent at heights below 2.30 Å. In this system, the RI is from 2.40 up to 2.80 Å. To investigate the effect of the second-order PT terms on the energies a similar procedure as section 4.2.2.1.1 is used here. It is found that going up to this PT term is enough to get a well-converged energy in all types of the interaction potentials except for EQels. In the next step, the effect of the third- and fourth- order EQels corrections are investigated and the results show that going up to fourth-order correction in EQels is necessary to get a well-converged energy in the RI range. In order to see the effect of the fifth-order interactions on the energy, these corrections are also estimated. The results are presented in Table C4-2. It is seen that at the height of 2.40 Å, the fifth-order energy is almost 7% of the total energy. This amount is not in the acceptable range and hence, at this height, going up to fourth order is not enough to obtain a well-converged 397 energy. It should be added that in the Heli/Mg2+ system, the energy difference between heights of 2.40 and 2.80 Å is 0.17806 kcal/mol (the energy barrier against the vertical motion of the molecule), which is about 27% of the minimum energy at 2.80 Å. This is a very large barrier that prevents the molecule to gain access to heights of 2.40 Å at low temperatures. The effect of not including the fifth-order EQels correction terms in the Heli/Mg2+ system (LBE) on the energy of the layer at heights equal to 2.40 Å and lower is discussed further in Chapter 6 and Chapter 7, where the simulation results are presented. It is interesting to note that, as expected, at the same heights, EQels(4)/EQels(tot) (or 4th/EQels(tot)) and EQels(5)/EQels(tot) (or 5th/EQels(tot)) ratios in Table C4-2 are the same as those in Table 4-8. However, a stronger binding energy (Etot) in the LBE case (Table C4-2) results in lower 4th/Etot and 5th/Etot ratios compared to the UBE case (Table 4-8). For example, at 2.40 Å, the fifth-order energy term in EQels contributes about 7% to the total energy in the LBE case compared to 13% in the UBE case. Table C4-2 also shows a decrease in both EDR(2)/Etot and Eind2(2)/Etot ratios compared to the same data in Table 4-8. Therefore, in the LBE case, a better convergence is achieved for the EDR and Eind2 energies as well. 398 Table C4-1 The energy results for the system of Heli/Mg2+ in the LBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 -0.56155 9.46251 -0.01164 -3.69926 -0.00099 -1.13657 -1.95980 3.01077 -2.79626 -3.56778 0.00000 0.17969 -0.04898 2.05 -0.30424 8.13136 -0.00835 -3.42457 -0.00086 -1.01937 -1.57646 2.17212 -1.80934 -2.88746 0.00000 0.14542 -0.03208 2.10 -0.20980 6.99567 -0.00605 -3.17541 -0.00075 -0.91460 -1.26906 1.56885 -1.17251 -2.33687 0.00000 0.11769 -0.02101 2.15 -0.20403 6.02588 -0.00444 -2.94874 -0.00066 -0.82087 -1.02228 1.13426 -0.76083 -1.89126 0.00000 0.09525 -0.01376 2.20 -0.24342 5.19663 -0.00329 -2.74198 -0.00059 -0.73696 -0.82396 0.82076 -0.49427 -1.53063 0.00000 0.07709 -0.00902 2.25 -0.30269 4.48656 -0.00246 -2.55296 -0.00052 -0.66179 -0.66445 0.59436 -0.32142 -1.23876 0.00000 0.06239 -0.00591 2.30 -0.36751 3.87764 -0.00185 -2.37978 -0.00047 -0.59443 -0.53606 0.43070 -0.20921 -1.00255 0.00000 0.05049 -0.00387 2.35 -0.43004 3.35479 -0.00141 -2.22083 -0.00042 -0.53402 -0.43265 0.31229 -0.13628 -0.81138 0.00000 0.04086 -0.00253 2.40 -0.48631 2.90524 -0.00107 -2.07470 -0.00038 -0.47984 -0.34930 0.22655 -0.08883 -0.65666 0.00000 0.03307 -0.00166 2.45 -0.53458 2.51827 -0.00083 -1.94014 -0.00034 -0.43121 -0.28210 0.16442 -0.05794 -0.53145 0.00000 0.02677 -0.00109 2.50 -0.57436 2.18481 -0.00064 -1.81607 -0.00030 -0.38757 -0.22788 0.11938 -0.03781 -0.43011 0.00000 0.02166 -0.00071 2.55 -0.60590 1.89717 -0.00049 -1.70153 -0.00027 -0.34838 -0.18413 0.08670 -0.02468 -0.34809 0.00000 0.01753 -0.00047 2.60 -0.62979 1.64882 -0.00038 -1.59564 -0.00025 -0.31318 -0.14881 0.06299 -0.01612 -0.28172 0.00000 0.01419 -0.00031 2.65 -0.64679 1.43420 -0.00030 -1.49766 -0.00022 -0.28157 -0.12028 0.04578 -0.01053 -0.22800 0.00000 0.01148 -0.00020 2.70 -0.65769 1.24858 -0.00023 -1.40690 -0.00020 -0.25317 -0.09724 0.03327 -0.00688 -0.18452 0.00000 0.00929 -0.00013 2.75 -0.66331 1.08790 -0.00018 -1.32273 -0.00018 -0.22765 -0.07862 0.02419 -0.00450 -0.14934 0.00000 0.00752 -0.00009 2.80 -0.66437 0.94872 -0.00014 -1.24462 -0.00017 -0.20471 -0.06358 0.01759 -0.00294 -0.12086 0.00000 0.00609 -0.00006 2.85 -0.66157 0.82804 -0.00011 -1.17205 -0.00015 -0.18409 -0.05141 0.01279 -0.00192 -0.09782 0.00000 0.00493 -0.00004 2.90 -0.65554 0.72334 -0.00009 -1.10458 -0.00014 -0.16556 -0.04158 0.00931 -0.00126 -0.07916 0.00000 0.00399 -0.00002 399 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.95 -0.64681 0.63243 -0.00007 -1.04179 -0.00012 -0.14890 -0.03364 0.00677 -0.00082 -0.06407 0.00000 0.00323 -0.00002 3.00 -0.63588 0.55343 -0.00005 -0.98331 -0.00011 -0.13392 -0.02721 0.00492 -0.00054 -0.05185 0.00000 0.00261 -0.00001 3.05 -0.62317 0.48473 -0.00004 -0.92881 -0.00010 -0.12045 -0.02201 0.00358 -0.00035 -0.04196 0.00000 0.00211 -0.00001 3.10 -0.60906 0.42494 -0.00003 -0.87797 -0.00009 -0.10834 -0.01781 0.00261 -0.00023 -0.03396 0.00000 0.00171 0.00000 3.15 -0.59387 0.37286 -0.00003 -0.83051 -0.00008 -0.09745 -0.01441 0.00190 -0.00015 -0.02749 0.00000 0.00138 0.00000 3.20 -0.57787 0.32746 -0.00002 -0.78617 -0.00008 -0.08765 -0.01166 0.00138 -0.00010 -0.02225 0.00000 0.00112 0.00000 3.25 -0.56131 0.28786 -0.00002 -0.74471 -0.00007 -0.07884 -0.00943 0.00101 -0.00006 -0.01800 0.00000 0.00091 0.00000 3.30 -0.54438 0.25327 -0.00001 -0.70593 -0.00006 -0.07092 -0.00763 0.00073 -0.00004 -0.01457 0.00000 0.00073 0.00000 3.35 -0.52725 0.22306 -0.00001 -0.66963 -0.00006 -0.06380 -0.00617 0.00053 -0.00003 -0.01179 0.00000 0.00059 0.00000 3.40 -0.51008 0.19663 -0.00001 -0.63561 -0.00005 -0.05739 -0.00500 0.00039 -0.00002 -0.00954 0.00000 0.00048 0.00000 3.45 -0.49297 0.17349 -0.00001 -0.60371 -0.00005 -0.05162 -0.00404 0.00028 -0.00001 -0.00772 0.00000 0.00039 0.00000 3.50 -0.47603 0.15323 -0.00001 -0.57379 -0.00004 -0.04644 -0.00327 0.00021 -0.00001 -0.00625 0.00000 0.00031 0.00000 3.55 -0.45934 0.13546 0.00000 -0.54570 -0.00004 -0.04177 -0.00265 0.00015 -0.00001 -0.00506 0.00000 0.00025 0.00000 3.60 -0.44297 0.11986 0.00000 -0.51931 -0.00004 -0.03758 -0.00214 0.00011 0.00000 -0.00409 0.00000 0.00021 0.00000 3.65 -0.42696 0.10616 0.00000 -0.49450 -0.00003 -0.03380 -0.00173 0.00008 0.00000 -0.00331 0.00000 0.00017 0.00000 3.70 -0.41136 0.09412 0.00000 -0.47116 -0.00003 -0.03041 -0.00140 0.00006 0.00000 -0.00268 0.00000 0.00014 0.00000 3.75 -0.39619 0.08352 0.00000 -0.44919 -0.00003 -0.02736 -0.00114 0.00004 0.00000 -0.00217 0.00000 0.00011 0.00000 3.80 -0.38148 0.07419 0.00000 -0.42849 -0.00003 -0.00176 0.00000 0.00009 0.00000 -0.00176 0.00000 0.00009 0.00000 3.85 -0.36725 0.06596 0.00000 -0.40899 -0.00002 -0.00142 0.00000 0.00007 0.00000 -0.00142 0.00000 0.00007 0.00000 3.90 -0.35349 0.05870 0.00000 -0.39059 -0.00002 -0.00115 0.00000 0.00006 0.00000 -0.00115 0.00000 0.00006 0.00000 4.00 -0.32744 0.04663 0.00000 -0.35683 -0.00002 -0.00075 0.00000 0.00004 0.00000 -0.00075 0.00000 0.00004 0.00000 400 Table C4-2 The energies and PEPR quantities for Heli/Mg2+ in the LBE case after adding the fourth-order EQels PT correction term to the energy. The RI for this system is from 2.40 up to 2.80 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy. 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels PT correction terms presented in Table C4-1. EQels(5) was estimated using the assumption explained in section 4.2.2.1. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %. Height Etot EQels(tot) EQels(5) 2nd/1st 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.00 -0.56155 -2.88187 2.59704 172.43 153.63 92.88 97.03 90.12 497.95 462.47 1.04 8.72 2.05 -0.30424 -2.23305 1.50714 154.65 137.78 83.30 81.03 67.49 594.70 495.38 1.27 10.55 2.10 -0.20980 -1.78732 0.87630 138.76 123.62 74.74 65.60 49.03 558.88 417.69 1.21 10.02 2.15 -0.20403 -1.46973 0.51035 124.54 110.95 67.08 51.77 34.72 372.90 250.13 0.82 6.75 2.20 -0.24342 -1.23443 0.29765 111.80 99.61 60.22 40.04 24.11 203.05 122.28 0.45 3.70 2.25 -0.30269 -1.05330 0.17382 100.40 89.45 54.08 30.52 16.50 106.19 57.43 0.24 1.95 2.30 -0.36751 -0.90899 0.10162 90.18 80.35 48.57 23.02 11.18 56.93 27.65 0.13 1.05 2.35 -0.43004 -0.79065 0.05947 81.02 72.18 43.64 17.24 7.52 31.69 13.83 0.07 0.59 2.40 -0.48631 -0.69142 0.03483 72.80 64.86 39.21 12.85 5.04 18.27 7.16 0.04 0.34 2.45 -0.53458 -0.60682 0.02041 65.42 58.29 35.24 9.55 3.36 10.84 3.82 0.02 0.20 2.50 -0.57436 -0.53388 0.01197 58.80 52.39 31.67 7.08 2.24 6.58 2.08 0.01 0.12 2.55 -0.60590 -0.47049 0.00703 52.85 47.09 28.47 5.25 1.49 4.07 1.16 0.01 0.08 2.60 -0.62979 -0.41512 0.00413 47.51 42.33 25.59 3.88 0.99 2.56 0.66 0.00 0.05 2.65 -0.64679 -0.36661 0.00242 42.72 38.06 23.01 2.87 0.66 1.63 0.37 0.00 0.03 2.70 -0.65769 -0.32401 0.00142 38.41 34.22 20.69 2.12 0.44 1.05 0.22 0.00 0.02 2.75 -0.66331 -0.28657 0.00084 34.54 30.77 18.60 1.57 0.29 0.68 0.13 0.00 0.01 2.80 -0.66437 -0.25363 0.00049 31.06 27.67 16.73 1.16 0.19 0.44 0.07 0.00 0.01 2.85 -0.66157 -0.22464 0.00029 27.93 24.88 15.04 0.86 0.13 0.29 0.04 0.00 0.01 2.90 -0.65554 -0.19910 0.00017 25.12 22.38 13.53 0.63 0.09 0.19 0.03 0.00 0.00 401 Height Etot EQels(tot) EQels(5) 2nd/1st 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.95 -0.64681 -0.17659 0.00010 22.59 20.13 12.17 0.47 0.06 0.13 0.02 0.00 0.00 3.00 -0.63588 -0.15674 0.00006 20.32 18.10 10.94 0.34 0.04 0.08 0.01 0.00 0.00 3.05 -0.62317 -0.13923 0.00003 18.27 16.28 9.84 0.25 0.02 0.06 0.01 0.00 0.00 3.10 -0.60906 -0.12377 0.00002 16.44 14.64 8.85 0.19 0.02 0.04 0.00 0.00 0.00 3.15 -0.59387 -0.11011 0.00001 14.78 13.17 7.96 0.14 0.01 0.03 0.00 0.00 0.00 3.20 -0.57787 -0.09803 0.00001 13.30 11.85 7.16 0.10 0.01 0.02 0.00 0.00 0.00 3.25 -0.56131 -0.08733 0.00000 11.96 10.66 6.44 0.07 0.00 0.01 0.00 0.00 0.00 3.30 -0.54438 -0.07786 0.00000 10.76 9.59 5.80 0.05 0.00 0.01 0.00 0.00 0.00 3.35 -0.52725 -0.06946 0.00000 9.68 8.62 5.22 0.04 0.00 0.01 0.00 0.00 0.00 3.40 -0.51008 -0.06201 0.00000 8.71 7.76 4.70 0.03 0.00 0.00 0.00 0.00 0.00 3.45 -0.49297 -0.05539 0.00000 7.83 6.98 4.22 0.02 0.00 0.00 0.00 0.00 0.00 3.50 -0.47603 -0.04951 0.00000 7.05 6.28 3.80 0.02 0.00 0.00 0.00 0.00 0.00 3.55 -0.45934 -0.04428 0.00000 6.34 5.65 3.41 0.01 0.00 0.00 0.00 0.00 0.00 3.60 -0.44297 -0.03961 0.00000 5.70 5.08 3.03 0.01 0.00 0.00 0.00 0.00 0.00 3.65 -0.42696 -0.03546 0.00000 5.13 4.57 2.78 0.01 0.00 0.00 0.00 0.00 0.00 3.70 -0.41136 -0.03176 0.00000 4.61 4.11 2.43 0.00 0.00 0.00 0.00 0.00 0.00 3.75 -0.39619 -0.02845 0.00000 4.15 3.70 2.14 0.00 0.00 0.00 0.00 0.00 0.00 3.80 -0.38148 -0.02550 0.00000 3.73 3.33 1.96 0.00 0.00 0.00 0.00 0.00 0.00 3.85 -0.36725 -0.02286 0.00000 3.36 3.00 1.79 0.00 0.00 0.00 0.00 0.00 0.00 3.90 -0.35349 -0.02050 0.00000 3.02 2.69 1.85 0.00 0.00 0.00 0.00 0.00 0.00 4.00 -0.32744 -0.01650 0.00000 2.45 2.18 1.16 0.00 0.00 0.00 0.00 0.00 0.00 402 C.4.2. Para/Mg2+ Table C4-3 shows the data for this system. The RI for this system is from 2.50 up to 2.90 Å. Similar to the Heli/Mg2+ system, the only difference between the LBE and UBE energies is the repulsion energy (compare the energies in Table 4-9 with the data in Table C4-3). Since the EQels PT terms are the same in both the LBE and the UBE cases, the 4th/EQels(tot) and 5th/EQels(tot) ratios stay the same (compare the ratios in Table 4-10 with those in Table C4-4). However, due to a stronger binding energy (Etot) in the LBE case, at the same heights, the 4th/Etot and 5th/Etot ratios are smaller in the LBE case. In addition, as seen in Table C4-4 and based on our criteria, the energy is not converged enough at heights below 2.65 Å. The energy difference between this height and the optimum height of 2.90 Å is around 0.08878 kcal/mol which is only 16% of the minimum energy. This small barrier against the vertical motion of the molecule may affect the average height of the molecules in the layer. This means that the convergence of the H2-surface energy may have an impact on the energy of the layer. This impact will be fully discussed in Chapter 7. Table C4-4 also shows a decrease in both EDR(2)/Etot and Eind2(2)/Etot ratios compared to the same data in Table 4-8, which means that a better convergence for these two components of the energy is obtained in the Para/Mg2+ system of the LBE case. 403 Table C4-3 The energy results for Para/Mg2+ in the LBE case. The numbers in brackets (superscripts) denote the order of the perturbation term in the corresponding PT series. ER, ED, EQels, Eind0, and Eind2 are, in turn, the repulsion, dispersion, H2 quadrupole-surface electrostatic, isotropic induction, and anisotropic induction energies. Heights are in Å and the energies are in kcal/mol. Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.00 20.60146 9.55000 -0.03771 -3.72478 -0.00320 0.00000 -6.35122 10.14026 14.73268 -3.56778 0.00000 0.00000 -0.15874 2.05 13.49283 8.20548 -0.02704 -3.44837 -0.00278 0.00000 -5.10891 7.31568 9.53286 -2.88746 0.00000 0.00000 -0.10398 2.10 8.79673 7.05880 -0.01961 -3.19769 -0.00244 0.00000 -4.11270 5.28389 6.17762 -2.33687 0.00000 0.00000 -0.06810 2.15 5.68485 6.07995 -0.01438 -2.96965 -0.00215 0.00000 -3.31293 3.82016 4.00860 -1.89126 0.00000 0.00000 -0.04461 2.20 3.61634 5.24317 -0.01065 -2.76166 -0.00190 0.00000 -2.67024 2.76431 2.60415 -1.53063 0.00000 0.00000 -0.02922 2.25 2.23706 4.52680 -0.00796 -2.57152 -0.00169 0.00000 -2.15331 2.00181 1.69349 -1.23876 0.00000 0.00000 -0.01914 2.30 1.31432 3.91258 -0.00600 -2.39732 -0.00151 0.00000 -1.73723 1.45061 1.10226 -1.00255 0.00000 0.00000 -0.01253 2.35 0.69497 3.38524 -0.00456 -2.23743 -0.00135 0.00000 -1.40210 1.05179 0.71800 -0.81138 0.00000 0.00000 -0.00821 2.40 0.27784 2.93186 -0.00348 -2.09042 -0.00122 0.00000 -1.13200 0.76301 0.46801 -0.65666 0.00000 0.00000 -0.00538 2.45 -0.00396 2.54160 -0.00268 -1.95505 -0.00109 0.00000 -0.91420 0.55377 0.30525 -0.53145 0.00000 0.00000 -0.00352 2.50 -0.19478 2.20530 -0.00207 -1.83022 -0.00098 0.00000 -0.73851 0.40206 0.19919 -0.43011 0.00000 0.00000 -0.00231 2.55 -0.32411 1.91521 -0.00160 -1.71496 -0.00089 0.00000 -0.59671 0.29202 0.13005 -0.34809 0.00000 0.00000 -0.00151 2.60 -0.41160 1.66472 -0.00124 -1.60841 -0.00080 0.00000 -0.48224 0.21216 0.08494 -0.28172 0.00000 0.00000 -0.00099 2.65 -0.47035 1.44824 -0.00097 -1.50980 -0.00072 0.00000 -0.38980 0.15418 0.05549 -0.22800 0.00000 0.00000 -0.00065 2.70 -0.50920 1.26099 -0.00076 -1.41844 -0.00066 0.00000 -0.31512 0.11207 0.03627 -0.18452 0.00000 0.00000 -0.00042 2.75 -0.53406 1.09888 -0.00059 -1.33372 -0.00059 0.00000 -0.25479 0.08148 0.02371 -0.14934 0.00000 0.00000 -0.00028 2.80 -0.54897 0.95843 -0.00046 -1.25508 -0.00054 0.00000 -0.20603 0.05925 0.01550 -0.12086 0.00000 0.00000 -0.00018 2.85 -0.55669 0.83665 -0.00036 -1.18201 -0.00049 0.00000 -0.16662 0.04309 0.01014 -0.09782 0.00000 0.00000 -0.00012 2.90 -0.55913 0.73098 -0.00029 -1.11406 -0.00044 0.00000 -0.13476 0.03134 0.00663 -0.07916 0.00000 0.00000 -0.00008 404 Height Etot ER(1) ER(2) ED(1) ED(2) EQels(1) EQels(2) EQels(3) EQels(4) Eind0(1) Eind0(2) Eind2(1) Eind2(2) 2.95 -0.55763 0.63920 -0.00023 -1.05082 -0.00040 0.00000 -0.10900 0.02280 0.00434 -0.06407 0.00000 0.00000 -0.00005 3.00 -0.55314 0.55944 -0.00018 -0.99192 -0.00036 0.00000 -0.08817 0.01659 0.00284 -0.05185 0.00000 0.00000 -0.00003 3.05 -0.54637 0.49007 -0.00014 -0.93701 -0.00033 0.00000 -0.07133 0.01207 0.00186 -0.04196 0.00000 0.00000 -0.00002 3.10 -0.53783 0.42969 -0.00011 -0.88579 -0.00030 0.00000 -0.05771 0.00878 0.00122 -0.03396 0.00000 0.00000 -0.00001 3.15 -0.52793 0.37708 -0.00009 -0.83796 -0.00027 0.00000 -0.04669 0.00639 0.00080 -0.02749 0.00000 0.00000 -0.00001 3.20 -0.51697 0.33122 -0.00007 -0.79328 -0.00025 0.00000 -0.03777 0.00465 0.00052 -0.02225 0.00000 0.00000 -0.00001 3.25 -0.50521 0.29120 -0.00005 -0.75150 -0.00023 0.00000 -0.03056 0.00338 0.00034 -0.01800 0.00000 0.00000 0.00000 3.30 -0.49284 0.25625 -0.00004 -0.71241 -0.00021 0.00000 -0.02473 0.00246 0.00022 -0.01457 0.00000 0.00000 0.00000 3.35 -0.48003 0.22571 -0.00003 -0.67581 -0.00019 0.00000 -0.02001 0.00179 0.00015 -0.01179 0.00000 0.00000 0.00000 3.40 -0.46692 0.19899 -0.00003 -0.64151 -0.00017 0.00000 -0.01619 0.00131 0.00010 -0.00954 0.00000 0.00000 0.00000 3.45 -0.45363 0.17560 -0.00002 -0.60935 -0.00016 0.00000 -0.01310 0.00095 0.00006 -0.00772 0.00000 0.00000 0.00000 3.50 -0.44026 0.15511 -0.00002 -0.57918 -0.00014 0.00000 -0.01060 0.00069 0.00004 -0.00625 0.00000 0.00000 0.00000 3.55 -0.42688 0.13713 -0.00001 -0.55085 -0.00013 0.00000 -0.00858 0.00050 0.00003 -0.00506 0.00000 0.00000 0.00000 3.60 -0.41358 0.12136 -0.00001 -0.52423 -0.00012 0.00000 -0.00694 0.00037 0.00002 -0.00409 0.00000 0.00000 0.00000 3.65 -0.40041 0.10750 -0.00001 -0.49921 -0.00011 0.00000 -0.00562 0.00027 0.00001 -0.00331 0.00000 0.00000 0.00000 3.70 -0.38743 0.09532 -0.00001 -0.47566 -0.00010 0.00000 -0.00455 0.00019 0.00001 -0.00268 0.00000 0.00000 0.00000 3.75 -0.37466 0.08459 -0.00001 -0.45350 -0.00009 0.00000 -0.00368 0.00014 0.00000 -0.00217 0.00000 0.00000 0.00000 3.80 -0.36215 0.07515 0.00000 -0.43262 -0.00008 0.00000 -0.00298 0.00010 0.00000 -0.00176 0.00000 0.00000 0.00000 3.85 -0.34992 0.06682 0.00000 -0.41294 -0.00008 0.00000 -0.00241 0.00007 0.00000 -0.00142 0.00000 0.00000 0.00000 3.90 -0.33799 0.05948 0.00000 -0.39438 -0.00007 0.00000 -0.00195 0.00005 0.00000 -0.00115 0.00000 0.00000 0.00000 4.00 -0.31510 0.04725 0.00000 -0.36031 -0.00006 0.00000 -0.00128 0.00003 0.00000 -0.00075 0.00000 0.00000 0.00000 405 Table C4-4 The energies and PEPR quantities for Para/Mg2+ in the LBE case after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.50 up to 2.90 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy (shown in Table C4-3). 1st, 2nd, 3rd, and 4th in the ratios refer to the corresponding EQels correction terms in Table C4-3. EQels(5) was estimated using the assumption explained in in section 4.2.1. 5th in the ratios refers to this estimated energy. Heights are in Å, energies are in kcal/mol, and the ratios are in %. Height Etot EQels(tot) EQels(5) 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.00 20.60146 18.52172 21.40498 159.66 145.29 79.54 115.57 71.51 103.90 0.09 0.77 2.05 13.49283 11.73964 12.42200 143.19 130.31 81.20 105.81 70.65 92.06 0.09 0.77 2.10 8.79673 7.34880 7.22252 128.48 116.91 84.06 98.28 70.23 82.10 0.09 0.77 2.15 5.68485 4.51583 4.20633 115.31 104.93 88.77 93.15 70.51 73.99 0.10 0.78 2.20 3.61634 2.69823 2.45327 103.52 94.21 96.51 90.92 72.01 67.84 0.10 0.81 2.25 2.23706 1.54199 1.43265 92.96 84.60 109.82 92.91 75.70 64.04 0.10 0.86 2.30 1.31432 0.81563 0.83756 83.50 75.99 135.14 102.69 83.87 63.73 0.11 0.95 2.35 0.69497 0.36769 0.49013 75.02 68.26 195.27 133.30 103.31 70.53 0.14 1.18 2.40 0.27784 0.09903 0.28707 67.40 61.34 472.61 289.89 168.45 103.32 0.21 1.94 2.45 -0.00396 -0.05519 0.16826 60.57 55.12 553.10 304.88 7704.40 4246.83 8.80 88.90 2.50 -0.19478 -0.13725 0.09869 54.44 49.54 145.13 71.90 102.26 50.66 0.10 1.18 2.55 -0.32411 -0.17465 0.05791 48.94 44.53 74.46 33.16 40.12 17.87 0.03 0.47 2.60 -0.41160 -0.18515 0.03400 43.99 40.03 45.88 18.37 20.64 8.26 0.01 0.24 2.65 -0.47035 -0.18013 0.01997 39.55 35.99 30.81 11.09 11.80 4.25 0.00 0.14 2.70 -0.50920 -0.16679 0.01174 35.56 32.36 21.75 7.04 7.12 2.31 0.00 0.08 2.75 -0.53406 -0.14960 0.00690 31.98 29.10 15.85 4.61 4.44 1.29 0.00 0.05 2.80 -0.54897 -0.13128 0.00406 28.76 26.17 11.81 3.09 2.82 0.74 0.00 0.03 2.85 -0.55669 -0.11339 0.00239 25.86 23.53 8.94 2.10 1.82 0.43 0.00 0.02 2.90 -0.55913 -0.09679 0.00140 23.26 21.16 6.85 1.45 1.19 0.25 0.00 0.01 406 Height Etot EQels(tot) EQels(5) 3rd/2nd 4th/3rd 4th/EQels(tot) 5th/EQels(tot) 4th/Etot 5th/Etot EDR(2)/Etot Eind2(2)/Etot 2.95 -0.55763 -0.08186 0.00083 20.92 19.03 5.30 1.01 0.78 0.15 0.00 0.01 3.00 -0.55314 -0.06875 0.00049 18.81 17.12 4.13 0.71 0.51 0.09 0.01 0.01 3.05 -0.54637 -0.05740 0.00029 16.92 15.40 3.24 0.50 0.34 0.05 0.01 0.00 3.10 -0.53783 -0.04771 0.00017 15.22 13.85 2.55 0.35 0.23 0.03 0.01 0.00 3.15 -0.52793 -0.03950 0.00010 13.69 12.46 2.02 0.25 0.15 0.02 0.01 0.00 3.20 -0.51697 -0.03260 0.00006 12.31 11.20 1.60 0.18 0.10 0.01 0.01 0.00 3.25 -0.50521 -0.02684 0.00003 11.08 10.08 1.27 0.13 0.07 0.01 0.01 0.00 3.30 -0.49284 -0.02204 0.00002 9.96 9.06 1.01 0.09 0.05 0.00 0.01 0.00 3.35 -0.48003 -0.01807 0.00001 8.96 8.15 0.81 0.07 0.03 0.00 0.01 0.00 3.40 -0.46692 -0.01479 0.00001 8.06 7.33 0.65 0.05 0.02 0.00 0.01 0.00 3.45 -0.45363 -0.01209 0.00000 7.25 6.60 0.52 0.03 0.01 0.00 0.01 0.00 3.50 -0.44026 -0.00987 0.00000 6.52 5.94 0.42 0.02 0.01 0.00 0.01 0.00 3.55 -0.42688 -0.00805 0.00000 5.87 5.34 0.33 0.02 0.01 0.00 0.01 0.00 3.60 -0.41358 -0.00656 0.00000 5.28 4.80 0.27 0.01 0.00 0.00 0.01 0.00 3.65 -0.40041 -0.00534 0.00000 4.75 4.31 0.22 0.01 0.00 0.00 0.01 0.00 3.70 -0.38743 -0.00434 0.00000 4.27 3.91 0.17 0.01 0.00 0.00 0.01 0.00 3.75 -0.37466 -0.00353 0.00000 3.84 3.47 0.14 0.00 0.00 0.00 0.01 0.00 3.80 -0.36215 -0.00287 0.00000 3.46 3.11 0.11 0.00 0.00 0.00 0.01 0.00 3.85 -0.34992 -0.00233 0.00000 3.11 2.80 0.09 0.00 0.00 0.00 0.01 0.00 3.90 -0.33799 -0.00189 0.00000 2.80 2.56 0.07 0.00 0.00 0.00 0.01 0.00 3.95 -0.32638 -0.00154 0.00000 2.52 2.27 0.06 0.00 0.00 0.00 0.01 0.00 4.00 -0.31510 -0.00125 0.00000 2.26 2.08 0.05 0.00 0.00 0.00 0.01 0.00 407 C.4.3. The rest of the systems (Cart/Mg2+, Heli/Saddle, Para/Saddle, etc.) Since in the rest of the systems, the optimum heights occur at distances greater than 3.10 Å from the surface, the RIs for these systems is far enough from the surface. Thus, even the stronger binding energy in the LBE case will have a negligible impact on the convergence of the potential energies. But in this section, only one of these systems is discussed as an example. Table C4-5 shows the energy for the Heli/Saddle system in the LBE case. The RI for this system is from 3.00 Å up to 3.40 Å. Although the difference in the data between the LBE and UBE cases is negligible, comparing the data in this table with the data in Table 4-10 shows that, in the LBE case, all the ratios shift to lower values. Also, the 4th/Etot ratios in the RI range suggest that, like the UBE case, even going up to third-order PT term in EQels is enough to get a well-converged energy. 408 Table C4-5 The energies and PEPR quantities for a system of Heli/Saddle (MgO) in the LBE case after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 3.00 Å up to 3.40 Å. EQels(tot) is the sum of all EQels PT terms up to the fourth-order PT energy. 4th in the ratios refers to the fourth order EQels correction term in its PT series. Heights are in Å, energies are in kcal/mol, and the ratios are in %. Height Etot EQels(tot) 4th/EQels(tot) 4th/Etot EDR(2)/Etot Eind2(2)/Etot 2.00 21.71433 -0.68874 17.10 0.54 1.36 0.00 2.05 17.61832 -0.57721 13.46 0.44 1.05 0.00 2.10 14.26976 -0.48037 10.65 0.36 0.82 0.00 2.15 11.53297 -0.39760 8.47 0.29 0.64 0.00 2.20 9.29667 -0.32767 6.76 0.24 0.51 0.00 2.25 7.46969 -0.26911 5.42 0.20 0.41 0.00 2.30 5.97746 -0.22042 4.35 0.16 0.33 0.00 2.35 4.75900 -0.18013 3.49 0.13 0.27 0.00 2.40 3.76446 -0.14695 2.81 0.11 0.22 0.00 2.45 2.95311 -0.11971 2.27 0.09 0.18 0.00 2.50 2.29168 -0.09740 1.83 0.08 0.15 0.00 2.55 1.75297 -0.07918 1.48 0.07 0.12 0.00 2.60 1.31472 -0.06431 1.19 0.06 0.11 0.00 2.65 0.95877 -0.05220 0.96 0.05 0.09 0.00 2.70 0.67023 -0.04235 0.78 0.05 0.08 0.00 2.75 0.43692 -0.03435 0.63 0.05 0.08 0.00 2.80 0.24888 -0.02785 0.51 0.06 0.08 0.00 2.85 0.09792 -0.02257 0.41 0.09 0.12 0.00 2.90 -0.02265 -0.01829 0.33 0.27 0.28 0.00 2.95 -0.11833 -0.01481 0.27 0.03 0.03 0.00 3.00 -0.19364 -0.01200 0.22 0.01 0.01 0.00 3.05 -0.25227 -0.00972 0.18 0.01 0.00 0.00 3.10 -0.29728 -0.00787 0.14 0.00 0.00 0.00 3.15 -0.33116 -0.00637 0.12 0.00 0.00 0.00 3.20 -0.35599 -0.00516 0.09 0.00 0.00 0.00 3.25 -0.37344 -0.00418 0.08 0.00 0.00 0.00 3.30 -0.38492 -0.00338 0.06 0.00 0.00 0.00 409 Height Etot EQels(tot) 4th/EQels(tot) 4th/Etot EDR(2)/Etot Eind2(2)/Etot 3.35 -0.39157 -0.00274 0.05 0.00 0.00 0.00 3.40 -0.39433 -0.00222 0.04 0.00 0.00 0.00 3.45 -0.39399 -0.00179 0.03 0.00 0.00 0.00 3.50 -0.39117 -0.00145 0.03 0.00 0.00 0.00 3.55 -0.38641 -0.00117 0.03 0.00 0.00 0.00 3.60 -0.38014 -0.00095 0.02 0.00 0.00 0.00 3.65 -0.37270 -0.00077 0.01 0.00 0.00 0.00 3.70 -0.36439 -0.00062 0.02 0.00 0.00 0.00 3.75 -0.35544 -0.00050 0.00 0.00 0.00 0.00 3.80 -0.34605 -0.00041 0.00 0.00 0.00 0.00 3.85 -0.33637 -0.00033 0.00 0.00 0.00 0.00 3.90 -0.32653 -0.00027 0.00 0.00 0.00 0.00 3.95 -0.31663 -0.00022 0.00 0.00 0.00 0.00 4.00 -0.30675 -0.00018 0.00 0.00 0.00 0.00 410 The PT series convergence for H2/NaCl systems (section 4.2.2.2) In this section, the results for the PT convergence series of the rest of the H2/NaCl systems are shown. So this section is an extended part to section 4.2.2.2 of the thesis. Other systems (Cart/Na+, Heli/Saddle, Para/Saddle, etc.) As stated at the beginning of section 4.2.2.2, for systems with optimum heights greater than 3.20 Å, EQels is well-converged even after adding the third-order correction terms. But again in this section, the results are shown only for Cart/Na+ as an example. This system is chosen here because the molecule in this system sits closest to the surface among all the systems considered in this section. Table C5-1 illustrates the results for this system, with an RI from 2.80 up to 3.20 Å. The 4th/Etot data in this table suggest that even adding the third-order PT terms is enough to get a well-converged energy in RI. Also, as seen in the table, the second-order PT energies are not required to be included in either Eind2 or EDR. Nonetheless, fourth-order EQels as well as second-order Eind2 and EDR correction terms have been included in the calculation of the total energy in this system. It should be added that, in the rest of the systems over NaCl, the optimum height occurs at heights greater than that in the Cart/Na+ system and hence, same conclusions can be made for those systems as well. 411 Table C5-1 The energies and PEPR quantities for the system of Cart/Na+ after adding the fourth-order EQels PT correction term to the total energy. The RI for this system is from 2.80 up to 3.20 Å. EQels(tot) is the sum of all EQels PT correction terms up to the fourth-order energy. 4th in the ratios refers to the fourth-order EQels correction term in the corresponding PT series. Heights are in Å, energies are in kcal/mol, and the ratios are in %. Height Etot EQels(tot) 4th/EQels(tot) 4th/Etot EDR(2)/Etot Eind2(2)/Etot 2.00 4.53095 0.37555 135.54 11.23 1.01 0.72 2.05 3.54564 0.38622 92.82 10.11 0.81 0.67 2.10 2.73480 0.39984 63.35 9.26 0.66 0.64 2.15 2.08008 0.41217 43.55 8.63 0.55 0.61 2.20 1.55803 0.42113 30.29 8.19 0.46 0.60 2.25 1.14502 0.42590 21.34 7.94 0.39 0.59 2.30 0.81993 0.42640 15.22 7.91 0.34 0.60 2.35 0.56472 0.42296 10.98 8.22 0.30 0.64 2.40 0.36460 0.41606 8.00 9.13 0.28 0.72 2.45 0.20767 0.40630 5.88 11.51 0.29 0.92 2.50 0.08453 0.39424 4.36 20.33 0.42 1.66 2.55 -0.01222 0.38042 3.25 101.22 1.64 8.36 2.60 -0.08835 0.36531 2.44 10.10 0.12 0.84 2.65 -0.14832 0.34932 1.84 4.34 0.04 0.37 2.70 -0.19554 0.33281 1.40 2.38 0.01 0.20 2.75 -0.23270 0.31606 1.06 1.45 0.00 0.12 2.80 -0.26184 0.29931 0.81 0.93 0.00 0.08 2.85 -0.28458 0.28274 0.62 0.62 0.00 0.05 2.90 -0.30215 0.26651 0.48 0.42 0.00 0.04 2.95 -0.31551 0.25072 0.37 0.29 0.00 0.03 3.00 -0.32545 0.23547 0.29 0.21 0.00 0.02 3.05 -0.33256 0.22080 0.22 0.15 0.00 0.01 3.10 -0.33732 0.20677 0.17 0.11 0.00 0.01 3.15 -0.34013 0.19340 0.13 0.08 0.00 0.01 3.20 -0.34130 0.18069 0.10 0.05 0.00 0.00 3.25 -0.34109 0.16866 0.08 0.04 0.00 0.00 412 Height Etot EQels(tot) 4th/EQels(tot) 4th/Etot EDR(2)/Etot Eind2(2)/Etot 3.30 -0.33972 0.15729 0.06 0.03 0.00 0.00 3.35 -0.33737 0.14657 0.05 0.02 0.01 0.00 3.40 -0.33419 0.13648 0.04 0.02 0.01 0.00 3.45 -0.33032 0.12701 0.03 0.01 0.01 0.00 3.50 -0.32586 0.11812 0.02 0.01 0.01 0.00 3.55 -0.32091 0.10980 0.02 0.01 0.01 0.00 3.60 -0.31556 0.10202 0.01 0.00 0.01 0.00 3.65 -0.30988 0.09474 0.01 0.00 0.01 0.00 3.70 -0.30393 0.08796 0.01 0.00 0.01 0.00 3.75 -0.29778 0.08162 0.01 0.00 0.01 0.00 3.80 -0.29146 0.07572 0.01 0.00 0.01 0.00 3.85 -0.28502 0.07023 0.00 0.00 0.01 0.00 3.90 -0.27850 0.06511 0.00 0.00 0.01 0.00 3.95 -0.27193 0.06035 0.00 0.00 0.01 0.00 4.00 -0.26535 0.05593 0.00 0.00 0.01 0.00 413 Appendix D In this appendix, all the supplementary tables and figures for Chapter 7 and Chapter 8 are presented. Table D-1 The calculated (this work) and experimentally measured (previous studies) vibrational energies in a layer of Para molecules at both 0.75 and 0.83 coverages over the MgO surface. In the calculation of all these energies Model II along with the LBE case were used. “disp. E” in brackets refers to the “dispersive energy” that is the vibrational energy of a collection of molecules over the surface. T-Mode Energy [meV] S-Mode Energy [meV] Experiment [meV]Ref 16 Direction O2--Mg2+-O2- O2--Sa-O2- Sa-Mg2+-Sa Above Mg2+ Above Saddle Along [100] 9.3 - - 12.3 10.6 10.1 and 8.1 Along [110] (Disp. E) - 3.8 9.6 12.3 10.6 10.1 and 8.1, (2.5 – 7) 414 Figure D-1 The minimum energy of the Heli and Cart molecules when up-to-first-order (1st) and up-to-fourth-order (4th) PT terms are used in the calculation of the total energy in the 𝑽෡𝒔𝒆𝒑 case of Model II. The data are along the saddle-anion-saddle channel of the MgO substrate. -0.45-0.35-0.25Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)Heli (4th)Cart (4th)AnionSaddle Saddle 415 Figure D-2 The comparison between total energies of the Heli and Cart molecules in the 𝑽𝒔𝒆𝒑 and 𝑽𝒕𝒐𝒕 cases of Model II for the MgO substrate and along the saddle-anion-saddle channel. All the energies are up-to-fourth-order (4th) PT energies. -0.45-0.35-0.25Energy [kcal/mol]Position of the moleculeHeli (Vsep)Cart (Vsep)Heli (Vtot)Cart (Vtot)SaddleSaddleAnion416 (a) (b) Figure D-3 The minimum energy of the Heli and Cart molecules when up-to-first-order (1st) and up-to-fourth-order (4th) PT terms are used in the calculation of the total energy in the 𝑽෡𝒔𝒆𝒑 case of Model II. The data are for: a. along the anion-cation-anion channel, and b. along the saddle-cation-saddle channel of the NaCl substrate. -0.85-0.65-0.45-0.25Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)Heli (4th)Cart (4th)Anion AnionCation-0.85-0.60-0.35Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)Heli (4th)Cart (4th)SaddleCationSaddle417 Figure D-4 The minimum energy of the Heli and Cart molecules when up-to-first-order (1st) and up-to-fourth-order (4th) PT terms are used in the calculation of the total energy in the 𝑽෡𝒔𝒆𝒑 case of Model II. The data are along the saddle-anion-saddle channel of the NaCl substrate. -0.40-0.30-0.20Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)Heli (4th)Cart (4th)AnionSaddle Saddle 418 (a) (b) Figure D-5 The comparison between total energies of the Heli and Cart molecules in the 𝑽𝒔𝒆𝒑 and 𝑽𝒕𝒐𝒕 cases of Model II for the NaCl substrate and along the channels of a. anion-cation-anion, and b. saddle-cation-saddle. All the energies are up-to-fourth-order (4th) PT energies. -0.95-0.75-0.55-0.35-0.15Energy [kcal/mol]Position of the moleculeHeli (Vsep)Cart (Vsep)Heli (Vtot)Cart (Vtot)Anion AnionCation-0.85-0.60-0.35Energy [kcal/mol]Position of the moleculeHeli (Vsep)Cart (Vsep)Heli (Vtot)Cart (Vtot)SaddleCationSaddle419 Figure D-6 The comparison between total energies of the Heli and Cart molecules in the 𝑽𝒔𝒆𝒑 and 𝑽𝒕𝒐𝒕 cases of Model II for the NaCl substrate and along the saddle-anion-saddle channel. All the energies are up-to-fourth-order (4th) PT energies. -0.45-0.35-0.25Energy [kcal/mol]Position of the moleculeHeli (Vsep)Cart (Vsep)Heli (Vtot)Cart (Vtot)AnionSaddleSaddle 420 Figure D-7 Comparison of the minimum energies of the two states of the ortho-H2 molecule in Model II (the 𝑽𝒔𝒆𝒑 case) and the three states of this molecule (MS1, MS2, and MS3) in Model III over the NaCl substrate. The energies presented in this figure are only up to first order and at positions along the saddle-anion-saddle channel. -0.45-0.40-0.35-0.30-0.25-0.20Energy [kcal/mol]Position of the moleculeHeli (1st)Cart (1st)MS1 (1st)MS2 (1st)MS3 (1st)SaddleAnionSaddle
Thesis/Dissertation
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Chemistry
Vancouver : University of British Columbia Library
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Graduate
Structure and stability of quantum H2 films on ionic surfaces : a theoretical study
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