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Application of density functional theory to the calculation of molecular core-electron binding energies Cavigliasso, German 1999

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APPLICATION  OF DENSITY FUNCTIONAL  THEORY  T O T H EC A L C U L A T I O N OF M O L E C U L A R CORE-ELECTRON BINDING  ENERGIES  by  German Cavigliasso B. Sc., Universidad Nacional de Cordoba (Argentina), 1995  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF MASTER OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMISTRY We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA July 1999 ©  German Cavigliasso, 1999  In  presenting this  thesis  in  degree at the University  partial  fulfilment  of British Columbia,  of  the  requirements  I agree that the  for  an  advanced  Library shall make it  freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by  his  or  her  representatives.  It  is  understood  that  publication of this thesis for financial gain shall not be allowed without permission.  Cfftm^  Department of  J~ ^j r  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  Z°  J  "/y  1 ^  copying  or  my written  Abstract  T h e procedure for calculating core-electron binding energies  (CEBEs),  based on the unrestricted generalized transition state ( u G T S ) model combined w i t h density functional theory ( D F T ) employing Becke's 1988 exchange (B88) and Perdew's 1986 correlation (P86) functionals, which has proven to yield highly accurate results for C , N , O , and F cases, was extended to boroncontaining molecules and to S i , P, S, CI, and A r cases. B o t h unsealed and scaled basis sets were used i n the studies of boroncontaining molecules. The s c a l e d - p V T Z basis set was as highly efficient for boron as it had been found to be for C , N , O, and F cases; the average absolute deviation ( A A D ) of the calculated C E B E s from experiment was 0.24 e V , compared to 0.23 e V for the much larger c c - p V 5 Z basis set.  A  generalization of the exponent-scaling methodology was proposed and tested on boron-containing molecules, and was found not to improve the original results to a significant extent. T h e preliminary calculations of S i , P, S, CI, and A r C E B E s indicated that, i n order to achieve the accuracy obtained for second-period elements, refinement of the basis sets and inclusion of relativistic effects are necessary. A s an additional application of the D F T / u G T S / s c a l e d - p V T Z approach, the C E B E s of four isomers of C H N O were calculated. T h e distinctive na3  5  ture of the core-ionization spectra of the isomers was depicted by the results, thus illustrating the potential utilization of accurate theoretical predictions  ii  as a complement to electron spectroscopy for chemical analysis. T h e model error i n u G T S calculations and the errors i n the functionals employed were calculated. It was observed that the high accuracy of the B 8 8 / P 8 6 combination was due to a fortuitous cancellation of the functional and model errors. In view of this finding, a K o h n - S h a m total-energy difference approach, which eliminates the model error, was investigated. Ten functional combinations and several basis sets (including unsealed, scaled, and core-valence correlated functions) were tested using a database of reliable observed C E B E s . T h e functionals designed by Perdew and W a n g (1986 exchange and 1991 correlation) were found to give the best performance w i t h an A A D from experiment of 0.15 e V . T h e scaled basis sets d i d not perform as well as they d i d i n the u G T S calculations, but it was found that the core-valence correlated c c - p C V T Z basis functions were an excellent alternative to the cc-pV5Z set as they provided equally accurate results and could be applied to larger molecules.  iii  Table of Contents  Abstract  ii  List of Tables  vii  List of Figures  xi  Preface  xii  Acknowledgements  xiii  1  Introduction 1.1  Core-Electron B i n d i n g Energies  1.1.1  Calculations of Core-Ionization Energies  1.2  2  1  Scope and Organization of this Thesis  Density Functional Theory in Quantum Chemistry 2.1  Traditional A b Initio Q u a n t u m Chemistry  2.1.1  The Hartree-Fock M e t h o d  2.1.2 . Electron Correlation 2.2 2.2.1  3 5 7  10 11 13 17  Density Functional Theory T h e K o h n - S h a m Formulation  iv  19 21  2.2.2  A Comparison between D F T and Traditional Methods  30  2.4  Computational Aspects of Q u a n t u m Chemical Methods  32  5  6  7  Basis Sets  34  Computational Approach  36  3.1  The u G T S and A E - K S Approaches  36  3.2  The Density Functional Program d e M o n  40  3.2.1  4  24  2.3  2.4.1  3  Density Functionals  Basis Sets  41  A p p l i c a t i o n s of the u G T S M o d e l  48  4.1  Boron-Containing Molecules  49  4.2  Isomers of C H N O  57  4.3  Core-Electron B i n d i n g Energies of S i , P, S, CI, and A r  62  3  5  T h e A E - K S A p p r o a c h : Test of Functionals  69  5.1  M o d e l E r r o r and Functional E r r o r  71  5.2  Functional Performance i n A E - K S Calculations  75  T h e A E - K S A p p r o a c h : Test of B a s i s Sets  79  6.1  Scaled Basis Sets  80  6.2  Core-Valence Correlated Basis Sets  84  Conclusion  91  v  References  94  Appendix  108  vi  List of Tables  3-1 Composition of cc-pVnZ basis sets  42  3-2 Scaling factors (uGTS) for second-period atoms  45  3- 3 Scaling factors (AE-KS) for second-period atoms  46  4- 1 Basis set convergence in the calculation of B Is energies with unsealed basis sets  50  4-2 Calculations of B core-electron binding energies with scaled basis sets  51  4-3 Basis set convergence in the calculation of C, N , O, and F Is energies with unsealed basis sets  52  4-4 Calculations of C, N , O, and F core-electron binding energies with scaled basis sets 4-5  53  Average Absolute Deviation and Maximum Deviation of calculated core-electron binding energies from experiment and from CBS  4-6  54  Calculations of core-electron binding energies with the gs-pVTZ basis set  56  vn  4-7  Deviation analysis of the g s - p V T Z basis set performance for the 12 cases reported i n Table 4-6  4-8  57  T o t a l energies, relative energies w i t h respect to ethyl isocyanate, and dipole moments for the isomers of C 3 H 5 N O  58  Core-electron binding energies for the isomers of C 3 H 5 N O  60  4-10  Basis set convergence i n the calculation of S i 2p energies  63  4-11  Basis set convergence i n the calculation of P 2p energies  64  4-12  Basis set convergence i n the calculation of S 2p energies  65  4-13  Basis set convergence i n the calculation of A r and CI 2p  4- 9  energies 4-14  66  Average Absolute Deviation of calculated core-electron binding energies from experiment sorted by element  4-18  Average Absolute Deviation of calculated core-electron binding energies from experiment for a l l test cases  5- 1 C o m p o s i t i o n of exchange-correlation functionals 5-2  68 70  E r r o r analysis for u G T S calculations w i t h the B 8 8 / P 8 6 functional  5-3  67  72  E r r o r analysis for u G T S calculations w i t h the P 8 6 / P 8 6 functional  73  viii  5-4 Error analysis for uGTS calculations with the LSD functional  74  5-5 Average Absolute Deviation and Maximum Deviation of the core-electron binding energies (calculated with the functionals in Table 5-1) from experiment  76  5- 6 Core-electron binding energies calculated with the P86/P91 functional  77  6- 1 Basis set convergence in AE-KS/P86-P91 calculations of core-electron binding energies  81  6-2 AE-KS/P86-P91 calculations of core-electron binding energies with scaled basis sets  82  6-3 Average Absolute Deviation of calculated core-electron binding energies from experiment and from CBS  83  6-4 AE-KS/P86-P91 calculations of core-electron binding energies with cc-pCVTZ basis sets  85  6-5 Average Absolute Deviation of calculated core-electron binding energies from experiment  86  6-6 Calculations of core-electron binding energies of larger molecules A-l  87  Observed ls energies for the molecules in the seventeen-case database  109  ix  A - 2 Observed Is energies for the aromatic compounds i n Table 6-6  114  A - 3 Observed Is energies for hydrogen fluoride  x  115  List of Figures  4-1 Numbering scheme for the isomers of C 3 H 5 N O  59  4-2 Calculated and available experimental C E B E s for the isomers of C 3 H 5 N O  61  6-1 Completeness profiles of cc-pVnZ and cc-pCVTZ basis sets  88  6-2 Comparison of completeness profiles for the s-, p-, and d-type functions of cc-pVTZ, cc-pV5Z, and cc-pCVTZ basis sets  xi  89  Preface  P a r t of the results reported in this thesis have been published, accepted, or submitted for publication. The corresponding references are  • G . Cavigliasso, D . P. Chong, Accurate Density Functional C a l c u l a t i o n of Core-Electron B i n d i n g Energies w i t h a Scaled Polarized Triple-Zeta Basis Set. V I . Extension to Boron-Containing Molecules, Can. J.  Chem.,  1999, 77-1, 24  • D . P. Chong, G . Cavigliasso, Density-Functional Calculation of CoreElectron B i n d i n g Energies of Isomers of C3H6O2 and C 3 H 5 N O , Int. Quantum  J.  Chem., accepted  • G . Cavigliasso, D . P. Chong, Accurate Density Functional C a l c u l a t i o n of Core-Electron B i n d i n g Energies by a Total-Energy Difference A p proach, J. Chem. Phys., submitted  Most of the data contained i n the A p p e n d i x were collected and organized by Prof. Delano P. Chong.  xii  Acknowledgements  I would like to express my gratitude to a l l the people who have contributed to my academic experience at the University of B r i t i s h C o l u m b i a . I am much obliged to  Professor D e l Chong, my research supervisor Dr. Ching-Hang H u Professors G r e n Patey and M a r k Thachuk Professors E l m e r Ogryzlo and Robert Thompson  I am also grateful to the academic and administrative personnel of the Department of Chemistry at U B C , i n particular the graduate secretaries T i l l y Schreinders, J u d y Wrinskelle and Diane Mellor, my instructors, classmates and research lab-mates.  I would like to acknowledge financial support from the Department of C h e m istry, the Faculty of Science and the Faculty of Graduate Studies.  I dedicate this thesis to my teachers.  xiii  Chapter 1  Introduction  The roots of Density Functional Theory (DFT) date back to the days when quantum mechanics was in its early stages of development [1]. It was, however, the work of Hohenberg, Kohn, and Sham [2, 3] in the 1960s that rigorously established density functional theory as a legitimate method of describing the electronic structure of matter. Applications of D F T to electronic structure calculations concentrated initially on solid-state systems and on problems which were of interest especially to physicists. The impact that D F T has had on theoretical and computational solid-state physics has been extremely significant as summarized by the remarks of Fulde [4] — " (DFT) has given (electronic structure) calculations a sounder theoretical basis than they had previously " — and Kohn [5] — " (DFT) vitalized first researches in the electronic structure of simple crystals and subsequently those on more complex systems (such as) defects,  1  Introduction  2  alloys, surfaces, superconductivity, magnetism ". Although the incorporation of density functional methods into the field of quantum chemistry (and computational molecular science in general) did not occur as rapidly as it had in the case of solid-state physics, over the past two decades there have been ever-increasing numbers of applications to chemical systems, applications which have been supported by continuing advances in both theoretical methodology and computational implementation [6-15]. Density functional calculations have become a highly effective method for studying the structure, properties, and dynamics of a wealth of molecular systems, and have been developed to a level where they challenge the accuracy of the conventional (more "sophisticated") quantum chemical techniques, which they already surpass in terms of computational efficiency and applicability to relatively large systems [16]. Density functional theory is also highly appealing, from a conceptual point of view, in that several important universal concepts of molecular structure and reactivity — such as chemical potential, electronegativity, hardness and softness, reactivity indices — are naturally involved in the density functional language [1, 9, 11, 15]. In an article entitled " The reachable dream: some steps toward the realization of quantum mechanics by computer " [17], Schaefer pointed out that "the dream" was in part " to make computational quantum chemistry sufficiently efficient and indispensable that experimental chemists would employ it routinely, much as they use N M R spectroscopy ". Density functional theory is playing and will continue to play a leading role in the realization of Schaefer's dream; the principal reason has been clearly stated by Ziegler  Introduction  3  [7], " ... popularity of approximate D F T stems i n large measure from its computational expedience which makes it amenable to large-size or real-life molecules ... ". T h e determination of core-electron binding energies ( C E B E s ) is one of the molecular problems which recently has been successfully addressed by density functional calculations [18-29]. T h e purpose of this work is to extend the applications of D F T to the calculation of molecular C E B E s by studying some additional systems w i t h the already established methodology and by exploring new computational approaches.  1.1  C o r e - E l e c t r o n B i n d i n g Energies  T h e binding energy (EB) of an electron i n a core level is the difference between the total energies of the initial (E-*) and final (Ej?~ ) states of the l  system, the former being the neutral molecule and the latter being a singly ionized cation (created by removal of the inner-shell electron) [30, 31],  iTi  rpN-l  rpl  (1.1)  where N is the total number of electrons. B i n d i n g energies of core electrons are determined experimentally by X - r a y Photoelectron Spectroscopy ( X P S ) [30-32] according to the relation [33]  KE  = hu  —  E  B  (1.2)  4  Introduction where K  E  is the kinetic energy of an electron that has been ejected by an  X-ray photon of energy hv. Core-ionization energies have been widely used for analytical purposes. The technique is known as Electron Spectroscopy for Chemical Analysis (ESCA) and utilizes data for electron binding energies as a means of determining elemental composition and of providing structural information via the so-called chemical shifts [30-33]. E S C A has proven useful for solid-state and surface science studies [30, 34-36] because the small escape depth for electrons renders it particularly suited to probing the outermost atomic layers of a solid-state material and also the adsorption of atoms or molecules on a surface. In addition to the practical uses of C E B E data, much attention has been focused on the chemical shifts for more fundamental reasons. A chemical shift is defined as a change in the C E B E of an atom due to a modification of its chemical environment [30-33]. In this sense, the analysis and understanding of chemical shifts constitute a valuable means of gaining insight into properties of significant chemical interest (such as the nature of chemical bonds in a molecule or solid). Some examples of investigations that have pursued this direction are the combination of core and valence ionization potential data to quantify the bonding or antibonding character of molecular orbitals [37-40], exploration of the connections between C E B E s and the concept of resonance [41] and of the correlation of C E B E s with activation energies for addition reactions to alkenes [42], and studies of electronic substituent effects in iron complexes of aromatic molecules [43].  Introduction  1.1.1  5  Calculations of Core-Ionization Energies  Core-electron spectroscopy has been recognized as a " field of research for which the interplay between theory and experiments has been of particular importance " [44].  Calculations of the binding energies of core electrons  and of their chemical-environment-dependent  shifts have proven useful i n  assisting the interpretation and understanding of various aspects of core-level ionization phenomena [26-28, 44-52]. A rigorous theoretical approach to determining C E B E s involves performing separate total-energy calculations for both the i n i t i a l ground-state and the final core-hole state of equation (1.1). T h i s total-energy difference procedure was introduced by Bagus [53] and has been applied [46, 48, 49, 54-57] m a i n l y at the Hartree-Fock level of theory (Chapter 2), but also considerably more sophisticated (and computationally expensive) quantum chemical calculations have been reported, including configuration interaction [46, 58], multiconfiguration [49], and second-order perturbation theory [49] investigations. Some calculations have employed the equivalent-core approximation — first introduced by Shirley [59] — i n which the exact calculation of the core-hole state is replaced by a ground-state calculation o f the equivalent system obtained by increasing the nuclear charge on the atom being ionized by one unit. T h e Hartree-Fock total-energy difference procedure — most frequently referred to as the ASCF  method — has not been particularly good at estimat-  ing absolute C E B E s , but has been remarkably good at predicting accurate values for the chemical shifts [48, 49]. Because the computational demands of high-quality quantum chemistry render the application of the total-energy  Introduction  6  difference method to even moderately-sized molecules and clusters (which are widely used as models for surfaces) impractical, and because most of the interest has focused on the chemical shifts (rather than the absolute CEBEs), several alternative models and methods have been devised [44, 54, 60, 61]. Some are specifically designed to avoid performing the core-ionized cation calculations, and have been employed extensively, often yielding good quantitative results [61]. Approaches that take advantage of Slater's transition state concept [62] — and thus rely on the use of fractional orbital occupation numbers to calculate absolute C E B E s — have also been proposed as an alternative to the procedures that require calculations on fully ionized final states. Following this line, Chong and coworkers [63-65] applied a transition-operatormoment/perturbation-theory approach with encouraging success (average deviation from experiment was 0.4 eV for eight cases involving small molecules [65]) but the computational effort was still too expensive for the method to be extended to larger systems. With the evolution of density functional theory into an accurate and reliable quantum chemical technique, applications of DFT-based methods to core-ionization phenomena have been actively pursued. Approaches that make use of the transition state model [47, 66], the total-energy difference procedure [67], and of combinations of calculations with observed data [68] have been employed, and they have been successfully applied to both discrete [47] and extended [66-68] systems. Chong [18, 19] has recently introduced methodology based on a combination of D F T with the generalized transition state model [69] — an exten-  Introduction  7  sion of Slater's original transition state idea — and has extensively tested [18-29] it on a wide variety of systems, including both closed-shell and openshell small molecules, transition metal complexes, and also model systems for polymers and adsorbate-surface interactions. H i g h quantitative accuracy has been accomplished (average deviations from experiment of 0.2 e V or lower have been routinely achieved [19, 20, 24, 27, 28, 29]) and the procedure has also proven useful from a qualitative point of view. For example, investigations of the C E B E s of isomers of C H 0 , C H N O , and C H 2  4  3  3  6  6  [20, 29] have  provided results which support the possible use of electron spectroscopy for chemical analysis. Also, surface-science-oriented studies have shown that employing C E B E s for the interpretation of X P S spectra on a molecular scale constitutes a useful method for tracing the interactions of molecules w i t h a surface [26], and is of valuable assistance i n the interpretation of X P S spect r a of compounds for which there is no gas-phase reference spectrum [27]. These examples support and confirm the significance of the role played by computational techniques i n the aforementioned theory-experiment interplay in core-electron spectroscopy.  1.2  Scope and Organization of this Thesis  T h i s thesis consists of seven chapters, which fall into two groups: background and reference material (Chapters 1, 2, and 3), and presentation and discussion of results (Chapters 4, 5, 6, and 7). Chapter 2 contains material of a theoretical nature which is intended to serve as introductory-level background to density functional theory and how  Introduction  8  it relates to and differs from more conventional ab initio quantum chemical methods. Chapter 3 will present a description of the computational approach, including the mathematical formalism, and some details of the calculation code and the basis sets employed. The density functional investigations of Chong and coworkers have addressed a large number of cases of core-electron binding energies of the second-period elements carbon, nitrogen, oxygen, and fluorine. A l l the studies have employed the unrestricted Generalized Transition State (uGTS) model [18]. Part of the work reported in this thesis was devoted to extending the applications of the u G T S / D F T approach to the calculation of molecular C E B E s . Thus, Chapter 4 will present the results of a study of core-ionization energies of boron-containing molecules, and of the third-period elements silicon, phosphorus, sulphur, chlorine, and argon. Also in Chapter 4, an investigation of the C E B E s of isomers of C H N O , which explores the analytical 3  5  applications of core-electron spectroscopy, will be reported. Chapters 5 and 6 contain the results of a different approach to the determination of C E B E s which is based on a density functional total-energy difference (AE) procedure. A n error-based comparison between the uGTS and the A E methods, and a test of functionals for use within the latter approach will be presented in Chapter 5, whereas the results of tests of various basis sets will be the subject of Chapter 6. All the calculated core-electron binding energies reported in this thesis are vertical ionization potentials in the sense that the calculations were carried out at the equilibrium geometries of the neutral molecules for both the initial state (neutral species) and the final state (ionized species) [48]. The results  Introduction  9  obtained w i l l be almost exclusively discussed and analyzed i n terms of comparisons of calculation w i t h experiment, and the extent of their agreement w i t h the observed values w i l l be the major factor i n assessing the performance of the computational procedures.  Chapter 2 Density Functional Theory in Quantum Chemistry  " A primary objective of molecular quantum mechanics is the solution of the non-relativistic, time-independent Schrodinger equation, and in particular the calculation of the electronic structures of atoms and molecules " [70]. In order to achieve the goal of calculating molecular electronic structure, two main approaches are widely employed. Semiempirical methods [7072] introduce a number of significant approximations in the solution of the Schrodinger equation, and rely upon adjustable parameters obtained from experimental information; their use is limited to the chemical systems for which they were parameterized.  Methods which do not resort to empiri-  cal data (except for the use of values of fundamental constants and nuclear atomic numbers) are in principle applicable to any molecular system, and are generally divided into traditional ab initio [73] and density functional [1, 74] approaches. In the former, the wavefunction is central to the description  10  Density Functional  Theory in Quantum  Chemistry  11  of electronic structure, whereas i n the latter the electron density plays the major role. In this chapter, a description of the principal concepts of density functional theory and quantum chemistry w i l l be presented, and some of the similarities and differences between D F T and traditional quantum chemistry w i l l be discussed.  2.1  Traditional A b Initio Quantum Chemistry  The study of the behaviour and properties of electrons i n molecules requires the solution of the time-independent Schrodinger equation which has the general eigenvalue-problem form [70, 73]  H$  = E$  (2.1)  where $ is the wavefunction, E is the total energy, and H is the (nonrelativistic) H a m i l t o n i a n operator, given by  H — T  nuc  + Ti + U e  nuc  + V t + U ex  ee  (2-2)  The terms on the right-hand side of equation (2.2) have the following physical interpretations and mathematical expressions (in atomic units [73]):  T c — — J2A 2m~^a nU  ^ ^ s  n e  nuclear kinetic energy operator,  Density Functional  Theory in Quantum  T i = — J2i \ Vf  12  is the electronic kinetic energy operator,  e  Unuc =  Chemistry  EB>/1 RAB  1S  Z  t  n  e  P ° * t i a l energy operator for nuclear repulen  sions,  Kx* —  YIA  is the potential energy operator for electron-nucleus  = S i Sj>i ^ :  is the potential energy operator for electron-electron re-  attractions,  U  ee  pulsions.  In the above expressions, M  A  mass of an electron, Z  is the ratio of the mass of nucleus A to the  is the atomic number of nucleus A, V  2  A  A  and Vf  are Laplacian operators [75] for nucleus A and electron i respectively, and Tij = |rj — Tj\, r  iA  — |rj — R ^ | , RAB — | R A — RB|> where r and R are  electronic and nuclear position vectors, respectively. Solving the Schrodinger equation is a formidable task even for the simplest molecular systems, so a number of approximations must be introduced to make calculations feasible. The vast majority of quantum chemical calculations are carried out within the Born-Oppenheimer approximation [76], which is based on the fact that electrons, being much lighter than nuclei, move considerably faster than nuclei do. Hence, it is reasonable to assume that electrons in molecules move in the field generated by fixed nuclei. This separation of the motion of electrons and nuclei leads to the "electronic" Schrodinger equation  Density Functional  Theory in Quantum  Ha^a  = (f  + V  el  ext  Chemistry  + U )^ i ee  e  13  = E <S> el  (2-3)  el  where \I/ ; is the wavefunction describing the behaviour of the electrons. B o t h e  and the electronic energy E \ depend parametrically on nuclear positions. e  A common quantum chemical solution to equation (2.3) utilizes the variational principle [73]. Given a t r i a l (almost always approximate) wavefunction ^ for a particular molecular system, the expectation value of the energy w i l l be greater than or equal to the true ground-state energy (E ), 0  (E)  E >  E, 0  or equivalently,  (2.4)  where D i r a c notation [73] has been used. The equality holds only when \1> is equivalent to the correct ground-state wavefunction ^o-  2.1.1  T h e Hartree-Fock  Method  T h e Hartree-Fock ( H F ) approximation is a mean-field method applied to the many-body electronic problem [16, 73, 77]. T h e central idea of the H F approach is the assumption that electrons move independently of one another and that a given electron interacts w i t h an average field produced by the rest of the electrons. Therefore, an explicit treatment of the instantaneous pairwise interaction, the so-called electron correlation, is lacking. For an N-electron system, an antisymmetrized product of N spin-orbitals (V'i), known as a Slater determinant, represents the simplest t r i a l wavefunc-  Density Functional  Theory in Quantum  t i o n which is physically acceptable.  Chemistry  14  T h i s determinant  satisfies the P a u l i  P r i n c i p l e [73], a necessary condition, and is given by  ^i(xi)  ^ (xi)  •07V (Xl)  ^l(x )  •02(X )  ^iv(x )  2  HF  2  2  (jV!)i/2  2  (2.5)  T h e spin-orbitals i n equation (2.5) are each a product of a spatial orbital Pi(r) and a spin function o(s) w i t h <r(s) being either a(s)  for "spin-up" or  P(s) for "spin-down", and x indicating both space (r) and spin (s) coordinates. T h e y are generally orthonormal, that is  < ipi\ipj >=  S  (2.6)  >= 1  (2.7)  i:j  so that the wavefunction is also normalized  < ^HF^HF  T h e H F ground-state energy Egp is obtained by a variational m i n i m i z a t i o n of the expectation value of the electronic H a m i l t o n i a n of equation (2.3) w i t h the wavefunction of equation (2.5) [73]. T h i s procedure involves optim i z a t i o n of the spin-orbitals that comprise the Slater determinant and yields the Hartree-Fock equations  flpi = €i1pi  (2.8)  Density Functional  Theory in Quantum  Chemistry  15  where ej is the orbital energy and / is an effective one-electron operator known as the Fock operator, given by  f = h + v HF  (2.9)  The operator h is defined as  (2.10)  and the effective one-electron H F potential operator v  HF  is given by a sum  of Coulomb (J) and exchange (K) contributions, that is  v  BF  = £ ( J -  (2.11)  k)  The definitions of the Coulomb and exchange operators are  Jjipi  I  ^•(x )^(x )—dx  Kjipi  1  •0*(x )^(x )— dx  2  2  2  2  2  2  4>i ( x i )  (2.12)  ^•(xi)  (2.13)  Kj is a non-local operator in the sense that the result of operating with it on an orbital ipii^i)  depends on the value of tpi throughout all space, not just at  Xi [73]. There are an infinite number of spin-orbitals which solve equation (2.8), so the N-electron H F ground-state wavefunction is formed from the N orbitals with the lowest energies, which are called the occupied spin-orbitals.  Density Functional  Theory in Quantum  Chemistry  16  The H F ground-state energy is then given by  EF H  = < *HF\Hei\*HF  >= £  #i + \ £ ( < ^ ~  (2-14)  where  Hi =< iPi\h\ipi >  Jij =< il>i\J\if>j >  Kij = < V i l ^ i  >  (2.15)  (2.16)  (2-17)  If the spin functions are integrated out i n equations (2.16) and (2.17), then Jij and  can be expressed as functions of the spatial orbitals ifi [73].  The resulting "spatial"  contributions may be thought of as originating  from the classical Coulomb repulsion between two charge clouds. However, no simple classical interpretation can be associated w i t h the exchange contributions. The energy of an individual spin-orbital is given by  ei = H + X)(J^ - K^) 3 so the H F energy can be also expressed as  (2.18)  Density Functional  Theory in Quantum  E  HF  Chemistry  —  17  (2.19)  which shows that the individual orbital energies do not add up to the total electronic energy of the system. given to  Instead, a physical interpretation can be  by means of K o o p m a n s ' theorem [73], which states that the ion-  ization potential (IP) for removal of an electron from a spin-orbital ipi i n an N-electron H F wavefunction is the negative of the energy of the spin-orbital, that is  IP = ~€i  (2.20)  Due to the approximate nature of the Hartree-Fock model, ionization potentials determined according to (2.20) are only of a qualitative or semiquantitative value [54].  2.1.2  Electron  Correlation  T h e Hartree-Fock method is the simplest of the wavefunction-based  ap-  proaches to molecular electronic structure, but it fails to provide a complete physical description of the many-electron problem because it disregards electron correlation. T h e correlation energy is defined i n conventional quantum  chemistry  as the difference between the exact non-relativistic ground-state energy  E  (within the Born-Oppenheimer approximation) and the H F energy, that is  0  Density Functional  Theory in Quantum  Chemistry  18  Ecorr — E — E 0  (2.21)  HF  A n exact procedure exists for representing any state of an N-electron system, called Configuration Interaction (CI) [73]. The infinite number of spinorbitals that are obtained as solutions to the H F equations can be viewed as two distinct sets of orbitals: the N lowest-energy occupied orbitals and the remaining higher-energy unoccupied or v i r t u a l orbitals. The former group is used to construct the ground-state wavefunction w i t h i n the H F approximation while the latter group provides a means of generating excited configurations  by promoting electrons from occupied to v i r t u a l orbitals.  An  exact N-electron wavefunction can be expressed as a linear combination of an infinite number of N-electron configurations (each represented by a Slater determinant), that is  * *a e  r f  = co*o + c  r a  £ ^ + C  ra  6  X  +  a<b,r<s  where \Po is the H F ground-state wavefunction, ty  r  figuration,  ^l r  a  (2-22)  a  is a singly-excited con-  is a doubly-excited configuration, and so on.  T h e c's are  expansion coefficients, the indices a and b label occupied orbitals, and the indices r and s label v i r t u a l orbitals. In practice, it is impossible to handle infinite numbers of spin-orbitals and configurations, yet due consideration of electron correlation is necessary to improve upon the H F description. Therefore, a number of methods [4, 16, 73, 77] which employ different approximate approaches to the treatment of correlation effects have been developed. They include several approximate versions  Density Functional  Theory in Quantum  Chemistry  19  of the CI method, the Multiconfigurational Self-consistent Field (MCSCF) method, the Coupled-Cluster (CC) method, and many-body perturbation techniques such as the M0ller-Plesset Perturbation (MPPT) method.  2.2  Density Functional T h e o r y  The fundamental variable of density functional theory is the electron density, p [1]. For a given state of an electronic system, the electron density is defined as the number of electrons per unit volume, and is mathematically represented in terms of the N-electron wavefunction \& by  p( )=Nj2 ri  J \V(r ,r ,---,r )\ dr ---dr 2  1  2  N  2  N  (2.23)  spin  The electron density is thus a simple non-negative function of three variables (compared with 3N variables in the case of the N-electron wavefunction) which integrates to the total number of electrons N, that is  J p(r) dr = TV  (2.24)  A non-negative, continuous electron density that satisfies equation (2.24) is said to be N-representable [1]. A definitive proof that the electron density could play the central role in the description of the electronic structure of matter was given by Hohenberg and Kohn [2] in the form of two theorems. The first theorem considers a system of N interacting electrons in a nondegenerate ground-state under the influence of an external potential v(r)  Density Functional  Theory in Quantum  Chemistry  20  (which is not restricted to Coulomb potentials) and states that the groundstate density p(r) uniquely determines v(r), to within an additive constant. The number of electrons N is also determined by p(r) via equation (2.24). Therefore, the electron density completely determines the Hamiltonian and, consequently, all properties of the system. For a given external potential, the energy functional is  E[p] = V[p] + F [p]  (2.25)  V[p] = J p{r)v{v)dv  (2.26)  HK  where  and  F [p]  = T [p] + U [p]  HK  el  ee  From equations (2.26) and (2.27) it follows that F  HK  (2.27)  is a universal functional  of the electron density, its form not being affected by the nature of the system. The second Hohenberg-Kohn theorem makes it possible to find the groundstate density by means of a variational minimization search. It states that given an N-representable trial density ptriai,  E[ptrial]>E  0  The true ground-state energy E  0  (2.28)  is obtained only if the true ground-state  Density Functional  Theory in Quantum  Chemistry  21  density is used as the trial density. The search for p(r) must be conducted subject to the constraint imposed by equation (2.24), that is  8^E[p]-p  jp{r)dv-  N J =0  (2.29)  The multiplier \i is the chemical potential, and is defined by [1] SE[p]  SF [p] HK  P = -=-f-r = V(T) +  (2.30)  Provided the exact FHK[p]  is known, equation (2.29) yields the exact  ground-state electron density.  As stated by Hohenberg and Kohn in their  original paper [2]: " if F[p) were a known and sufficiently simple functional of p, the problem of determining the ground-state energy and density in a given external potential would be rather easy since it requires merely the minimization of a functional of the three-dimensional density function. The major part of the complexities of the many-electron problems are associated with the determination of the universal functional F[p] ".  2.2.1  T h e K o h n - S h a m Formulation  The Kohn-Sham (KS) formulation [3] is of central importance to the application of density functional theory to quantum chemical problems [13]. The Kohn-Sham treatment of the complex many-electron problem posed by real systems is based on a simpler one-electron approach which is introduced by means of a reference system of non-interacting electrons which move under  Density Functional  Theory in Quantum  Chemistry  the influence of a local external potential  22  v f(x). re  The Hamiltonian for this reference system is a sum of one-electron Hamiltonians (due to the absence of electron-electron repulsion),  Href = J2 ref(i) k  = }Z  +  ref{*i)  V  (2.31)  and the wavefunction is correctly represented by a Slater determinant. The fundamental connection between the non-interacting reference system and the real system of interacting electrons lies in the fact that both systems possess the same ground-state electron density, that is,  Pre} = 2^2 |0i| = Pexact 2  (2.32)  i  where the Kohn-Sham orbitals </>j are the solutions to the one-electron equations associated with the reference system  Keffa  = U<l>i  (2.33)  From equations (2.25) and (2.27), the energy functional has the general form  E[p]=T [p} el  + V[p} + U [p] ee  (2.34)  With the introduction of the reference system, equation (2.34) for the real interacting system can be recast as  Density Functional  Theory in Quantum  Chemistry  23  E[p] = T [p] + V[p] + J[p] + E [p] ref  xc  (2.35)  where T f[p] is the kinetic energy of the reference system re  (2.36)  V[p] is the external potential energy for the interacting system, J[p] is the classical Coulomb potential energy, and E [p]  is the exchange-correlation  xc  energy, given by  E [p] = T[p] - T [p] + U [p] - J[p) xc  ref  ee  (2.37)  The Kohn-Sham equations for the real system are  h ff{i)(t>i e  ;V  =  +v,eff  where the Kohn-Sham effective potential v ff e  SJ[p]  <t>i -  U $i  (2.38)  is  5E [p] xc  (2.39)  and  v c(r) = X  5E [p] xc  Sp(v)  defines the exchange-correlation potential.  (2.40)  Density Functional  Theory in Quantum  Chemistry  24  The total energy is given by equation (2.35) but it can also be expressed in terms of the Kohn-Sham orbital energies as [1]  E[p] =  ' -J  dT + dv  Y,H-\I  l _r? r  u  -(r)p(r) dr  (2.41)  where  5 > = T [p] + [ v {r)p{v) ref  i  efS  dv  (2.42)  J  Equations (2.41) and (2.42) show that the total electronic energy is not given by the sum of the orbital energies, just as in the Hartree-Fock method. The significance of the Kohn-Sham approach has been clearly stated by Kohn, Becke, and Parr [11]: " . . . in spite of the appearance of simple, single particle orbitals, the KS equations are in principle exact provided that the exact E  xc  is used . . . the only error in the theory is due to approximations of  E " • xc LJ  2.2.2  Density Functionals  The fact that density functional theory is exact but the exact form of the energy functional is not known implies that approximate formulations of E  xc  have to be employed in order for density functional calculations to be  practicable.  The development and testing of functionals is thus a central  issue and a major challenge in modern D F T [9]. The solid foundations of density functional theory were established by the  Density Functional  Theory in Quantum  Chemistry  25  Hohenberg and Kohn theorems, but long before this work, Thomas, Fermi, and Dirac had already developed models that focused on the electron density as the central variable in the description of electronic structure and behaviour  The Thomas-Fermi (TF) energy functional is  ETF[P] =C  J p ( r ) / dv + J (T)V(T) 5  F  3  P  dv + J[p]  (2.43)  where the first term on the right-hand side represents the kinetic energy functional, taken from the theory of a non-interacting uniform electron gas [1, 4, 78], J[p] replaces U [p] of equation (2.34) — thus, non-classical electronee  electron interactions are neglected — , and CF — 2.8712 [1]. The Thomas-Fermi-Dirac (TFD) energy functional extends the T F model by including the exchange-energy functional of a uniform electron gas  (2.44)  where C = 0.7386 [1]. x  The T F method is a rather crude model. In fact, its accuracy for atoms is rather limited, and its application to molecules has been completely unsuccessful since it fails to predict bonding [1]. Nevertheless, the T F model does contain " all the important ingredients of a density functional theory " [4] and is the first example of application of " one of the most important ideas in modern density functional theory, the local density approximation "  Density Functional  Theory in Quantum  Chemistry  26  The Local Density Approximation (LDA) [3] is the simplest expression for the energy functional of D F T . The exchange-correlation energy is given in the L D A by  •E£ {p)= A  where e (p) xc  J P(r) e (p) dv xc  (2.45)  is the exchange-correlation energy per particle of a uniform  electron gas of density p [1, 4, 78]. Thus, the L D A applies locally the relations and results for a uniform electronic system to the description of systems which have inhomogeneous electron distributions, such as atoms, molecules, i and solids. Although there is no formal justification for this procedure, many successful applications support the use of L D A in chemistry [7]. Separation of the exchange and correlation contributions is usual in D F T , so e (p) may be expressed as xc  e {p) = e (p) + e (p) xc  x  c  .  (2.46)  where the exchange contribution is from Dirac's exchange functional of equation (2.44)  e( ) x  P  = -l(l) p(v)^ 1/3  (2.47)  Analytic forms for e (p) have been derived [79-81] from the results of Monte c  Carlo calculations on the electron gas [82]. Ignoring the correlation contribution in equation (2.45) leads to Slater's Xa  method [83] which was developed before the work of Hohenberg, Kohn,  Density  Functional  Theory in Quantum  Chemistry  27  and Sham. A simplification of the Hartree-Fock method is achieved by replacing the complicated non-local Fock operator by a simpler local operator, named the Xa  local potential and defined by 1/3  vxa(r) =  ~2  a  (2.48)  -AT)  7T  In atomic and molecular electronic structure calculations, a has been used as an adjustable parameter, the best results having been obtained w i t h a ~ 0.75 [I]'A generalization of the L D A to treat a-spin and /3-spin densities separately is the L o c a l Spin Density ( L S D ) approximation [1, 84] i n which  p(r)=p (r)-r-p"(r)  (2.49)  a  and the exchange energy becomes  E  LSD  =  2  l/3  Cx  J[ a( )4/3 p  r  +  ^( )4/3] r  ( .50) 2  T h e corresponding separation of correlation contributions is not possible because correlation effects, unlike exchange, involve both like-spin and unlikespin interactions. T h e L S D is needed for correctly describing systems under the influence of an external magnetic field, but it is also necessary (in the absence of a magnetic field) for the treatment of spin-polarized systems and relativistic effects. Furthermore, there is a more fundamental reason that favours L S D over L D A . Were the exact form of functionals known, L S D and L D A should  Density Functional  Theory in Quantum  Chemistry  28  produce the same results where no net magnetic effects are present. However, for calculations on real systems approximate functional expressions must be introduced, and it turns out that the performance of approximate LSD functionals usually surpasses that of approximate L D A functionals [1, 84]. A systematic extension [78] of the L D A is obtained by means of gradient expansions of the form  E [p] = A J x  where A  x  and B  x  X  p(r) / dv + B J 4  dv + • • •  3  x  (2.51)  are constants. Although Gradient Expansion Approxima-  tions (GEA) may appear to be the natural step for improving upon the L D A description, it has been found that they fail to yield quantitatively significant results [78]. The failure of the G E A led to the development of methods based on a different type of gradient-based corrections, the so-called Generalized Gradient Approximation (GGA) [11, 15, 78, 85]. In the G G A , the exchange-correlation energy is given by  Eg  GA  = / f( , a  P  / , V p , V / ) dv Q  (2.52)  where / represents a function of the density and density gradients. Among the most popular G G A functionals are those developed by Perdew [87], Becke [88], Perdew and Wang [78, 86], and Lee, Yang, and Parr [89]. The L D A and LSD have been described as "remarkably useful structural, though not thermochemical, tools" [11]. With the development of the GGAs, D F T has also become a good approach to chemical energetics. However, the  Density Functional  Theory in Quantum  Chemistry  29  G G A has not been clearly superior in applications of D F T to the study of properties of solid state systems [90]. The G G A functionals have certainly broadened the spectrum of chemical problems to which D F T can be reliably applied, but they still have deficiencies [11]. Extensive effort has been devoted to improving the G G A by incorporating some exactly computed exchange [91-94]. The resulting scheme (usually called hybrid methods) has proven remarkably successful in a wide variety of chemical applications [95, 96]. Approaches to the exchange-energy functional that go beyond the G G A are also being explored.  Becke [97] has recently introduced second-order  gradient corrections through a new inhomogeneity parameter q°'. The new functional takes the general form  (2.53)  where q and q@ depend on the density p and the first-order and second-order a  gradients, V p and V p [97]. 2  For some particular systems, highly accurate conventional quantum chemical results are available. Therefore, there is interest in using these accurate electron densities to find highly accurate expressions for the exchangecorrelation potential. High-quality information on a specific system can shed light on how to systematically upgrade the existing approximate functionals, and may eventually lead to the exact treatment of exchange and correlation [9, 13].  Density  2.3  Functional  Theory in Quantum  Chemistry  30  A C o m p a r i s o n between D F T and Traditional M e t h o d s  T h e fundamental conceptual difference between D F T and traditional quant u m chemistry has been stated i n the introduction to this chapter and some of the details have been discussed i n the preceding sections. B o t h the Hartree-Fock method and the K o h n - S h a m method constitute one-electron approaches to the many-body electronic problem. T h e i r basic formalism is indeed very similar, as revealed by inspection of the Hartree-Fock equations, (2.8) and (2.9), and of the K o h n - S h a m equations, (2.38). However, the H F effective potential, equation (2.11), contains a (complicated) non-local exchange operator, equation (2.13), and lacks electron correlation effects. O n the other hand, the exchange-correlation potential, equation (2.40), i n the K S effective potential, equation (2.39), is a (simple) local operator that explicitly includes the effects of both electron exchange and correlation. In traditional quantum chemistry, the H F approximation can be used as a starting point, and it can be improved upon by systematically incorporating increasingly accurate treatments of electron correlation effects v i a configuration interaction, many-body perturbation, or coupled-cluster techniques. Achieving high accuracy is, however, very costly since the computational dependence of p o s t - H F methods on the molecular size M is of the order of M  5  - M  1  [16]. A c c u r a c y i n D F T lies i n the expression for the exchange-  correlation functional. T h e closer to the exact E , xc  the more accurate the  results but there is no clear, systematic procedure that can be followed to bring approximate functionals closer to the exact E . xc  A major advantage of  D F T is the fact that it can yield results that are as accurate as (and sometimes more accurate than) those obtained from more conventional ab initio  Density  Functional  Theory in Quantum  Chemistry  31  techniques, w i t h considerably reduced computational demands [95]. In fact, density functional methods that depend linearly on the molecular size have already been developed [9]. A physical interpretation can be easily associated w i t h the Hartree-Fock single-orbital energies by recourse to K o o p m a n s ' theorem (Section 2.1.1). O n the other hand, the physical interpretation of the K o h n - S h a m orbitals has been a rather controversial subject. It is clear that the energy of the highest occupied K S orbital is equal to the exact first ionization energy of the system, enoMO = —IP  e x a c t  [13], but opinions have been varied regarding the general  significance of the K S orbitals. Some examples are Levine [98]: " T h e K o h n - S h a m orbitals have no physical significance other than i n allowing the exact p to be calculated . . . the K o h n - S h a m orbital energies should not be confused w i t h molecular-orbital energies. " P a r r and Y a n g [1]: " Given the auxiliary nature of the K S orbitals — just N orbitals the sum of squares of which add up to the true total electron density — one should expect no simple physical meaning for the K o h n - S h a m orbital energies. There is none. " K o h n , Becke, P a r r [11]: " T h e individual eigenfunctions and eigenvalues, 4>j and €j, of the K S equations have no strict physical significance . . . A t the same time, a l l €j and (f)j are of great semiquantitative value, much like the Hartree-Fock energies and wavefunctions, often more so, because they reflect also correlation effects, and are consistent w i t h the exact physical density " Baerends, Gritsenko, van Leeuwen [99]: " T h e theoretical status of the K o h n Sham model has received comparatively little attention, as may be evident  Density Functional  Theory in Quantum  Chemistry  32  from the frequently voiced opinion that the K o h n - S h a m orbitals are just a means to generate electron densities but do not have any physical meaning themselves.  However, this is a far too restricted view on the K o h n - S h a m  model . . . T h e K o h n - S h a m orbitals represent electrons that move i n a potent i a l that is certainly as realistic as the Hartree-Fock 'potential' and indeed has some advantages.  There is no reason to believe that the K o h n - S h a m  orbitals are any less 'physical' or useful than the Hartree-Fock orbitals ".  2.4  C o m p u t a t i o n a l Aspects of Q u a n t u m C h e m i c a l M e t h o d s  P r a c t i c a l strategies are essential to make the computation of electronic structure feasible. Numerical methods are common for atomic systems, but the majority of molecular calculations are performed using basis-set expansion methodology [4, 70, 73]. A finite set of appropriate basis functions {p;} is introduced and the spatial function of the molecular spin-orbitals is expressed as a linear combinations of these basis functions, that is  <Pk = J29iCik  (2.54)  i  where the Cik are expansion coefficients determined i n the calculation process. T h e introduction of the basis set transforms the problem of solving the molecular electronic structure equations into a m a t r i x eigenvalue problem [73, 100]. In the Hartree-Fock method, this m a t r i x eigenvalue problem takes the form [73]  Density Functional  Theory in Quantum Chemistry  F C = SCe  33  (2.55)  The equivalent expression in the Kohn-Sham method is [100]  H C  = SCe  KS  (2.56)  In equations (2.55) and (2.56), C is the matrix of the coefficients  dk,  the orbital energies 6{ are the elements of the matrix e, and the elements of the matrix S are the overlap integrals (the basis functions are usually non-orthogonal)  Sij = J gi9jdr  (2.57)  The Fock matrix elements are given by  Fij = J 9if9jdv  (2.58)  and the corresponding Kohn-Sham matrix elements are  H%  s  = J  g i  h  e f f g j  dT  (2.59)  Both the H F and the KS equations depend on the orbitals which are the solutions to the equations.  This apparent dilemma is solved by means of  a self-consistent field (SCF) approach. The calculations are started with a reasonable guess for the orbitals and the equations are solved in an iterative  Density Functional  Theory in Quantum  Chemistry  34  fashion. The computational procedure is stopped when a given convergence criterion is met (for example, the total energy difference between two consecutive cycles has to remain smaller than a specified value for a certain number of iterations), at which point the solutions are said to be self-consistent.  2.4.1  Basis Sets  The most widely used basis functions for molecular electronic structure calculations are Slater-type orbitals (STO) and Gaussian-type orbitals (GTO) [4, 70, 73]. The general form of an S T O is  9STO(T) = C r - e~^ n  Y (9, <f>)  l  N  (2.60)  lm  Cartesian G T O s are defined by  g To(r) = C x y z e-^ a  G  b  c  (2.61)  2  N  In equations (2.60) and (2.61) GV is a normalization constant, Y (9,(f)) lm  a spherical harmonic [70], and £ is a positive exponent.  is  The sum of the  non-negative integers a, b, and c defines the type of Cartesian Gaussian: an s-type Gaussian has a + b + c = 0, a p-type Gaussian has a + b + c = 1, a d-type Gaussian has a + b + c = 2, and so on. Spherical Gaussians — which include spherical harmonics — can also be used. STOs provide a better representation of the electronic wavefunctions near the nucleus of an atom, but they render the evaluation of electron integrals complicated. On the other hand, the mathematical form of a G T O is con-  Density Functional  Theory in Quantum  Chemistry  35  venient for computational purposes. GTOs are then usually preferred, especially for calculations on relatively large systems. There is however a disadvantage in using GTOs in that they are functions of inferior quality when compared with STOs. Therefore, a larger number of G T O s are necessary to obtain results of the same accuracy as that achieved by S T O basis sets. A practical solution to this problem which permits to take advantage of the computational convenience of GTOs is the use of basis sets consisting of contracted Gaussians. A contracted Gaussian function is a fixed linear combination of primitive Gaussian functions, and can be constructed through a least-square fit of primitive Gaussians to STOs which have been already optimized in an atomic calculation [70, 73]. In molecular calculations the choice of basis sets is a compromise between accuracy (which demands large basis sets) and computational requirements (which favour smaller basis sets). A minimal basis set is the simplest form as only one basis function is used to represent each orbital, but it is seldom capable of yielding accurate results. In order to improve upon the minimal basis set performance, the set is extended by including twice, three times, four times, five times, and so on, as many functions as there are orbitals thus generating the so-called double-zeta, triple-zeta, quadruple-zeta, quintuplezeta, and so on, basis sets.  Incorporation of functions of higher angular  momentum quantum number [70] than that of the valence electrons is also important. These functions are called polarization functions because they are required for the description of the polarization of an atom in a molecular field.  Chapter 3  Computational Approach  In this chapter, the general methodology involved i n the calculation of the core-electron binding energies reported i n Chapters 4, 5, and 6 w i l l be described. T h e formalism associated w i t h the unrestricted generalized transition state ( u G T S ) model and the A E - K S approach w i l l be presented first, followed by a description of some of the details of the density functional program and the basis sets employed for the computation of the C E B E s .  3.1  T h e u G T S and A E - K S Approaches  T h e energetics of an electron removal process, such as the ionization from a core level, can be generically represented as the difference i n the total energy of the i n i t i a l and final states of the electronic system  36  Computational  Approach  37  AE  — Efi i  — Ei iu i  na  n  (3.1)  a  T h e total energy can be expressed as an analytic function of the occupation numbers  of a set of one-electron molecular orbitals 4>i [62], that is  E = £ ( n i , n , n , . . . ,n ) 2  3  (3.2)  N  and the electron density of the electronic system is defined by  p= Y , i  n  M  (-) 3  3  *  E q u a t i o n (3.3) is a generalization of (2.32) and allows for fractional occupation of the molecular orbitals [1]. F r o m (3.1) and (3.2), a core-ionization energy is given by  A£ = £(0,l,l,...,l)-£(1,1,1,...,1)  (3.4)  T h e calculation of core-electron binding energies according to equation (3.4) using K o h n - S h a m D F T constitutes the A E - K S approach. T h e results of the application of this method to the determination of molecular C E B E s w i l l be presented i n Chapters 5 and 6. Slater's transition state (TS) [62] model and its generalization by W i l l i a m s , deGroot, and Sommers [69] are approximations to equation (3.4). If the total energy is represented as a function of a continuous variable A (which can be related to orbital occupation) by the series expansion [18]  Computational  Approach  E(X)  38  = E + XEi + X E  + XE  2  0  3  2  + A £ + XE 4  3  + •••  5  4  5  (3.5)  then equation (3.4) becomes  AE  = E(l) - E(0) = E + E + E + E + E + --1  2  3  i  5  (3.6)  where -E'(O) and E(l) give the energies of the initial and final states, respectively. In D F T , according to Janak's theorem [101], the first derivative of the total energy with respect to the occupation number of an orbital is equal to the energy of that orbital, that is  £ r «  <-> 3  7  For the ionization of an electron from the ith molecular orbital fa, X represents the fraction of an electron removed. Therefore, application of Janak's theorem leads to  From equation (3.5), the first derivative is given by  BE — = F(X) = E + 2XE '+ 1  2  3X E 2  + AX E 3  3  + 5A £ + ••• 4  4  Slater's TS model uses A = | to approximate AE as  5  (3.9)  Computational  Approach  39  = E + E + -E  F  l  2  A  + -E, + ^ E L  Z  5  + •••  (3.10)  Therefore, the error of the TS model is  S  TS  = -\E  - -E± - ^E L  3  5  + •••  (3.11)  In the generalized transition state (GTS) model, AE is calculated by  AE=  \F(0)  + \ F Q  (3.12)  Thus, AE is approximated as AE  Q OA = E + E + E, + -Ei + —E 9 ol 1  5  2  + •••  (3.13)  with an error  SGTS = ~ E  4  - - ^ E  5  + ---  (3.14)  Calculations using the TS and G T S models can be performed in a restricted or an unrestricted fashion.  In the former, \ spin-a electron and  \ spin-/? electron (TS) or | spin-a electron and | spin-/? electron (GTS) are removed, whereas in the latter | spin-a//? electron (TS) or | spin-a//? electron (GTS) are removed. A study by Chong [18] showed that the uGTS approach gives the best results in density functional determinations pf molecular C E B E s .  Chapter 4 of this thesis contains the results of a number of  applications of the uGTS model.  Computational  3.2  Approach  40  The Density Functional Program deMon  All the density functional calculations reported in this thesis were performed with the program deMon [102-104], which makes use of Gaussian-based density functional methodology in the form of Gaussian orbital and auxiliary basis sets. The molecular orbitals are expanded in the orbital basis set as in equation (2.54), whereas both the density and the exchange-correlation potential are expanded in the auxiliary basis sets, that is  P = J2 i9i  (3-15)  a  i  V  xc  = Y,bi9i  (3-16)  C  i  where p and V  xc  are fits to p and V , respectively. xc  The density fitting coefficients a; are obtained analytically via a least squares fitting procedure by requiring that the error in the Coulomb energy  [ f [p(r0 - p(n)] —[p(r ) - p(r )] dv dr 2  2  x  2  (3.17)  be minimized and the fitted density p be normalized to the total number of electrons. The potential fitting coefficients b{ are determined by performing a least squares fit over a set of grid points centered about each atom. This procedure minimizes the error in the fitted potentials over the sum of the grid points.  Computational  Approach  41  B y representing the density and the exchange-correlation potential as l i n ear combinations of Gaussian functions, the integrals i n which p and V  xc  are  involved are given a simpler mathematical form and the computational demands are alleviated. In fact, if no is the number of orbital basis functions and riA is the number of auxiliary basis functions, the computational effort i n G T O - b a s e d D F T scales formally as (no)  2  Hartree-Fock methods and at least (no)  5  n, A  compared w i t h (no)  4  for  for conventional correlated meth-  ods.  3.2.1  Basis Sets  In d e M o n , the Gaussian functions used are s-, p-, and d-type G T O s (orbitals of higher angular momentum quantum number such as / - t y p e functions cannot be included). T h e auxiliary basis sets are described by the general notation (j, k; m, n), where j and k are, respectively, the number of s-type G T O s and the number of sets of s-, p-, and d-type G T O s for the density fit, and m and n are the corresponding number of basis functions and number of sets for the exchangecorrelation potential fit. T h e auxiliary fitting functions denoted by (3,1;3,1) for hydrogen, (4,4;4,4) for boron, carbon, nitrogen, oxygen, and fluorine, (5,4;5,4) for silicon, phosphorus, sulphur, chlorine, and argon, and (5,5;5,5) for chromium, vanadium, germanium, bromine, and iodine were employed i n the calculations. T h e orbital basis sets used were the s, p, d parts of the correlationconsistent polarized valence basis set given by D u n n i n g and coworkers [105,  Computational  Approach  42  106]. These sets are described by the general notation c c - p V n Z , where n i n dicates the quality of the basis functions: T for triple-zeta, Q for quadruplezeta, and 5 for quintuple-zeta. T h e c c - p V n Z basis sets were used for hydrogen, the second-period elements boron through fluorine, and the third-period elements silicon through argon. T h e basis sets of double-zeta quality denoted by D Z V P or D Z V P 2 [107] — which are included i n d e M o n — were employed for chromium, vanadium, germanium, bromine, and iodine. For hydrogen, c c - p V n Z basis sets consisted of n sets of s-type functions and p-type functions were used as polarization functions. For second- and third-period elements, the composition of the basis sets was as given i n Table 3-1. T h e d-type basis functions consisted of six cartesian components.  T a b l e 3-1. Composition of cc-pVnZ basis sets.  basis set  number of functions s-type  p-type  d-type  second-period elements cc-pVTZ  4  3  2  cc-pVQZ  5  4  3  cc-pV5Z  6  5  4  third-period elements cc-pVTZ  5  4  2  cc-pVQZ  6  5  3  cc-pV5Z  7  6  4  Computational  Approach  43  The initial investigations carried out by Chong [18, 19] showed that u G T S / D F T calculations of C E B E s using the cc-pV5Z basis set were in excellent agreement with experimental results.  Nevertheless, application of  this approach to relatively large molecules — and eventually to species large enough to be realistic models for extended systems — becomes prohibitively expensive as far as computational demands are concerned. A viable alternative is to use smaller basis sets modified to improve the description of the core-hole state involved in C E B E calculations.  Chong and coworkers [20]  proposed a modification of the cc-pVTZ basis set based on an exponent scaling factor designed to provide a more efficient representation of the shielding effects associated with the partially ionized state. The scaled basis sets were thoroughly tested on a variety of molecules [20, 24, 25, 29] — including relatively large transition metal carbonyls and nitrosyls [22] — with remarkably good results.  In fact, the performance of the scaled-pVTZ basis set was  almost as good as that of the much larger cc-pV5Z set. For G T O basis functions, the scaling factor for the exponent is the square of the ratio of the effective nuclear charge  (3.18)  where Z is the nuclear charge of the atom considered, o and o' are the screening constants of the neutral molecule and the (partially) core-ionized cation, respectively.  The screening constants were determined using the formulas  provided by Clementi and Raimondi [108]. For boron through neon, they are given by  Computational  Approach  o(ls)  44  = 0.3 + 0 . 0 0 7 2 ( Z - 2)  (3.19)  CT(2S) = 1.7208 + 0.3601(Z - 3)  (3.20)  o(2p) = 2.5787 + 0.3326(Z - 5)  (3.21)  In the u G T S approach, the modified screening constants (due to the removal of two thirds of an electron from a core Is orbital) were calculated via  (3.22)  (3.23)  1.8585 + 0.7202 + 0.3326(Z - 5)  (3.24)  In equation (3.24) the screening factor 2.5787 was arbitrarily partitioned into 1.8585 and 0.7202 in order to consider l s electron and 2s electron effects separately.  A n equivalent partition into 1.7208 ( l s ) and 0.8579 (2s) had  negligible effects [20]. The scaling factors determined from equation (3.18) [20] are shown i n Table 3-2. Results of calculations performed w i t h scaledp V T Z and scaled-pVQZ basis sets w i l l be presented i n Chapter 4.  Computational  Approach  45  T a b l e 3-2. Scaling factors (uGTS) for second-period atoms.  atom  Is  2s  2p  B  1.0873268  1.4985434  1.5771700  C  1.0717755  1.3907759  1.4413679  N  1.0609247  1.3211702  1.3570659  0  1.0529234  1.2725484  1.2997082  F  1.0467796  1.2366817  1.2581833  Scaled basis sets were also employed w i t h i n the A E - K S approach. In this case, screening constants for a fully ionized cation are needed. For I s and 2s orbitals, they were calculated by  o'(ls) = 0.0072(Z - 2)  a'(2s) =  1.7208 + 0 . 3 6 0 1 ( 2 - 3 )  (3.25)  (3.26)  Three different 2p scaling factors were obtained and tested. T h e screening constants were determined using different partition schemes as follows  a  J(2p) =  1.8585 + 0.7202 + 0.3326(Z - 5)  (3.27)  Computational  Approach  46  o' (2p)=(^j  1.7208 + 0.8579 + 0 . 3 3 2 6 ( 2 - 5 )  n  o' (2p) HI  =  2.5787 + 0.3326(zT - 5)  (3.28)  (3.29)  T h e corresponding scaling factors from equation (3.18) are given i n Table 3-3.  T a b l e 3-3. Scaling factors (AE-KS) for second-period atoms.  atom  ls  2s  2p(I)  2p(II).  2p(III)  B  1.1323609  1.7854975  1.9147939  1.8369636  1.6034193  C  1.1085960  1.6102784  1.6922230  1.6347254  1.4610248  N  1.0920626  1.4985005  1.5560007  1.5106066  1.3727422  0  1.0798969  1.4211246  1.4642723  1.4268460  1.3127300  F  1.0705703  1.3644409  1.3983809  1.3665784  1.2693126  Another type of basis set, labeled as c c - p C V T Z [109], was also tested i n A E - K S calculations. It represents an extension of the c c - p V T Z set developed to treat core a n d core-valence correlation effects more efficiently. T h e new basis set of triple-zeta quality adds two s-type, two p-type, and one d-type functions to the original set. Therefore, the number of basis functions grows  Computational  Approach  47  from [4s3p2d] (Table 3-1) for cc-pVTZ to [6s5p3d] for cc-pCVTZ, but the latter is still smaller than a cc-pV5Z set, and it is a legitimate option for the treatment of fairly large systems. The results of the A E - K S calculations with scaled-pVTZ and cc-pCVTZ sets will be discussed in Chapter 6.  Chapter 4  Applications of the u G T S Model  T h e results of calculations of core-electron binding energies using the unrestricted generalized transition state approach described i n Section 3.1 are contained i n this chapter. Extensive calculations by C h o n g and coworkers [18-25] have already demonstrated that the u G T S / D F T approach yields C , N , O , and F C E B E s in very good agreement w i t h experiment. Thus, the i n vestigations reported i n this chapter represent an extension of this approach to boron-containing molecules, to some additional isomer cases, and to the third-period elements S i , P, S, CI, and A r . A l l calculations were performed using experimental geometries [110] of the neutral parent molecules and the functional labeled B 8 8 / P 8 6 , which consists of Becke's 1988 exchange functional [88] and Perdew's 1986 correlation functional [87]. T h e B 8 8 / P 8 6 combination and some other functional choices available i n d e M o n were tested by C h o n g [18].  48  The results obtained showed  Applications  of the uGTS  Model  49  that the B 8 8 / P 8 6 functional delivers the best performance i n u G T S - b a s e d calculations of C E B E s . Relativistic effects must be taken into account when calculating large binding energies such as those of core-level electrons because the velocity of an electron i n an atom's inner shell is not negligible compared to the velocity of light [33]. T h e results of the calculations for second-period elements were modified w i t h approximate relativistic corrections, based on the studies by Pekeris [111] for two-electron ions. The corrections were estimated w i t h the following empirically derived equation [19]  Crel = A I*  where C i  (4.1)  a  is the relativistic correction, I i nre  is the non-relativistic C E B E  i n e V , A = 2.198 • 1 ( T , and B = 2.178.  A and B are two fitting pa-  re  7  rameters of the relation between the relativistic correction and the nonrelativistic C E B E s for Pekeris' two-electron ions.  Relativistic corrections  for core-ionization energies of third-period elements were not available.  4.1  Boron-Containing Molecules  The results for boron C E B E s are summarized i n Tables 4-1 and 4-2.  The  range of experimental C E B E s [40, 112] is almost 10 e V , from 192.9 e V for BH P(CH ) 3  3  3  to 202.8 e V for B F . The C E B E s of C , N , O , and F were also 3  calculated and are shown i n Tables 4-3 and 4-4.  Applications  of the uGTS  Model  50  T a b l e 4-1. Basis set convergence in the calculation of B ls energies (in eV) with unsealed basis sets. Calculated CEBEs include relativistic corrections from equation (4.1).  molecule BF  3  cc-pVTZ  cc-pVQZ  cc-pV5Z  experiment  202.44  202.21  202.17  202.80  BC1  3  199.83  199.50  199.45  199.80  BBr  3  199.19  198.84  198.80  199.00  198.25  197.86  197.80  197.80  196.84  196.59  196.56  196.50  195.59  195.34  195.31  195.15  195.16  194.84  194.79  194.69  194.46  194.17  194.14  193.73  194.07  193.82  193.78  193.60  199.54  199.31  199.28  BI  3  B BH CO 3  BH PF 3  3  B H NH 3  3  BH CNCH 3  B F PH 3  3  3  Average absolute deviations ( A A D ) and m a x i m u m deviations ( M D ) of the calculated C E B E s from the experimental values and from the estimated complete basis set ( C B S ) limits are given i n Table 4-5. T h e estimated C B S limits for 19 of the test cases were determined from the results of the corresponding c c - p V T Z , c c - p V Q Z , and cc-pV5Z calculations, according to the empirically derived equation [113]  A(x)  = A(oo) + B e "  (4.2)  Applications  of the uGTS  Model  51  where x is the cardinal number of the basis set (3, 4, 5 for T Z , Q Z , 5 Z , respectively) and A(o6) is the estimated C B S limit for the property.  T a b l e 4-2. Calculations of B core-electron binding energies (in eV) with scaled basis sets. Calculated CEBEs include relativistic corrections from equation (4.1).  molecule  scaled-pVTZ  scaled-pVQZ  202.23  202.19  202.80  199.54  199.50  199.80  198.86  198.83  199.00  197.89  197.84  197.80  BH3CO  195.37  195.33  195.15  B H3PF3  194.90  194.83  194.69  B H3NH3  194.20  194.16  193.73  BH3CNCH3  193.85  193.81  193.60  BF  3  BC1  3  BBr BI  3  3  experirru  3  193.57  193.37  BH P(CH )3  193.00  192.93  B  196.63  B 5H9 — base  196.27  196.10  B 5H9 — apex  194.26  194.20  1,5- B C H  5  196.28  196.00  1,6 — B 4C2H6  195.67  195.40  B(OCH )  197.52  197.80  3  195.91  196.40  B F3PH3  199.30  B 3N H6  196.08  BH N(CH ) 3  3  3  3  3  3  B(CH ) 3  3  2  3  196.58  199.28  196.50  Applications  of the uGTS  Model  52  T a b l e 4-3. Basis set convergence in the calculation of C, N, O, and F Is energies (in eV) with unsealed basis sets. Calculated CEBEs include relativistic corrections from equation (4.1).  molecule  cc-pVTZ  cc-pVQZ  cc-pV5Z  experiment  BH CO  296.03  295.74  295.70  296.18  BH3CNCH3  294.21  293.91  293.87  294.06  BH C NCH  293.36  293.12  293.07  293.43  BH3NH3  408.64  408.25  408.20  408.41  BH3CNCH3  407.43  407.08  407.03  407.13  BH3CO  542.29  541.87  541.80  542.05  BF  695.24  694.72  694.65  694.80  BH3PF3  695.41  694.89  694.82  694.30  BF3PH3  692.58  692.10  692.03  3  3  3  3  For the 9 cases studied with the cc-pVQZ and cc-pV5Z basis sets and the 17 cases investigated with the cc-pVTZ basis set, the A A D from experiment for the boron C E B E s was smallest for cc-pV5Z, and the cc-pVQZ performance was almost as good as that of cc-pV5Z. It is also observed that with exponent scaling the results from the triple-zeta basis set (scaled-pVTZ) significantly improved upon those from the corresponding unsealed basis set (cc-pVTZ), and they were brought much closer to the results obtained from cc-pV5Z calculations. Exponent scaling did not have an appreciable effect on the performance of the pVQZ basis set as far as agreement with experimental  Applications  of the uGTS  T a b l e 4-4.  Model  53  Calculations of C, N , O, and F core-electron binding energies (in  eV) with scaled basis sets. Calculated C E B E s include relativistic corrections from equation (4.1).  molecule  scaled-pVTZ  scaled-pVQZ  experiment  BH CO  295.74  295.72  296.18  BH3CNCH3  293.93  293.90  294.06  293.12  293.10  293.43  3  BH C NCH 3  3  292.36  292.20  292.27  291.97  291.59  291.30  BH P(CH ) ;  291.28  290.96  1,5-B C H  290.52  290.20  B(OCH ) 3  3  BH N(CH ) 3  3  1,6 —  B4  3  C  3  3  3  3  2  5  BH3NH3  408.28  408.23  408.41  BH3CNCH3  407.08  407.05  407.13  BH N(CH ) 3  3  B(OCH ) 3  3  3  406.68  407.23 541.86  BH3CO  BF  3  541.83  538.30  538.49 694.71  694.69  694.80 694.30  BH PF 3  3  694.88  694.85  BF PH  3  692.03  692.06  B N H6  405.01  3  3  3  542.05  results is concerned. However, comparisons w i t h the estimated C B S limits are more meaningful i f the a i m is to design an efficient basis set applicable  Applications  of the uGTS  Model  54  T a b l e 4-5. Average Absolute Deviation (in eV) and Maximum Deviation (in eV) of calculated core-electron binding energies from experiment and from C B S . The number of test cases is given in parentheses.  basis set  deviations from experiment AAD  MD  deviations from C B S AAD  MD  boron test cases cc-pVTZ  0.36 (17)  + 0.73  0.34 (10)  0.46  cc-pVQZ  0.24 ( 9)  - 0.59  0.05 (10)  0.07  cc-pV5Z  0.23 ( 9)  - 0.63  0.01 (10)  0.01  scaled-pVTZ  0.24 (17)  - 0.57  0.07 (10)  0.10  scaled-pVQZ  0.24 ( 9)  - 0.64  0.03 (10)  0.06  CBS  0.23 ( 9)  - 0.64 all test cases  cc-pVTZ  0.41 (32)  + 1.11  0.38 (19)  0.60  cc-pVQZ  0.25 (17)  ± 0.59  0.06 (19)  0.08  cc-pV5Z  0.26 (17)  - 0.63  0.01 (19)  0.01  scaled-pVTZ  0.26 (32)  + 0.58  0.07 (19)  0.10  scaled-pVQZ  0.24 (17)  - 0.61  0.04 (19)  0.06  CBS  0.26 (17)  - 0.64  to large molecules. Also, experimental observations are affected by experimental error whereas the C B S l i m i t gives a fixed value. Table 4-5 shows that the c c - p V 5 Z basis set approximated the C B S l i m i t very well and that  Applications  of the uGTS  Model  55  the scaled-pVTZ basis set also performed satisfactorily. In the case of the scaled-pVQZ basis set, results do show some improvement upon those from cc-pVQZ calculations when compared with the CBS limit. The effect of relativistic corrections is either very small or negligible for boron C E B E s . This was expected from the fact that the relativistic correction calculated via equation (4.1) was found to be only 0.02 eV for the boron ls energies. The results for the C E B E s of all five second-period elements investigated — 32 cases with p V T Z and 17 cases with pVQZ and pV5Z — gave an A A D from experiment of 0.26 eV for both the cc-pV5Z and scaled-pVTZ basis sets. This compares well with a previous study [24] of 66 C E B E s of small closed-shell molecules, in which the A A D was found to be 0.22 eV. In the calculations with scaled basis sets, only the atom with the core hole is treated differently by making use of exponent scaling. Equations (3.19) through (3.21) can be rewritten as  CT(1S) = 0.3 + 0.0072(n + n ) 2s  o{2s) = 1.7208 + 0.3601(n + n  2p  2s  2p  - 1)  o{2p) = 2.5787 + 0.3326(n - 1) 2p  where n  2s  and n  2p  (4.3)  (4.4)  (4.5)  can be obtained from population analysis [73], and a  general-scaled basis set can be constructed by calculating new screening constants from equations (4.3) through (4.5) and the corresponding scaling fac-  Applications  of the uGTS  Model  56  tors from equation (3.18), for each of the atoms in a given molecule. The resulting basis set was labeled gs-pVTZ when obtained from the cc-pVTZ set, and was tested in calculations of C E B E s for 12 of the test cases reported in this section. The results are given in Table 4-6, where they are compared with those from scaled-pVTZ basis sets. Table 4-7 presents an AAD-based analysis of the performance of the gs-pVTZ basis set.  T a b l e 4-6. Calculations of core-electron binding energies (in eV) with the gsp V T Z basis set. C E B E s include relativistic corrections from equation (4.1).  molecule  scaled-pVTZ  gs-pVTZ  202.23  202.22  B  196.63  196.60  BH3CO  195.37  195.33  B H3NH3  194.20  194.15  BH3CNCH3  193.85  193.81  BH3CO  295.74  295.74  BH3CNCH3  293.93  293.92  BH C NCH  293.12  293.11  BH3NH3  408.28  408.25  BH3CNCH3  407.08  407.07  BH3CO  541.86  541.84  BH3PF3  694.71  694.67  BF  3  3  3  Applications  of the uGTS  Model  57  T a b l e 4-7. Deviation analysis (in eV) of the gs-pVTZ basis set performance for the 12 cases reported in Table 4-6.  basis set  deviations from experiment  deviations from C B S  AAD  AAD  scaled-pVTZ  0.25  0.07  gs-pVTZ  0.25  0.04  cc-pV5Z  0.27  0.01  Although the use of generalized scaling helped to improve upon the performance of the single-atom scaling approach for the 12 cases investigated (this is reflected by the A A D from CBS), the gain is relatively small and is not justified by the large amount of additional effort that is required for obtaining the scaling factors and gs-pVTZ basis sets for all the atoms.  4.2  Isomers of C H N O 3  5  The total energies, relative energies with respect to the most stable isomer (ethyl isocyanate), and dipole moments of four isomers of  C3H5NO  are sum-  marized in Table 4-8. Figure 4-1 shows the structures and atom numbering schemes for the isomers. For the three species with lowest energies (ethyl isocyanate, 2-azetidinone, 3-hydroxypropanenitrile) it is observed that the calculated dipole moments compare very well with the corresponding experimental values [114].  Applications  of the uGTS  Model  58  Table 4-8. Total energies (in eV), relative energies (in eV) with respect to ethyl isocyanate, and dipole moments (in D) for the isomers of C 3 H 5 N O .  isomer  E  AE  p ;  ethyl isocyanate  -6732.0304  (0.0)  2.848  2.81±0.02  2- azetidinone  -6731.5353  0.4951  3.783  3.828  3- hydroxypropanenitrile  -6731.2750  0.7554  3.122  3.166  lactonitrile B  -6731.2302  0.8002  3.139  lactonitrile A  -6731.1962  0.8342  2.982  a  ca  p  a obs  reference 114  Two conformers have been reported [115] for the fourth isomer, lactonitrile. They were labeled A and B by the authors. Their microwave spectroscopic measurements indicated that conformer A was more stable than conformer B. D F T calculations were performed for both conformers using the experimental geometries reported by the authors, but the results did not agree with those from the microwave experiment, the total energy of conformer B having been found to be lower than that of conformer A . In addition, geometry optimizations were carried out (with the semiempirical method Austin Model 1, or A M I , [116] available in the software package called W i n M O P A C Version 2 [117]) with both conformer A and B structures as starting geometries, and in both cases the calculations converged to the geometry of conformer B.  Applications  of the uGTS Model  59  A : 2-azetidinone  B : e t h y l isocyanate  F i g u r e 4-1. Numbering scheme for the isomers of C 3 H 5 N O . Hydrogen atoms are not shown.  Table 4-9 summarizes the C E B E s of the  C3H5NO  isomers. T h e y are also  shown schematically i n Figure 4-2. O n l y the C E B E s of 2-azetidinone have been determined experimentally [41], and i n this case the agreement between  Applications  of the uGTS  Model  60  T a b l e 4-9. Core-electron binding energies (in eV) for the isomers of C 3 H 5 N O . Calculated CEBEs include relativistic corrections from equation (4.1)  isomer 2-azetidinone  ethyl isocyanate  3-hydroxypropanenitrile  lactonitrile B  lactonitrile A  atom  CEBE  experiment  deviation  0  537.35  537.32  +0.03  N  405.95  405.76  +0.19  CI  293.77  C2  292.43  C3  291.41  0  539.44  N  405.96  CI  294.60  C2  292.59  C3  291.28  0  539.41  N  405.71  C3  293.11  CI  292.85  C2  292.70  0  539.57  N  405.67  C2  294.09  CI  292.70  C3  291.67  0  539.60  N  405.65  C2  294.05  CI  292.67  C3  291.65  Applications  of the uGTS  Model  61  1 I | 1 1 1 11 1  Carbon  A B C D  i  i  291  i  i  293  295  Nitrogen  i 1 i 1  1 1  A B C D i  i  404  i  i| i il  A  i  406  i  408  Oxygen  1 1  B C D i  i  537  i  539  i  i  541  Core-Electron Binding Energy / e V  Figure 4-2. Calculated (solid lines) and available experimental (dashed lines) CEBEs for the isomers of C3H5NO.  Applications  of the uGTS  62  Model  calculated and observed energies is very good. All isomers possess a distinctive core-ionization spectrum, a demonstration of the analytical potential of E S C A . For 2-azetidinone, ethyl isocyanate, and lactonitrile, three clearly distinct C Is energies are observed. In the case of 3-hydroxypropanenitrile, the C Is energies are still different but much closer to one another. The N Is energies are all similar, as are the O Is energies, except for the C E B E value of 2-azetidinone which is considerably lower — about 2 eV — than those of the other three species. It should be noted that, with the increasing availability of high resolution synchrotron radiation facilities, the experimental observation of very close C E B E values has become possible, as evidenced by the results of a recent investigation of the photoelectron spectra of propene and 2-methylpropene [51]. For the two reported conformers of lactonitrile, only a very small difference between their corresponding C E B E s was found. E S C A is, however, not capable of detecting small conformational changes [32], since they do not represent a substantial modification of the chemical environment.  4.3  C o r e - E l e c t r o n B i n d i n g Energies of Si, P, S, C I , and A r  The results of the calculations of C E B E s of Si, P, S, CI, and Ar are summarized in Tables 4-10 through 4-13.  A l l the values reported correspond  to ionization from 2p orbitals, which is the most common experimentally studied transition in the case of third-period elements. Most experimental data [40, 112] are for 2p / energies — 3/2 is the value of the total angular 3  2  momentum quantum number j, in the case of a 2p orbital j (= l±s)  is either  Applications  of the uGTS  Model  63  T a b l e 4-10. Basis set convergence in the calculation of Si 2p energies (in eV). Experimental results are for 2p transitions.  molecule  cc-pVTZ  cc-pVQZ  cc-pV5Z  experiment  112.72  113.03  111.86  111.75  3  110.37  110.88  109.79  109.44  SiH Cl  108.95  109.70  108.65  108.11  SiH Br  108.91  109.51  108.45  108.08  109.01  109.58  108.51  108.01  Si H 0  108.83  109.41  108.36  107.81  Si H S  108.39  109.00  107.96  107.45  SiH  108.13  108.76  107.71  107.30  107.68  108.30  107.25  106.89  SiF  4  SiHCl 3  3  SiH FCH 2  2  6  2  6  4  SiH CH 3  3  3  3/2 or 1/2 (1 = 1, s = 1/2) — except for Si cases and some of the P, S, and CI cases where weighted averages — these are referred to as 2p energies — of spin-orbit doublets have been reported [112]. Irregularities associated w i t h the convergence of the c c - p V n Z basis sets are observed especially in the case of Si C E B E s but also i n most P cases. It should also be noticed that although S and CI C E B E s show the expected trend ( T Z > Q Z > 5Z), the convergence is considerably slower than it is i n the case of calculations for second-period elements (Section 4.1). T h e existence of irregularities has been confirmed by D u n n i n g [118], who has pointed out that some of the problems are due to deficiencies i n the d-set, and has suggested augmenting the c c - p V n Z sets v i a addition of high-exponent d-functions. A  Applications  of the uGTS Model  64  T a b l e 4-11. Basis set convergence in the calculation of P 2p energies (in eV). Experimental results are for 2 p / transitions unless otherwise indicated. 3  molecule PF  5  POF  3  2  cc-pVTZ  cc-pVQZ  cc-pV5Z  experiment  145.39  145.35  144.25  144.65 < )  144.20  144.18  143.10  143.25  2p  ^  143.00 P SF PF  3  3  143.41  143.43  142.38  142.68  ^  142.78  142.87  141.93  142.05  ^  141.78 PF BH 3  POCl  3  3  143.46  143.53  142.53  141.79  142.04  142.09  141.05  141.35 t ^ 2  141.02 PC1  3  140.83  140.96  140.05  140.15 ( > 2p  139.75 P CI2CH3  139.79  139.95  139.04  138.88  PH  138.21  138.44  137.57  137.33 ^ )  3  ^  137.05 PH2CH3  137.83  137.98  137.10  136.55  P  137.68  137.88  13.6.98  136.20  4  set of augmenting (^-functions for sulphur (provided by D u n n i n g [118]) was incorporated into the original c c - p V n Z basis sets (Section 3.2.1, Table 3-1) and was found to yield more accurate results. Therefore, a l l S C E B E s were  Applications  of the uGTS  Model  65  calculated w i t h the augmented c c - p V n Z basis sets which differ from those described i n Section 3.2.1 by the inclusion of one more d-function.  T a b l e 4-12. Basis set convergence i n the calculation of S 2p energies (in eV). Experimental results are for 2p$/2 transitions unless otherwise indicated.  molecule SF  6  cc-pVTZ  cc-pVQZ  cc-pV5Z  180.69  180.52  179.90  experiment 181.00 <> 2 p  180.29 SF C1  179.93  179.77  179.15  179.27  SF  178.32  178.22  177.63  178.20 ( >  178.52  178.39  177.78  177.67  177.79  177.67  177.07  176.97  177.55  177.43  176.84  176.67  177.01  176.90  176.31  176.20  176.91  176.71  176.13  176.05  175.83  175.63  175.06  174.82  175.47  175.33  174.72  174.53  (CH3) S 0  173.01  172.94  172.35  171.91  s C1  172.63  172.48  171.88  171.57  ocs  172.35  172.22  171.63  170.69  H S  171.81  171.70  171.11  170.32  171.35  171.20  170.61  170.30 < )  2  171.54  171.42  170.82  169.92  CH SH  171.10  170.98  170.39  169.40  (CH ) S  170.55  170.44  169.85  169.06  (SiH ) S  170.36  170.22  169.64  168.60  5  4  S0 F 2  NSF S0  2  3  3  SOF  2  S0 C1 2  so  2  2  SOCl  2  2  2  2  2  PSF CS  3  3  3  3  2  2  2p  2p  Applications  of the uGTS  Model  T a b l e 4-13. Basis set convergence in the calculation of A r and CI 2p energies eV). Experimental results are for 2 p / transitions unless otherwise indicated. 3  molecule  2  cc-pVTZ  cc-pVQZ  cc-pV5Z  Ar  249.91  249.48  249.18  248.62  CIFO3  217.96  217.33  216.39  216.27  CIF3  214.19  213.87  213.35  213.02  C1F  211.16  210.84  210.39  209.19  CI  209.51  209.15  208.71  207.82  208.84  208.50  208.07  207.49  SF CI  208.84  208.48  208.04  207.44  POCI3  208.67  208.32  207.89  207.40 < )  2  S0 C1 2  2  5  experiment  2p  207.32 HC1  209.11  208.78  208.34  207.39  CCI4  208.38  208.07  207.62  207.02  BCI3  208.36  208.06  207.62  207.00  Si C I 4  208.18  207.85  207.42  206.92  CHCI3  208.34  208.04  207.60  206.83  I CI  208.22  207.85  207.40  206.68  208.21  207.89  207.45  206.66  207.97  207.62  207.19  206.65  207.83  207.44  206.99  206.60  CH  CI  2  GeCl  2  4  PCI3  206.42 SOCla  207.88  207.52  207.08  206.55  CH  CI  208.00  207.67  207.22  206.25  SiH CI  207.88  207.53  207.10  206.22  S2 C I 2  207.48  207.12  206.67 .  206.21  207.87  207.47  207.04  206.18  207.61  207.23  206.79  206.10  207.31  206.98  206.56  205.67  3  3  CrC-2 C I VO CI  3  GeH CI 3  2  ( 2 p )  Applications  of the uGTS  Model  67  Average absolute deviations from experimental results are given in Tables 4-14 and 4-15. The AADs for Si and P C E B E s and the A A D s from 2p energies (the vast majority of which are Si and P cases) clearly show the convergence irregularities in the basis sets.  Calculations are in better agreement with  experimental 2p energies than they are with 2p / energies. This is related 3  2  to the fact that the relativistic spin-orbit coupling effects which cause the splitting of the 2p transition into a doublet cannot be explicitly incorporated or treated in deMon calculations.  T a b l e 4-14. Average Absolute Deviation (in eV) of calculated core-electron binding energies from experiment sorted by element. The number of test cases is given in parentheses.  basis set  Si  P  S  CI  AAD from 2p energies cc-pVTZ  0.91 (9)  0.79 (8)  cc-pVQZ  1.48 (9)  0.87 (8)  cc-pV5Z  0.41 (9)  0.22 (8)  AAD from 2p / energies 3  2  cc-pVTZ  1.24 (8)  1.12 (17)  1.50 (23)  cc-pVQZ  1.35 (8)  0.99 (17)  1.14 (23)  cc-pV5Z  0.40 (8)  0.45 (17)  0.68 (23)  Applications  of the uGTS  Model  68  Table 4-15. Average Absolute Deviation (in eV) of calculated core-electron binding energies from experiment for all test cases. The number of test cases is given in parentheses.  basis set  A A D from 2p energies A A D from 2p / energies 3  2  cc-pVTZ  0.84 (22)  1.32 (49)  cc-pVQZ  1.06 (22)  1.12 (49)  cc-pV5Z  0.38 (22)  0.55 (49)  In general, the results obtained with the cc-pV5Z basis sets are reasonably good, especially if the comparison is made with experimental 2p energies. In fact, the A A D (from 2p energies) for P cases (0.22 eV) is almost as good as the A A D obtained in calculations of C E B E s of second-period elements [18, 19, 20, 24], in which a larger number of cases were explored. However, the strongest (and, as reflected by the number of cases in Table 4-15, most frequently reported) transition is usually the ionization from the 2p / level, 3  2  with which calculated C E B E s do not agree so well, particularly for the CI cases. It is expected that the performance of the uGTS approach to the C E B E s of third-period elements will improve once the deficiencies in the basis set have been corrected and relativistic effects can be calculated or be available to be included as corrections to the calculated CEBEs.  Chapter 5  The A E - K S Approach: Test of Functionals  The unrestricted generalized transition state model, combined w i t h density functional theory, has been shown to be an excellent approach to the calculation of core-electron binding energies of second-period elements (Chapter 4). Nevertheless, as discussed i n Chapter 3, the u G T S model is an approximation to the exact core-ionization energies. T h i s and the next chapter w i l l explore the application of D F T to the determination of molecular C E B E s using the A E - K S method, i n which no model error is introduced because calculations are performed for a fully ionized final state rather than for a transition state (as i n the u G T S ) . A set of seventeen cases, representing a reliable database of observed C E B E s , was selected to perform a l l the calculations reported i n this chapter. The purpose of using this small database was to reduce experimental error. E a c h of the observed C E B E s has been measured  69  (or recalibrated) at least  The AE-KS  Approach:  Test of  Functionals  70  four times, and the value that is used for comparison w i t h calculations was obtained by taking a weighted average of the corresponding experimental results, based on the reported (or estimated) uncertainties. Complete details about the C E B E database are given i n the A p p e n d i x . A l l calculations were carried out at the experimental geometries [110]. Ten functional combinations — a l l available i n d e M o n - K S [119] —  were  studied using the A E - K S procedure. T h e functional compositions are given in Table 5-1.  Table 5-1. Composition of exchange-correlation functionals.  functional  exchange  correlation  P86/P91  Perdew-Wang (1986)  Perdew-Wang (1991)  P86/P86  Perdew-Wang (1986)  Perdew (1986)  B88/P86  Becke (1988)  Perdew (1986)  P91/P86  Perdew-Wang (1991)  Perdew (1986)  B88/P91  Becke (1988)  Perdew-Wang (1991)  P91/P91  Perdew-Wang (1991)  Perdew-Wang (1991)  B88/LAP  Becke (1988)  Laplacian  P91/LAP  Perdew-Wang (1991)  Laplacian  P86/LAP  Perdew-Wang (1986)  Laplacian  LSD  LSD  Vosko-Wilk-Nusair  The AE-KS  Approach:  Test of  Functionals  71  T h e functionals developed by Becke (B88 [88]), Perdew (correlation P86 [87]), and Perdew and W a n g (exchange P86 [86], P91 [78]) are of the G G A type (discussed i n Chapter 2, Section 2.2.2). T h e L a p l a c i a n functional [120] is a non-local generalization of a gradient-free correlation functional [121] designed to involve the kinetic energy density and hence the L a p l a c i a n of the electron density. T h e L S D functional tested employs the V o s k o - W i l k - N u s a i r parametrization [80] for the correlation energy.  5.1  M o d e l E r r o r and Functional E r r o r  In the u G T S / D F T method, the deviation of the calculated C E B E s from the observed values can be represented as follows  deviation  = EE + ME + RCE + BSE  + FE  (5.1)  where the terms on the right-hand side are the experimental error ( E E ) , the model error ( M E ) , the relativistic correction error ( R C E ) , the basis set error ( B S E ) , and the functional error ( F E ) , respectively. M E is obtained as the difference between the u G T S result and the A E - K S result. R C E is assumed to be negligible, and so are E E and B S E . T h e justification for neglecting the experimental and basis set errors is that the calculations were limited to the aforementioned database of reliable observed C E B E s , and were carried out w i t h the c c - p V 5 Z basis sets (which have been shown to perform almost as efficiently as a complete basis set [20, 24]). F E is calculated from equation (5.1) after the deviation and the model error have been determined from the  The AE-KS  Approach:  Test of  Functionals  72  results of the D F T calculations.  T a b l e 5-2. Error analysis (in eV) for uGTS calculations with the B88/P86 functional.  molecule  deviation  model error  functional error  +0.17  0.49  -0.32  -0.35  0.53  -0.88  +0.12  0.65  -0.53  -0.59  0.62  -1.21  CCI4  +0.01  0.74  -0.73  C  +0.12  0.57  -0.45  +0.09  0.49  -0.40  NNO  -0.08  0.58  -0.66  NNO  -0.04  0.53  -0.57  0.00  0.67  -0.67  +0.17  0.66  -0.49  +0.10  0.66  -0.56  Ho  +0.14  0.67  -0.53  HCOOH  -0.08  0.70  -0.78  HCOOH  +0.19  0.70  -0.51  CH3OH  +0.08  0.80  -0.72  CF  -0.42  0.77  -1.19  CO C0 CH CF  N  2  4  4  2  CH CN 3  CO  co  2  2  4  The AE-KS  Approach:  Test of Functionals  73  Table 5-3. Error analysis (in eV) for u G T S calculations with the P86/P86 functional.  deviation  model error  functional error  0.85  0.57  +0.28  co  0.35  0.51  -0.16  CH  0.87  0.66  +0.21  0.11  0.58  -0.47  0.69  0.75  -0.06  0.85  0.60  +0.25  0.93  0.65  +0.28  NNO  0.79  0.63  +0.16  NNO  0.79  0.64  +0.15  CH CN  0.88  0.67  +0.21  CO  1.10  0.66  +0.44  co  0.87  0.51  +0.36  H 0  1.10  0.76  +0.34  HCOOH  0.88  0.71  +0.17  HCOOH  1.12  0.71  +0.41  CH3OH  1.01  0.77  +0.24  CF  0.63  0.74  -0.11  molecule CO 2  CF  4  4  CC1  4  C N  2  3  2  2  4  In the original study conducted by C h o n g [18], three of the functionals given i n Table 5-1 were tested: B 8 8 / P 8 6 , P 8 6 / P 8 6 , and L S D . A n error anal-  The AE-KS  Approach:  Test of  Functionals  74  ysis based on equation (5.1) was carried out for each of them and the results are shown i n Tables 5-2, 5-3, and 5-4.  T a b l e 5-4. Error analysis (in eV) for u G T S calculations with the L S D functional.  deviation  model error  functional error  -3.05  0.44  -3.49  -3.65  0.48  -4.13  -3.45  0.53  -3.98  -3.93  0.46  -4.39  -3.31  0.43  -3.74  C H6 2  -3.40  0.51  -3.91  N  2  -3.87  0.49  -4.36  NNO  -4.03  0.49  -4.52  NNO  -4.02  0.49  -4.51  CH CN  -4.02  0.54  -4.56  CO  -4.44  0.55  -4.99  -4.55  0.55  -5.10  H 0  -4.72  0.61  -5.33  HCOOH  -4.73  0.55  -5.28  HCOOH  -4.52  0.57  -5.09  CH3OH  -4.67  0.62  -5.29  CF  -5.68  0.60  -6.28  molecule CO  co  2  CH  4  CF  4  CC1  4  3  co  2  2  4  The AE-KS  Approach:  Test of  Functionals  75  T h e reason the B 8 8 / P 8 6 functional performs so well i n u G T S calculations of C E B E s is clear from examination of Table 5-2. T h e model error is always a positive quantity whereas the functional error is always negative.  This  results i n a fortuitous partial cancellation of errors which helps to produce calculated energies i n impressive agreement w i t h observed values. For the 17 cases i n the database, the A A D from experiment is only 0.17 e V . T h e functional error of the P 8 6 / P 8 6 combination is smaller than that of the B 8 8 / P 8 6 functional but it is a positive quantity for most of the cases investigated (Table 5-3). Therefore, the functional error adds to the model error and causes the performance of P 8 6 / P 8 6 to be considerably inferior — the A A D from experiment is 0.82 e V — to that of B 8 8 / P 8 6 . T h e L S D had been found to be incapable of yielding C E B E s w i t h acceptable accuracy [18]. Table 5-4 shows that this is due to a large error associated w i t h the functional itself, which leads to a notable underestimation of the core-ionization energies.  5.2  Functional Performance in A E - K S Calculations  T h e results of the study of the ten functional combinations using the A E - K S procedure are presented in this Section. Table 5-5 shows their performance analyzed on the basis of the deviations of the calculated C E B E s from the corresponding experimental values. T h e combination of the Perdew-Wang functionals ( P 8 6 / P 9 1 ) yielded the best results w i t h an A A D from experiment of 0.15 e V , and surpassed the performance of B 8 8 / P 8 6 i n the u G T S approach whose  A A D was 0.17 e V  The AE-KS  Approach:  Test of  Functionals  76  T a b l e 5-5. Average Absolute Deviation (in eV) and Maximum Deviation (in eV) of the core-electron binding energies (calculated with the functionals in Table 5-1) from experiment. A l l results include data from the 17 cases in the database, except for the P91/P86 and P 9 1 / L A P results which include 15 cases (calculations for C C U and  C2H6  failed to converge).  functional  (Section 5.1).  AAD  MD  P86/P91  0.15  -0.66  P86/P86  0.26  -0.47  B88/P86  0.65  -1.21  P91/P86  0.66  -1.11  B88/P91  0.81  -1.44  P91/P91  0.87  -1.40  B88/LAP  0.88  +1.14  P91/LAP  0.91  +1.16  P86/LAP  1.67  +2.05  LSD  4.64  -6.27  T h i s is a particularly important result because the model  error had already been eliminated (by employing the A E - K S method) and a functional has been found that leads to a sufficiently small error to provide highly accurate C E B E s . The core-ionization energies for each of the database cases (obtained w i t h the P 8 6 / P 9 1 functional) are shown i n Table 5-6. Except for the carbon cases in C 0  2  and C F , and the fluorine case, all the C E B E s 4  The AE-KS  Approach:  Test of  Functionals  77  deviate from the observed values by 0.20 e V at most. In fact, i f these three "problem" cases were ignored, the A A D would drop to 0.08 e V .  T a b l e 5-6. Core-electron binding energies (in eV) calculated with the P86/P91 functional. Calculated CEBEs include relativistic corrections from equation (4.1).  molecule  CEBE  experiment  CO  296.25  296.21  297.28  297.69  290.87  290.84  301.23  301.89  296.35  296.36  C H6 2  290.75  290.72  N  2  410.01  409.98  NNO  412.52  412.59  NNO  408.59  408.71  CH CN  405.53  405.64  CO  542.73  542.55  541.36  541.28  H 0  539.98  539.90  HCOOH  538.86  538.97  HCOOH  540.83  540.63  CH OH 3  539.09  539.11  CF  4  695.14  695.56  co  2  CH  4  CF  4  CC1  4  3  co  2  2  The AE-KS  Approach:  Test of  Functionals  78  T h e performance of the Perdew-Wang-Perdew ( P 8 6 / P 8 6 ) combination is also good and represents a remarkable improvement upon the u G T S results obtained w i t h this functional ( A A D of 0.26 e V for A E - K S compared w i t h A A D of 0.82 e V for u G T S ) . O n the other hand, the performance of the B 8 8 / P 8 6 degraded considerably (from an A A D of 0.17 e V for u G T S to an A A D of 0.65 e V for A E - K S ) . B o t h results are consistent w i t h the observations made i n the previous section.  Chapter 6  The A E - K S Approach: Test of Basis Sets  The results presented i n Chapter 5 have shown that the P 8 6 / P 9 1 functional combination is the best option for D F T calculations of C E B E s w i t h i n the A E - K S approach. T h e tests were carried out w i t h a highly efficient though large (computationally demanding) basis set. It was pointed out i n Chapter 3 that the use of smaller basis sets is essential to extend calculations to increasingly large systems as the ultimate goal is to be able to treat systems which can serve as realistic models for extended structures such as polymers and surfaces, on which most current experimental investigations are being focused. A number of possible alternatives to the cc-pV5Z set (which was the only basis used i n the i n i t i a l tests) were considered i n Section 3.2.1, and their performances w i l l be presented and discussed in this chapter. A l l calculations reported i n this chapter were performed w i t h the P 8 6 / P 9 1  79  The AE-KS  Approach:  Test of Basis Sets  80  functional at the experimental geometries of the neutral parent molecules [110]. T h e database introduced in Chapter 5 was employed as well as some additional fluorine cases (there is only one case in the database) and some boron cases (not represented i n the database). O n l y for H F was it possible to obtain a weighted average (details are given i n the A p p e n d i x ) .  6.1  Scaled Basis Sets  A convergence study was carried out for the c c - p V n Z basis sets and the results are shown i n Table 6-1. The estimated complete basis set limits were calculated using equation (4.2). It is observed that the c c - p V 5 Z set is indeed highly efficient as evidenced by the fact that except for the O case in C O (with a deviation of only 0.01 eV) all the C E B E s are equal to the corresponding C B S limits. Table 6-2 summarizes the results of calculations carried out w i t h the scaled basis sets constructed by means of the three different scaling procedures described by equations (3.25) through (3.29), and compares their performances w i t h the C B S limits (from Table 6-1) and w i t h the experimental energies. A more general analysis, based on deviations from b o t h experiment and C B S , of a l l six basis sets studied is given in Table 6-3. T h e average absolute deviations indicate that exponent scaling is not an effective means of describing the core-hole state i n the A E - K S method, i n contrast to the results obtained i n u G T S calculations (Section 4.1). In fact, only one of the scaled basis sets ( I I I - p V T Z ) has consistently improved upon the A A D s  of the original c c - p V T Z  sets, but the extent of improvement is  The AE-KS  Approach:  Test of Basis Sets  81  T a b l e 6 - 1 . Basis set convergence in AE-KS/P86-P91 calculations of core-electron binding energies (in eV). Calculated CEBEs include relativistic corrections from equation (4.1).  molecule  cc-pVTZ  cc-pVQZ  cc-pV5Z  CBS  experiment  296.55  296.28  296.25  296.25  296.21  297.54  297.31  297.28  297.28  297.69  291.19  290.90  290.87  290.87  290.84  301.46  301.24  301.23  301.23  301.89  296.77  296.39  296.35  296.35  296.36  291.08  291.78  290.75  290.75  290.72  410.34  410.04  410.01  410.01  409.98  NNO  412.83  412.54  412.52  412.52  412.59  NNO  408.90  408.61  408.59  408.59  408.71  CH CN  405.88  405.56  405.53  405.53  405.64  CO  543.17  542.78  542.73  542.72  542.55  541.77  541.40  541.36  541.36  541.28  H 0  540.29  539.98  539.98  539.98  539.90  HCOOH  539.24  538.88  538.86  538.86  538.97  HCOOH  541.23  540.86  540.83  540.83  540.63  CH3OH  539.47  539.11  539.09  539.09  539.11  CF  695.60  695.18  695.14  695.14  695.56  HF  694.62  694.27  694.26  694.26  694.23  CIF  694.80  694.36  694.32  694.32  694.36  695.10  694.65  694.60  694.59  694.80  696.98  696.56  696.52  696.52  696.69  202.36  202.13  202.10  202.10  202.80  B 2H6  196.67  196.42  196.40  196.40  196.50  BH3CO  195.40  195.16  195.14  195.14  195.10  BH3NH3  194.25  193.97  193.94  193.94  193.73  CO C0  2  CH CF  4  4  CC1  4  C N  2  3  co  2  2  BF F  4  3  2  BF  3  The AE-KS  Approach:  Test of Basis Sets  82  T a b l e 6-2. A E - K S / P 8 6 - P 9 1 calculations of core-electron binding energies (in eV) with scaled basis sets. Calculated C E B E s include relativistic corrections from equation (4.1).  molecule  I-pVTZ  II-pVTZ  III-pVTZ  CBS  experiment  CO  296.68  296.67  296.65  296.25  296.21  C0  297.55  297.55  297.54  297.28  297.69  291.03  291.02  291.01  290.87  290.84  301.35  301.35  301.35  301.23  301.89  296.45  296.44  296.43  296.35  296.36  290.87  290.86  290.86  290.75  290.72  410.42  410.40  410.33  410.01  409.98  NNO  412.73  412.72  412.71  412.52  412.59  NNO  408.88  408.84  408.76  408.59  408.71  CH CN  405.79  405.76  405.71  405.53  405.64  CO  543.20  543.17  543.12  542.72  542.55  2  541.79  541.77  541.70  541.36  541.28  H0  540.39  540.35  540.28  539.98  539.90  HCOOH  539.27  539.23  539.16  538.86  538.97  HCOOH  541.22  541.19  541.12  540.83  540.63  CH3OH  539.50  539.47  539.39  539.09  539.11  CF  695.68  695.64  695.54  695.14  695.56  HF  694.78  694.73  694.63  694.26  694.23  C1F  694.96  694.92  694.82  694.32  694.36  695.15  695.11  695.01  694.59  694.80  697.19  697.14  697.04  696.52  696.69  202.44  202.44  202.42  202.10  202.80  B H6  196.67  196.67  196.67  196.40  BH3CO  195.39  195.39  195.39  195.14  195.10  BH3NH3  194.25  194.21  194.21  193.94  193.73  2  CH CF  4  4  CC1  4  C N  2  3  co 2  BF F  4  3  2  BF  3  2  •  196.50  The AE-KS  Approach:  Test of Basis Sets  83  T a b l e 6-3. Average Absolute Deviation (in eV) of calculated core-electron binding energies from experiment and from C B S . The number of test cases is given in parentheses.  basis set  A A D from experiment  A A D from C B S  database cases (17) cc-pVTZ  0.30  0.33  cc-pVQZ  0.16  0.03  cc-pV5Z  0.15  0.00  I-pVTZ  0.33  0.32  II-pVTZ  0.30  0.30  III-pVTZ  0.26  0.25  CBS  0.15 all test cases (25)  cc-pVTZ  0.35  0.34  cc-pVQZ  0.16  0.03  cc-pV5Z  0.16  0.00  I-pVTZ  0.35  0.36  II-pVTZ  0.33  0.34  III-pVTZ  0.29  0.29  CBS  0.16  still not significant (especially if compared w i t h u G T S results). E x a m i n a t i o n of the i n d i v i d u a l molecules reveals that, in general, B , O , and F C E B E s are difficult cases for the scaled basis sets. The I - p V T Z and I I - p V T Z sets do  The AE-KS  Approach:  Test of Basis Sets  84  not perform satisfactorily at a l l , and the performance of the I I I - p V T Z set is acceptable only for some of the cases.  Results are reasonably good for  "sp " C cases and for most N cases (the N C E B E is the only definitely poor 3  2  result among the four cases studied), but C atoms involved i n multiple bonds appear to be problematic (the C O molecule i n particular). T h e difference between the I/II sets and the III sets lies i n that I and II separate the effects of l s - and 2s-electrons on the screening factor for the 2p functions whereas no such partition is included i n III. T h e fact that the performance of the I I I - p V T Z basis set is better than that of the I / I I - p V T Z sets suggests that, as far as shielding effects on 2p-electrons are concerned, treating l s - and 2s-electrons as a whole may be a more efficient way of describing the core-hole state i n the A E - K S method.  6.2  Core-Valence Correlated Basis Sets  It was mentioned i n Chapter 3 that the core-valence correlated basis functions labeled as c c - p C V T Z [109] were another possible alternative to the use of the large c c - p V 5 Z set. Therefore, they were also tested i n A E - K S calculations of core-electron binding energies. A summary of the results obtained is given i n Table 6-4 and a comparison w i t h c c - p V T Z and cc-pV5Z results is presented in Table 6-5. A significant improvement upon the c c - p V T Z results was achieved when the calculations were carried out using c c - p C V T Z basis functions (the A A D decreased more than half for a l l test cases i n Table 6-4). It should be noted that although the c c - p C V T Z basis set is an augmented version of the original  The AE-KS  Approach:  Test of Basis Sets  85  T a b l e 6-4. A E - K S / P 8 6 - P 9 1 calculations of core-electron binding energies (in eV) with c c - p C V T Z basis sets. Calculated C E B E s include relativistic corrections from equation (4.1).  molecule  cc-pCVTZ  experiment  CO  296.27  296.21  co  297.28  297.69  290.93  290.84  301.17  301.89  296.45  296.36  290.80  290.72  410.05  409.98  NNO  412.55  412.59  NNO  408.62  408.71  CH CN  405.60  405.64  CO  542.84  542.55  co  541.44  541.28  H0  539.95  539.90  HCOOH  538.90  538.97  HCOOH  540.88  540.63  CH3OH  539.12  539.11  CF  695.23  695.56  HF  694.26  694.23  C1F  694.43  694.36  694.74  694.80  696.61  696.69  202.12  202.80  B H6  196.43  196.50  BH3CO  195.17  195.10  BH3NH3  193.98  193.73  2  CH CF  4  4  cci  4  C N  2  3  2  2  BF F  4  3  2  BF  3  2  The AE-KS  Approach:  Test of Basis Sets  86  T a b l e 6-5. Average Absolute Deviation (in eV) of calculated core-electron binding energies from experiment. The number of test cases is given in parentheses.  basis set  A A D from experiment database cases (17)  cc-pVTZ  0.30  cc-pCVTZ  0.17  cc-pV5Z  0.15 all test cases (25)  cc-pVTZ  0.35  cc-pCVTZ  0.17  cc-pV5Z  0.16  c c - p V T Z set (Section 3.2.1) the results are still remarkable i n that they almost reproduce the A A D s of the cc-pV5Z set which is considerably more demanding i n computational terms. Moreover, for the database cases, the A A D of the c c - p C V T Z results is equal to the A A D obtained i n the u G T S / B 8 8 - P 8 6 calculations w i t h the cc-pV5Z basis sets (Chapter 5). T h e results of additional calculations performed w i t h c c - p V T Z and ccp C V T Z basis set are reported in Table 6-6. Seven molecules — a l l of which are larger than the species that comprise the test cases of the previous section — were studied, and the results obtained were highly accurate w i t h an A A D from experiment of 0.07 e V and of 0.19 e V for c c - p C V T Z and c c - p V T Z , respectively.  The AE-KS  Approach:  Test of Basis Sets  87  If a l l 32 cases studied w i t h the c c - p C V T Z basis set are considered, then the A A D from experiment is 0.15 e V , exactly the same as the A A D obtained for the 17 database cases w i t h the cc-pV5Z basis set.  T a b l e 6-6. Calculations of core-electron binding energies (in eV) of larger molecules. Calculated CEBEs include relativistic corrections from equation (4.1). The experimental data for the aromatic compounds are a weighted average of the observed CEBEs (details are given in the Appendix).  molecules  cc-pVTZ  cc-pCVTZ  experiment  B 5H9 — apex  194.27  194.11  194.20  B 5H9 — base  196.30  196.11  196.10  290.58  290.33  290.39  291.51  291.25  291.37  292.91  292.66  292.75  405.72  405.42  405.40  693.15  692.79  692.92  C H NH 6  5  2  C 6H5F  C H NH 6  5  2  C6H5 F  Further insight into the performance of basis sets can be gained by means of completeness profiles [122, 123]. T h e completeness profile of a basis set is defined as the s u m of the squares of the overlap of a test normalized Gaussian function w i t h an orthonormalized basis [122]. If the test Gaussian is represented as G(a)  — where a is a variable exponent — and {ipk} is a  Figure 6-1. Completeness profiles of cc-pVnZ and c c - p C V T Z basis sets; s-functions: solid, p-functions: dash, d-functions: dot-dash.  The AE-KS  Approach:  Test of Basis Sets  89  log (a)  F i g u r e 6-2. Comparison of completeness profiles for the s-, p-, and d-type functions of cc-pVTZ (dot-dash), cc-pV5Z (solid), and c c - p C V T Z (dash) basis sets.  generic set of orthonormalized basis functions, then the completeness profile is a plot of Y(o) as a function of x = log(a),  w i t h Y(a) given by  The AE-KS  Approach:  } »  Test of Basis Sets  = £  <G(a)|^><^  90  f c  |G(a)>  (6.1)  Completeness profiles for the cc-pVroZ and c c - p C V T Z basis sets are shown in Figure 6-1. A comparison of the profiles for the s-, p-, and d-type functions is presented i n Figure 6-2. T h e profiles have been calculated only for the carbon atom, but they are expected to be qualitatively similar for the other second-period elements [123]. T h e closer the value of Y(a) spanned by the basis.  is to 1.0, the more completely is the space  T h e tight region is represented by high x values  whereas low x values are associated w i t h the diffuse (bonding) region. It is observed that at high x = log (a) the c c - p C V T Z set shows appreciably higher completeness than does the c c - p V T Z set — this tends to lead to better energies [123] — and also provides more adequate coverage than the c c - p V 5 Z set.  Thus, the behaviour of the basis sets, as displayed by the complete-  ness profiles, is i n accord w i t h the A A D - b a s e d results and supports the fact that the c c - p C V T Z basis set is an appropriate choice for density functional calculations of C E B E s . A l i m i t e d number of tests were carried out w i t h a core-valence correlated basis set of double-zeta quality. T h i s c c - p C V D Z set [109] is similar i n composition to the c c - p V T Z set — [4s3pld] compared to [4s3p2d] — so it was thought to be perhaps capable of providing good C E B E s at reduced computational effort. However, the results for seven of the database cases were rather unsatisfactory, the deviations from experiment ranging from 0.68 e V to 1.65 e V .  Chapter 7  Conclusion  T h i s thesis has extended the computational approach to the determination of molecular core-electron binding energies introduced by Chong, the unrestricted generalized transition state model combined w i t h density functional theory, by applying it to the calculation of C E B E s of boron-containing molecules, of isomers of C 3 H 5 N O , and of the third-period elements silicon, phosphorus, sulphur, chlorine, and argon. T h e results obtained for boron were i n very good agreement w i t h experimental observations, for calculations performed both w i t h large basis sets (cc-pV5Z) and w i t h smaller but efficient basis sets (scaled-pVTZ). T h e s c a l e d - p V T Z calculations for boron l s energies gave an average absolute deviation of 0.07 e V from the estimated complete basis set l i m i t , confirming that exponent-scaling is a highly effective method for improving basis-set performance i n the treatment of partial core-hole states.  91  Conclusion  92  T h e extension of the original scaling methodology, i n which the use of scaled basis functions is restricted to the atom w i t h the core hole, to a generalized-scaling method, requiring scaled basis sets for every atom, was found not to be advantageous. T h e effort involved i n calculating scaling factors for a l l the atomic centers i n a molecule is too intensive to justify the l i m i t e d improvement attained. T h e calculated core-ionization energies of the isomers of C 3 H 5 N O clearly revealed the distinctive nature of the core-electron spectrum of the i n d i v i d ual species.  T h i s and previous results, i n conjunction w i t h the fact that  synchrotron-radiation instrumentation has already achieved sufficiently high resolution to distinguish atoms i n remarkably similar environments, continue to support the traditional application of combined experimental and theoretical approaches to core-electron spectroscopy for the purpose of chemical and structural analysis. T h e results for C E B E s of the third-period elements were i n most cases of acceptable accuracy for calculations performed w i t h the c c - p V 5 Z basis sets, but the deviations from observed values were found to be, i n general, more than twice as large as those obtained for the second-period elements. A number of factors were recognized as (partially) responsible for the deficiencies detected, notably irregularities associated w i t h the basis sets employed, and also absence of a (at least approximate) relativistic treatment. T h i s thesis has also tested a total-energy difference approach, w i t h i n K o h n - S h a m density functional theory, to calculating core-ionization energies. It was found that the remarkable success of u G T S / D F T calculations employing Becke's 1988 exchange functional and Perdew's 1986 correlation  Conclusion  93  functional was due to a fortuitous cancellation of the errors associated w i t h the model ( u G T S ) and the functional. For the A E - K S method, the combination of Perdew and Wang's 1986 exchange and 1991 correlation functionals proved the most accurate among the ten functional options tested. Exponent scaling was found not to be an adequate procedure for i m proving the performance of basis sets i n A E - K S calculations. 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C h e m . Phys.  107  64, 3690  (1976) 158. K . D . B o m b e n , W . L . Jolly, unpublished data quoted i n reference 112  Appendix  It was pointed out in Chapter 5 that calculations using the A E - K S approach were i n i t i a l l y limited to a database of reliable observed C E B E s which was selected i n order to reduce as much as possible the effect of experimental errors on the assessment of functional performance. T h e database consisted of seventeen cases, for each of which at least four experimental C E B E s have been documented. T h e experimental uncertainty 6 was sometimes reported along w i t h each observed value. Where 8 was not given, a reasonable estimate was assumed. Table A - l summarizes the data for the seventeen cases which were studied in Chapter 5. A d d i t i o n a l cases were considered for the basis-set tests reported in Chapter 6, and where experimental results were available, weighted averages were obtained, the corresponding data being given i n Tables A - 2 and  A-3.  108  Appendix  109  T a b l e A - l . Observed Is energies (in eV) for the molecules in the seventeen-case database.  molecule  year  CEBE  6  CO  1969  295.9  0.2 °  133  1970  296.2  0.1  134  1973  296.1  1974  296.2  0.1 °  127  1976  296.24  0.03  136  1984  296.19  0.05  0.1  b  average  a  2  137  a  4  296.21  1969  297.5  0.2 °  133  1972  297.5  0.1  133, 141  1973  297.69  0.14  135  1974  297.69  0.14 °  142  1974  297.75  0.07  127, 143  1974  297.71  0.05  127  1969  290.7  0.2  1970  290.8  0.1  1974  290.73  0.2  1974  290.91  0.05  1976  290.83  1984  290.90  297.64 CH  reference  133, 135  a  296.14 C0  w e i g h t e d average  297.69 133  a  124 131  a  127  0.02 c  0.05  132 112, 131  a  290.81  290.84  Appendix  110  Table A-l continued  molecule  year  CEBE  S  CF  1969  301.8  1970  4  average  4  133  301.8  0.1  a  138  1974  301.68  0.2  Q  131  1974  301.9  0.2  1974  301.96  0.05  1984  301.85  1970  296.3  0.1  a  1974  296.22  0.2  a  1974  296.38  0.05  1976  296.3  0.1  1984  296.39  1970  290.6  0.1  a  1974  290.57  0.3  a  1974  290.76  0.05  126,127  1976  290.71  0.02  132  1984  290.74  0.05  112, 131  1969  c  127, 144 127  a  0.05  112, 131  c  2 I 1 6  c  301.89 138 131 139  a  140  a  0.05  112, 131  a  296.32 C  reference  0.2  301.83 CC1  w e i g h t e d average  a  296.36 124 131  290.68  290.72  409.9  0,2  a  133  1973  409.93  0.10  135  1974  409.95  0.20  127,  1974  409.93  0.20  142  1974  409.93  0.05  127  1980  410.0  0.03  150 409.94  409.98  Appendix  111  Table A-l continued  molecule  year  CEBE  5  average  N N O  1969  412.5  0.2  1974  412.62  0.21  1974  412.5  0.1  1974  412.62  0.05  1969  408.5  0.2  1974  408.75  0.22  127, 143  1974  408.6  0.1  127, 149  1974  408.75  0.05 °  127, 143 127, 149  a  126, 127  a  3  412.59 133  a  a  126, 127 408.65  CH CN  reference 133  412.56 NNO  w e i g h t e d average  a  408.71  1972  405.9  0.3  145  1976  405.6  0.2  1982  405.74  0.03  147  1984  405.60  0.02  148  1969  542.1  1970  542.3  1973  542.6  1974  542.82  0.12  a  127, 143  1976  542.40  0.11  a  155  1976  542.57  1984  542.51  146  a  405.71  6  405.64  0.5  a  133  0.1  a  134  0.1  a  133, 135  0.03 c  0.05  136 112, 155  Q  542.47  542.55  Appendix  112  Table A-l continued  molecule  year  CEBE  S  C 0  1969  540.8  0.5 °  133  1972  541.1  0.1  141  1973  541.28  0.12  135  1974  541.28  0.12  142  1974  541.32  0.05  127  1974  541.32  0.09  127, 143  2  average  541.18 H O 2  0.2 °  133  1974  539.88  0.07  127  1974  539.93  0.05  1976  539.67  0.2  142  a  133, 152  a  3  539.90  1974  538.93  0.09  127, 143  1975  538.92  0.05  1976  538.75  0.2 °  133, 152  1978  539.00  0.03  153  153  a  538.97  1974  540.55  0.09  127,143  1975  540.60  0.1  1976  540.45  0.2 °  133,152  1978  540.65  0.03  153  151  a  540.56 CH OH  541.28  539.7  538.90  HCOOH  reference  1969  539.80 HC O OH  w e i g h t e d average  540.63  1969  538.9  0.2 °  133  1974  539.08  0.12  127, 143  1976  539.09  0.08  127, 152  1984  539.2  0.1  154  a  539.07  539.11  Appendix  113  Table A-l continued  molecule  year  CEBE  S  CF  1969  695.2  0.2  a  133  1970  695.0  0.1  a  138  1973  695.52  0.14  135  1974  695.60  0.2 °  131  1974  695.52  0.14  142  1974  695.52  0.05  1974  695.57  1984  695.77  4  average  weighted average  139  a  0.05 c  0.05  reference  127 112,131  a  695.46  695.56  a  assumed  b  recalibrated, based on the improved measurement on C O 2 [135]  c  correction applied according to reference 112  Appendix  114  T a b l e A - 2 . O b s e r v e d I s e n e r g i e s ( i n e V ) for t h e a r o m a t i c c o m p o u n d s i n T a b l e 6-6.  molecule  year  CEBE  S  average  C  1970  290.4  0.1  1971  290.2  0.1  1974  290.42  0.05  1975  290.3  0.2  128  1975  290.38  0.07  129  1975  290.42  0.05°  130  6  5  2  1975  291.38  0.05  1975  291.2  0.2  125 126,  a  6H5F  1975  292.85  1975  130 128  6  5  2  0.05  130, 144  292.70  0.05  a  130  1975  292.9  0.2  128  1978  292.5  0.1  156  1969  405.5  0.1  1975  405.32  0.05  1975  405.3  0.2  (?)  405.31  0.05  1980  405.45  0.03  133 130  a  128 112  a  150  assumed  405.40  1974  692.88  0.05  a  139, 144  1975  692.93  0.05  a  130  1975  693.3  0.2  128 693.04  a  292.75  a  405.38 C6H5 F  291.37  a  292.74 C H N H  127  290.39  Q  291.38 C  reference 124  a  290.35 C H NH  w e i g h t e d average  692.92  Appendix  115  T a b l e A - 3 . Observed Is energies (in eV) for hydrogen fluoride.  molecule  year  CEBE  S  average  HF  1974  694.22  0.05 °  127  1976  694.0  0.2  157  1984  694.31  0.1 °  112, 158 694.18  a  assumed  A.  w e i g h t e d average  694.23  reference  

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