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Mathematical aspects and chemical applications of density functional theory Zhang, Yu 2009

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Mathematical Aspects and Chemical Applications of Density Functional Theory by Yu Zhang B. Sc., Xiamen University, 1999 M. Sc., Peking University, 2002 Ph. D., University of British Columbia, 2009 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Chemistry)  The University of British Columbia (Vancouver) February, 2009 © Yu Zhang 2009  Abstract This is a dissertation about the theoretical basis and applications of Density Functional Theory (DFT). With the rapid increase in computing power, modern chemists have routinely used all kinds of theoretical methods to explore what interests them in chemical studies. DFT demonstrates a good balance between the computing cost and accuracy. So it has become one of the most popular daily-used quantum chemistry methods. In Chap. 1 an introduction of the theoretical basis as well as the important concepts of DFT is given. Historical developments of the theory and recent studies on hot topics are reviewed. Remarks on disputed topics are also provided. In Chap. 2 we talk about the asymptotic behavior of finite-system wavefunctions. The exponential decaying asymptotic behavior is confirmed and the structure of the prefactors is further explored. By comparing the asymptotic behavior of Dyson orbitals and Kohn-Sham orbitals, we also provide a physical interpretation of Kohn-Sham orbital energies. Chap. 3 is about rebutting the theory of “unconventional density variation” proposed more than 20 year ago. Supported by theoretical analyses as well as numerical evidence, we prove that all density variations are the same in nature. In Chap. 4 we extend two total energy functionals suggested before to the Hartree-Fock method. Numerical tests on different molecules show these functionals are very promising in accelerating the SCF convergence of quantum chemistry calculations. In Chap. 5 we completed a comprehensive theoretical study on the tautomers of pyridinethiones. All sorts of molecular properties predicted from theory are compared with those got from experiments. The dominant forms of the tautomers are confirmed to be the thione forms. This demonstrates the power of DFT methods, and this work can serve as a reference for studying similar molecules. In the last chapter conclusions from previous chapters are drawn and possible future developments are discussed. In Appendix A we present another method to solve the coupled system ii  Abstract of differential equations appeared in Chap. 2. In Appendix B the coordinate scaling transform of the eigenequation in Chap. 2 is explained. In Appendix C the derivation of the method of dominant balance solution in Chap. 2 is shown. Appendix D contains all the supporting information of Chap. 5. The last appendix is a thorough review on the mathematical concepts and methods relating to all chapters of this thesis. Rigorous definitions and illustrative examples can be found there.  iii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv  Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . xv 1 Introduction to Density Functional Theory . . . . . . 1.1 Wave-Function Theory for Many-Electron Systems . . . 1.2 Density Is Everything: The Hohenberg-Kohn Theorems 1.3 Implementation of DFT . . . . . . . . . . . . . . . . . . 1.3.1 Thomas-Fermi Model . . . . . . . . . . . . . . . 1.3.2 Kohn-Sham Formulation . . . . . . . . . . . . . 1.4 Mathematical Foundation of DFT . . . . . . . . . . . . 1.4.1 υ-representability of an Electron Density . . . . 1.4.2 Levy Functional and Lieb Functional . . . . . . 1.4.3 Functional Derivative of Density Functional . . 1.5 Key Concepts and Relations in DFT . . . . . . . . . . . 1.5.1 XC Hole and Pair-Correlation Function . . . . . 1.5.2 Scaling Relations . . . . . . . . . . . . . . . . . 1.5.3 Adiabatic Connection Formula . . . . . . . . . . 1.5.4 Bounds on Exact Functionals . . . . . . . . . . . 1.5.5 Functional Derivative Discontinuity . . . . . . . 1.5.6 Asymptotic Behavior of the XC Potential . . . .  . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . .  1 1 2 3 3 4 6 6 7 9 11 11 13 16 17 18 20 iv  Table of Contents 1.6 1.7 1.8  Kinetic Energy Functional . . . . . . . . . . . . . . . . . . . Exchange-Correlation Energy Functional . . . . . . . . . . . Applications of DFT . . . . . . . . . . . . . . . . . . . . . .  23 27 31  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  33  2 Asymptotic Behavior of Finite-System Wave Functions 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Asymptotic Behavior of the Dyson Orbitals . . . . . . . . 2.3 Extenstion to Non-Coulombic Systems . . . . . . . . . . 2.4 Interpretation of the Kohn-Sham Orbital Energies . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . .  46 46 48 56 59 64  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  66  3 Density Variations in DFT 3.1 Background . . . . . . . 3.2 The Model System . . . . 3.3 Numerical Tests in Hilbert 3.4 Theoretical Analysis . . . 3.5 More Numerical Evidence 3.6 Density Variations in Fock 3.7 Conclusions . . . . . . . .  . . . . . . . .  71 71 72 76 79 86 89 94  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  96  . . . . . . . . . . . . Space . . . . . . . . Space . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . .  . . . . . . . .  4 Perturbative Total Energy Evaluation in SCF Iterations 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Perturbation Expansions of Total Energy Functionals . . . 4.3 Implementation and Computational Details . . . . . . . . . 4.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . .  98 98 100 106 107 114  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5 Theoretical Studies of the Tautomers of 5.1 Introduction . . . . . . . . . . . . . . . 5.2 Details of Computational Methods . . . 5.3 Results and Discussions . . . . . . . . . 5.4 Conclusions . . . . . . . . . . . . . . . .  Pyridinethiones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . .  117 117 120 121 132  v  Table of Contents Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6 Conclusions and Perspective . . . . . . . . . . . . . . . . . . . 140 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142  Appendices A Another Way to Solve the Coupled System  . . . . . . . . . 143  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 B Coordinate Scaling Transform of the Eigenequation . . . . 146 C Method of Dominant Balance Solution D Supporting Information of Chapter 5  . . . . . . . . . . . . 147 . . . . . . . . . . . . . 149  E General Mathematical Review . . . . . . . . . E.1 Basic Concepts Related to Functional . . . . . E.1.1 Map . . . . . . . . . . . . . . . . . . . . E.1.2 Field and Scalar . . . . . . . . . . . . . E.1.3 Vector Space . . . . . . . . . . . . . . . E.1.4 Definitions of Functional . . . . . . . . E.1.5 Linear and Antilinear Map . . . . . . . E.1.6 Linear Functional . . . . . . . . . . . . E.2 Functional Calculus . . . . . . . . . . . . . . . E.2.1 An Nonrigorous Definition of Functional E.2.2 Properties of Functional Derivative . . E.2.3 Euler-Lagrange Equation . . . . . . . . E.2.4 Evaluation of Functional Derivative . . E.3 Basic Concepts Related to Function Spaces . . E.3.1 Sesquilinear Form . . . . . . . . . . . . E.3.2 Inner Product Space . . . . . . . . . . E.3.3 Metric Space . . . . . . . . . . . . . . . E.3.4 Complete Metric Space . . . . . . . . . E.3.5 Normed Space . . . . . . . . . . . . . . E.4 More Concepts Related to Functional . . . . . E.4.1 Convex Set and Convex Functional . . E.4.2 Convergence and Weak Convergence . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . .  163 163 163 163 164 165 166 166 167 167 167 168 170 170 170 171 171 172 172 173 173 173 vi  Table of Contents  E.5  E.6  E.7  E.8  E.4.3 Continuity and Lower Semicontinuity E.4.4 Convex Envelope . . . . . . . . . . . E.4.5 Tangent Functional . . . . . . . . . . Important Function Spaces . . . . . . . . . . E.5.1 Hilbert Space . . . . . . . . . . . . . E.5.2 Banach Space . . . . . . . . . . . . . E.5.3 Lp space . . . . . . . . . . . . . . . . E.5.4 Sobolev Space . . . . . . . . . . . . . Functional Differentiation . . . . . . . . . . . E.6.1 Gˆ ateaux Derivative . . . . . . . . . . E.6.2 Fr´echet Derivative . . . . . . . . . . . Legendre Transform . . . . . . . . . . . . . . E.7.1 Legendre Transform on Functions . . E.7.2 Legendre Transform on Functionals . Asymptotic Analysis . . . . . . . . . . . . . . E.8.1 Notation of Asymptotic Analysis . . . E.8.2 Asymptotic Expansion . . . . . . . . E.8.3 Method of Dominant Balance . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  174 174 175 175 175 176 176 178 179 179 180 180 180 181 182 182 183 183  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186  vii  List of Tables 1.1 1.2  Functionals Summary . . . . . . . . . . . . . . . . . . . . . . Popular XC Functionals . . . . . . . . . . . . . . . . . . . . .  9 31  3.1  Values of (4IJ − IJ−1 ) . . . . . . . . . . . . . . . . . . . . . .  93  4.1  Convergence of the SCF process of DFT calculations on CrC 113  5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9  Bond lengths and bond angles . . . . . . . . . . . . . . . . Theoretical dipole moments . . . . . . . . . . . . . . . . . Total energies and Gibbs free energies . . . . . . . . . . . Theoretical IR peak positions . . . . . . . . . . . . . . . . Theoretical NMR chemical shifts of Hmppt-1 in DMSO . Theoretical NMR chemical shifts of Hmppt-2 in DMSO . Theoretical NMR chemical shifts of Hmppt-d-1 in DMSO Theoretical NMR chemical shifts of Hmppt-d-2 in DMSO Theoretical UV-Vis absorption positions . . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  122 123 127 128 130 131 132 133 133  D.1 More theoretical dipole moments . . . . . . . . . . . . . . D.2 Theoretical NMR chemical shifts of Hmppt-1 in vacuum . D.3 Theoretical NMR chemical shifts of Hmppt-2 in vacuum . D.4 Theoretical NMR chemical shifts of Hmppt-d-1 in vacuum D.5 Theoretical NMR chemical shifts of Hmppt-d-2 in vacuum D.6 Theoretical CSGT NMR chemical shifts of Hmppt-1 . . . D.7 Theoretical CSGT NMR chemical shifts of Hmppt-2 . . . D.8 Theoretical CSGT NMR chemical shifts of Hmppt-d-1 . . D.9 Theoretical CSGT NMR chemical shifts of Hmppt-d-2 . . D.10 More theoretical UV-Vis absorption positions . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  150 151 152 153 153 154 155 155 156 157  viii  List of Figures 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9  Exact reproduction of Fig. 1 of the Perdew-Levy paper The fine detail of the “unconventional density variation” Detailed energy density functional curve for Z = 0.25 . . Detailed energy density functional curve for Z = 0.5 . . Detailed energy density functional curve for Z = 0.67 . . Detailed energy density functional curve for Z = 1.0 . . Detailed energy density functional curve for Z = 1.34 . . Energy density functional curve for Z = 2 . . . . . . . . Another energy density functional curve for Z = 0.67 . .  4.1 4.2 4.3 4.4  4.5  Convergence of the KS total energies of HF and H2 O . . . . . 108 Convergence of the HF total energies of HF and H2 O . . . . . 109 Convergence of the KS total energies of HF and H2 O (damping)110 Convergence of the total energy (in Hartree) of a SiH4 molecule with an elongated Si−H bond, evaluated with the HKS (circles), the Harris (hollow triangles), the cHarris (squares), and the cHKS (hollow stars) functionals during the SCF iterations of a Kohn-Sham calculation (top) and a Hartree-Fock calculation (bottom). For the Kohn-Sham calculation, the elongated Si−H bond length is 4 ˚ A and the damping factor is 50%. For the Hartree-Fock calculation (only shown the data after 100th iteration), the elongated Si−H bond length is 12 ˚ A, and the damping factor is 40% and 10% before and after the 100th iteration, respectively. The Hartree-Fock data of first 140 iterations behave very similarly. . . . . . . . . . . . . 111 Convergence of the KS total energies of 1 CrC and 3 CrC . . . 112  5.1 5.2 5.3 5.4 5.5  Tautomeric forms of pyridinones and pyridinethiones Proposed mechanism for the dimerization . . . . . . Synthesis of the pyridinethiones . . . . . . . . . . . . ORTEP diagram of Hdppt-dimer . . . . . . . . . . . NBO charge anlysis (I) . . . . . . . . . . . . . . . . .  . . . . .  . . . . .  . . . . . . . . .  . . . . .  . . . . . . . . .  . . . . .  . . . . . . . . .  . . . . .  73 74 75 76 77 87 88 90 91  118 119 120 121 124 ix  List of Figures 5.6 5.7  NBO charge anlysis (II) . . . . . . . . . . . . . . . . . . . . . 125 NBO charge anlysis (III) . . . . . . . . . . . . . . . . . . . . . 126  D.1 Geometries of Hmppt-2 dimers from different calculations . . 149 D.2 Geometries of Hmppt-2 dimers from different calculations . . 150  x  List of Acronyms AC ADA B3LYP B88 CDV cHarris cHKS CI CSGT DFT DIIS DMSO DV EDIIS E-vrepresentable FA FDD GEA GGA GIAO GS HCTH Hdppt Hdppt-d HF HK HKS Hmppt Hmppt-d HOKSO HOMO  Asymptotic Correction Average Density Approximation Becke’s three-parameter hybrid functional Becke’s 1988 exchange functional Conventional Density Variation Corrected Harris Corrected Hohenberg-Kohn-Sham Configuration Interaction Continuous Set of Gauge Transformations Density Functional Theory Direct Inversion of the Iterative Subspace Dimethyl Sulfoxide Density Variation Energy-DIIS Ensemble-v-representable Fermi-Amaldi Functional Derivative Discontinuity Gradient Expansion Approximation General Gradient Approximation Gauge-independent Atomic Orbital Ground-state Hamprecht-Cohen-Tozer-Handy 3-Hydroxy-1,2-dimethyl-4(para)-pyridinethione Hdppt-dimer Hatree-Fock Hohenberg-Kohn Hohenberg-Kohn-Sham 3-Hydroxy-2-methyl-4(para)-pyridinethione Hmppt-dimer Highest Occupied Kohn-Sham Orbital Highest Occupied Molecular Orbital xi  List of Acronyms IR KS LC LDA LR LUMO LYP MP2 NBO NMR OF-DFT OPM PBE PCM PES PS-vrepresentable SCF SDFT SIC TDDFT TF TFD TFDvW TPSS UDV UEG UV-Vis VSXC vW VWN WDA XC  Infrared Kohn-Sham Long-range Correction Local Density Approximation Linear Response Lowest Unoccupied Molecular Orbital Lee-Yang-Parr Møller-Plesset Second Order Perturbation Natual Bond Orbital Nuclear Magnetic Resonance Orbital-free Density Functional Theory Optimized Potential Method Perdew-Burke-Ernzerhof Polarizable Continuum Model Photoelectron Spectrum Pure-state-v-representable Self-consistent Field Spin Density Functional Theory Self-interaction Correction Time-dependent Density Functional Theory Thomas-Fermi Thomas-Fermi-Dirac Thomas-Fermi-Dirac-von-Weizs¨acker Tao-Perdew-Staroverov-Scuseria Unconventional Density Variation Uniform Electron Gas Ultra Violet-Visable van-Voorhis-Scuseria Exchange-Correlation von Weizs¨acker Vosko-Wilk-Nusair Weighted Density Approximation Exchange-Correlation  xii  Acknowledgements The author would like to thank Prof. R. Kuske of the UBC mathematics department for giving directions on the mathematical references in Chap. 2; Dr. Yongbin Chang for help with Mathematica calculations in Chap. 3; Dr. F. E. Zahariev for generating Figs. 3.1––3.7 initially and his constructive discussions on the topic of Chap. 3; Prof. Weiquan Tian of Jilin University, and finally Dr. Wei (David) Deng of Gaussian Inc. for help with some of the calculations in Chap. 5. WestGrid and C-HORSE (CFI) have provided us the necessary computational resources. Financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada, Canadian Institutes of Health Research (CIHR), and Canada Foundation for Innovation (CFI) is gratefully acknowledged.  xiii  Dedication To my parents, my brother, and all my friends who always love, encourage and support me.  xiv  Statement of Co-Authorship • Chap. 2 is coauthored with Prof. Y. A. Wang. I contributed to the theoretical analysis and preparation of the manuscript. • Chap. 3 is coauthored with Prof. Y. A. Wang. I contributed to all the numerical tests and preparation of the manuscript. Dr. F. E. Zahariev generated Figs. 3.1––3.7 initially and I independently replotted the same figures later with higher accuracy and a better algorithm. • Chap. 4 is coauthored with Prof. Y. A. Wang. I contributed to the programming, all the calculations, and the preparation of the manuscript. • Chap. 5 is coauthored with Dr. V. Monga, Prof. C. Orvig, and Prof. Y. A. Wang. I contributed to all the calculations and preparation of the manuscript.  xv  Chapter 1  Introduction to Density Functional Theory 1.1  Wave-Function Theory for Many-Electron Systems  The Schr¨odinger equation of a N -electron system reads   N N 1 1  Ψk (r1 σ1 , · · · , rN σN ) − 1 Vext (ri ) + ∇2i + 2 2 |ri − rj | i=1  i=1  i  j=i  = Ek Ψk (r1 σ1 , · · · , rN σN ),  (1.1)  where Vext denotes the external potential, ri and σi denote the spacial and spin coordinates, respectively. Different approximations to the N -electron wave-function Ψk (r1 σ1 , · · · , rN σN ) lead to different levels of theory. For examples, if Ψ is simplified as a single determinant constructed from all of the N one-electron orbitals, one gets the Hartree-Fock (HF) theory; and if Ψ is written in the form of a linear combination of all possible Slater determinants generated from some one-electron basis set, one has the full configuration interaction (CI) theory. All the wave-function-based methods have a serious computing scaling problem when they are applied to large systems consist of thousands of electrons [1]. The computing efforts of these methods scale as O(N m ) (N is the number of the electrons or the number of the basis functions that used, and usually m 4). Some modifications of these methods might help to reduce the bad scaling indices but would not change the whole situation, since one has to deal with the 3N spacial coordinates (spin coordinates omitted for simplicity). When N is large, the problem becomes unconquerable.  1  Chapter 1. Introduction to Density Functional Theory  1.2  Density Is Everything: The Hohenberg-Kohn Theorems  The computing difficulty mentioned above prompts researchers to find alternative approaches to circumvent it. Stimulated by the early idea of Thomas [2] and Fermi [3–5], Hohenberg and Kohn [6] proposed that electron density alone can be chosen as the essential variable to handle in quantum mechanical calculations. That was the famous Hohenberg-Kohn (HK) theorems came into being. The First Hohenberg-Kohn Theorem. The external potential is determined within an additive constant by the electron density, or alternatively, the ground-state wave-function of a system can be determined uniquely once the electron density of the system is known. This theorem sets up an almost “one-to-one” mapping between the external potential and the electron density. That is to say, instead of dealing with the formidable 3N degrees of freedom, only 3 degrees of freedom are necessary to be studied in any physical or chemical system since the electron density normally is a 3-dimensional variable. It seems that a computing scheme using the electron density as the essential variable will greatly reduce the computational expense. So this theorem grantees that a quantum theory based on the particle density is not only possible but also promising. The Second Hohenberg-Kohn Theorem. For any electron density function ρ(r) which fulfills the conditions that ρ(r) > 0 and ρ(r)dr = N , where N is the number of the electrons in the system studied, the following variational inequality holds: Ev [ρ(r)]  Ev [ρ0 (r)] = E0 ,  (1.2)  where Ev [ρ(r)] is the total energy which has a functional dependency on the ground-state electron density ρ0 (r) and E0 is the ground-state energy. The second theorem provides a variational principle to carry out practical calculations using the electron density as the essential variable. If the space which all physical electron densities are in is known, the density variation can be done and the ground-state energy can be located as the minimum of the energy functional in principle. In this theorem a universal energy functional Ev [ρ(r)], which has a parametric dependency on the external potential v, is defined as Ev [ρ] = T [ρ] + Vne [ρ] + Vee [ρ] ,  (1.3) 2  Chapter 1. Introduction to Density Functional Theory here T [ρ], Vne [ρ] and Vee [ρ] are the kinetic energy functional, the nuclearelectron interaction energy functional and the electron-electron interaction energy functional, respectively. Obviously the nuclear-electron interaction energy functional has a simple explicit expression like Vne [ρ] =  ρ(r)vne (r)dr.  (1.4)  So only the other two terms in Eq. (1.3) need to be considered. Another universal energy functional FHK (HK functional) can be defined as FHK [ρ] = T [ρ] + Vee [ρ] ,  (1.5)  and Eq. (1.3) can be re-written as Ev [ρ] = FHK [ρ] +  ρ(r)v(r)dr.  (1.6)  Finding out the explicit expression of FHK might be infeasible because of the difficulty of representing a many-body operator with an one-body operator, but obtaining more accurate and computational less expensively approximate energy functionals is the major aim of all researchers in DFT field.  1.3 1.3.1  Implementation of DFT Thomas-Fermi Model  The theorems in last section are fundamental for DFT. But without further formulation no practical calculations can be done, since it is difficult to write down an explicit expression for the two terms in the definition of FHK . To make the problem simple, the noninteracting uniform electron gas (UEG) model was proposed by Thomas and Fermi [2, 3]. Together with some statistical mechanic considerations, a problem of particles in a 3-dimensional infinite potential well is solved, and the kinetic energy expression is TTF [ρ] = CF  5  ρ 3 dr,  (1.7)  3 where CF = 10 (3π 2 )2/3 ≈ 2.8712 in atomic units. Then Thomas and Fermi used the classic Coulomb interaction energy functional J[ρ] to approximate Vee [ρ], such as  Vee [ρ] ≈ J[ρ] =  1 2  ρ(r1 )ρ(r2 ) dr1 dr2 . |r1 − r2 |  (1.8) 3  Chapter 1. Introduction to Density Functional Theory So the Thomas-Fermi (TF) total electronic energy functional is ETF [ρ(r)] = CF  5  ρ(r) 3 dr − Z  1 ρ(r) dr + r 2  ρ(r1 )ρ(r2 ) dr1 dr2 , (1.9) |r1 − r2 |  where Z is the nuclear charge. When the TF functional is applied in the molecular electronic structure calculations, it behaves so badly that no molecule is stable in this model [7–10]. For other limitations of this model please see Chap. 6 of Ref. [11]. A natural improvement of TF model might be introducing the exchange interaction. Dirac [12] derived the exchange energy functional of the UEG model, namely 4  ρ(r) 3 dr,  KD [ρ] = Cx  (1.10)  where Cx = 43 ( π3 )1/3 ≈ 0.7386 in atomic units. Adding this term to the TF functional in Eq. (1.9) makes the Thomas-Fermi-Dirac (TFD) functional. However, this functional does not improve the TF functional but actually gives even worse results (see Chap. 6 of Ref. [11]). Another attempt to improve the TF or TFD model is introducing density gradient correction into the kinetic energy functional expression. The simplest example is the von Weizs¨acker (vW) correction [13] TvW [ρ] =  1 8  |∇ρ(r)|2 dr. ρ(r)  (1.11)  So the corrected kinetic energy functional is TTFD-λW [ρ] = TTFD [ρ] + λTvW [ρ],  (1.12)  where λ can have different values (usually in the range 1/9 ∼ 1) [13–19]. The Thomas-Fermi-Dirac-von-Weizs¨acker (TFDvW or TFD-λ W) model does improve the original TF or TFD model [20] to some degree, but is still lack of chemical accuracy [21].  1.3.2  Kohn-Sham Formulation  The failure of all TF-type functionals comes from the lack of an accurate kinetic energy functional, which accounts for the largest part of the total electronic energy. Instead of searching for a better kinetic energy functional, Kohn and Sham [22] assumed a noninteracting axillary system which has the same electron density as that of the real system. They used the occupied 4  Chapter 1. Introduction to Density Functional Theory orbitals to construct the kinetic energy functional for the noninteracting system, such as occ  Ts [ρ] = i=1  ψi | − 21 ∇2 |ψi .  (1.13)  And then they put everything unknown into a quantity called exchangecorrelation (XC) energy functional, that is Exc [ρ] = T [ρ] − Ts [ρ] + Vee [ρ] − J[ρ].  (1.14)  So the energy functional defined in Eq. (1.5) can be rewritten as F [ρ] = Ts [ρ] + J[ρ] + Exc [ρ].  (1.15)  The total energy functional will be E[ρ] = Ts [ρ] +  Vext (r)dr + J[ρ] + Exc [ρ].  (1.16)  Here Vext (r) is the external potential. According to the second HK theorem (see last section), the orbitals can be varied and the ground-state energy should be the minimum of the following orbital functional: occ  L[ρ] = E[ρ] −  i,j  ǫij ψi∗ |ψj .  (1.17)  The second term appears because of the orthonormal conditions obeyed by the orbitals ψi and ǫij serves as the Lagrangian multiplier. The necessary condition for L[ρ] to reach its minimum is δL = 0, δψi  (1.18)  so that after taking the functional derivative the equation for a single orbital ψi can be obtained as 1 − ∇2 + Vext (r) + 2  ρ(r′ ) dr′ + Vxc (r) ψi = |r − r′ |  occ  ǫij ψj ,  (1.19)  j  where  δExc (1.20) δρ is the XC potential. Similar to the canonicalization process of Hartree-Fock equation [23], a unitary transformation can be applied on both sides of Eq. Vxc (r) =  5  Chapter 1. Introduction to Density Functional Theory (1.19) so that the canonical form of the famous Kohn-Sham equation looks like 1 ρ(r′ ) − ∇2 + Vext (r) + dr′ + Vxc (r) ψi = ǫi ψi . (1.21) 2 |r − r′ |  This is the working equation for most of the DFT calculations in current days. It gives much more accurate results than do the TF-type models since Ts [ρ] accounts for the most of the kinetic energy of a real system. Just like the Hartree-Fock equation, the Kohn-Sham equation appears as a pseudo-single-particle equation, although the potential terms in both equations do depend on all occupied orbitals. In the Hartree-Fock equation, the potential terms for different orbitals are different; while in the Kohn-Sham equation, they are the same. Reintroducing the using of wave functions in the equation deviates the spirit of DFT to some extent, and also reduces the computational advantages of DFT.  1.4 1.4.1  Mathematical Foundation of DFT υ-representability of an Electron Density  The HK functional in Eq. (1.5) has its own limitation: It is only defined on the electron densities associated with some antisymmetric ground-state wave-functions, and these wave-functions are solutions of the Schr¨odinger equation (1.1) which corresponds to an external potential υ(r) (that is why we call these densities υ-representable). Strictly speaking, the HK theorems are valid only on pure-state-υ-representable (PS-υ-representable) densities, that is, the wave-function which generates the electron density must represent a pure quantum state. This is a serious restriction. Actually the HK theorems can be generalized to degenerate ground states whose electron densities are not PS-υ-representable but ensemble-υ-representable (Eυ-representable) [24]. In quantum mechanics wave-functions are usually in the Hilbert space1 L2 (R3N ). So a reasonable electron density can be obtained from some antisymmetric wave-function in the Hilbert space. This is a condition weaker than the υ-representability condition. We call it N representability condition. Gilbert [25] showed that the N -representability condition is equivalent to the following: ρ(r) 1  0,  ρ(r)dr = N, and  |∇ρ(r)1/2 |2 dr < ∞,  (1.22)  For all mathematical concepts see Appendix D.  6  Chapter 1. Introduction to Density Functional Theory where N is the number of electrons of the system. The third inequality in the equation above guarantees that the kinetic energy of the system should be finite. Although Hohenberg and Kohn originally thought that the N representable but not E-υ-representable densities are only some “pathological” examples, researchers later discovered that some densities which look “reasonable” actually give out infinitely negative potential energies [24]. As the KS implementation of the HK theory is concerned, an even stronger condition of noninteracting-υ-representability is required for the densities.  1.4.2  Levy Functional and Lieb Functional  The unclearness of the υ-representability problem poses a serious challenge to the HK theory. Strictly speaking, almost all practical DFT calculations are not really in the spirit of the HK theorems. Fortunately this difficulty can be circumvented by redefining the universal energy functional. The total Hamiltonian of the system can be decomposed as ˆυ = H ˆ 0 + Vˆ , H  (1.23)  where Vˆ = N i=1 υ(ri ) is the external potential operator. So the total energy functional of potential is E[υ(r)] = inf  ˆ υ |ψ |ψ ∈ QN , ψ|H  (1.24)  where QN =  |ψ|2 dr3N = N, T [ψ] < ∞  ψ  .  (1.25)  Here T [ψ] is the kinetic energy functional of the wave-function. Let AN denote the set which consists of all υ-representable densities and BN to be the set which consists of all N -representable densities. Levy defined his functional (or called the Levy constrained search formulation) as [26] FLevy [ρ] = inf  ˆ 0 |ψ |ψ → ρ, ψ ∈ QN . ψ|H  (1.26)  According to his idea, the minimization process to reach the ground-state is divided into two steps: E0 = inf ρ  inf  ψ→ρ  FLevy [ρ] +  υ(r)ρ(r)dr  ,  (1.27)  where E0 is the ground-state energy. The first step is the minimization over the wave-functions which correspond to some density (that is why this 7  Chapter 1. Introduction to Density Functional Theory approach is called “constrained search”); and the second step is the minimization over all possible densities. Obviously the Levy functional is defined on the densities in the set of BN . Since the necessary and sufficient condition to judge whether a density is in BN or not is known (see Eq. (1.22)), the υ-representability difficulty has been circumvented. In addition, FHK [ρ] = FLevy [ρ], if ρ ∈ AN .  (1.28)  The Levy functional is in principle an orbital functional and only an implicit functional of the electron density. To apply Levy’s theory a constrained optimization must be performed. Practically it is not easy to do so directly [27–30]. Lieb also sugested another functional in his pioneering paper [31]: FLieb [ρ] = sup E[υ(r)] −  υ(r)ρ(r)dr υ ∈ L3/2 + L∞ ,  (1.29)  where L3/2 + L∞ means the sum of two Lebesgue spaces. Different from the Levy functional which optimizes on a set of orbitals, the Lieb functional is defined with the potential as the axillary variable. If an infinite positive value is allowed for FLieb , the density ρ is in the set CN =  ρ ρ(r) ≥ 0,  ρ(r)dr3 = N, ρ ∈ L3 (R3 )  .  (1.30)  Without the positive and normalization requirements of ρ, Eq. (1.29) actually defines FLieb [ρ] on the much larger set D = L3 ∩ L1 than CN . If ρ ∈ D but ∈ / CN , FLieb [ρ] = +∞. For the relation between FLieb [ρ] and FLevy [ρ], the following inequality holds: FLieb [ρ]  FLevy [ρ] for all ρ ∈ BN .  (1.31)  Furthermore, FLieb [ρ] is a convex functional while FLevy [ρ] is not. For more mathematical properties of these two functionals please see Ref. [31]. The Lieb functional is a potential functional and like the Levy functional, only an implicit functional of electron density. The potential functional presented in Wu and Yang’s paper [30] can be thought as a special case of the Lieb functional. A general discussion of potential functionals can be found in Ref. [32]. In addition, the Lieb functional can also be derived from the total energy variational principle and the Legendre transformation [33] from the external potential to the electron density. This idea can be used to construct functionals for spin-density-fucntional theory [34, 35] and variational 8  Chapter 1. Introduction to Density Functional Theory principles for density matrices and electron distribution functions [36]. For a good explaining of the applications of Legendre transform in DFT, please see Ref. [37]. In all, three functionals, such as the HK functional, the Levy functional and the Lieb functional can be defined in different manners. Their properties are summarized in the following table: Table 1.1: Functionals Summary  Functional  Density Domain  Convex?  υ-Representable Problem  FHK  AN ⊂ BN  No  Not solved  FLevy  BN ⊂ D  No  Solved  FLieb  D = L3 ∩L1  Yes  Solved  *  Remark Density functional Orbital functional* Potential functional*  Implicit density functional.  1.4.3  Functional Derivative of Density Functional  The functional derivative of the universal density functional is very important in the framework of DFT, since it gives the negative of the external potential [38]: δF = −V (r). (1.32) δρ(r) But before we start to calculate the functional derivative we should make sure the differentiability of density functionals. Actually a rigorous mathematical proof for the differentiability is not straightforward. Rigorously speaking there are two types of derivatives for a functional, one is the Gˆ ateaux derivative and the other is the Fr´echet derivative. Over years different researchers have contributed to the proof of the Gˆ ateaux differentiability of the Levy functional or the Lieb functional at all E-υ-representable densities [38–46]. But Lammert [47] argues all the existing proofs are flawed since at any density there are “bad directions” for which the Lieb functional has no directional derivative. So FLieb is not Gˆ ateaux differentiable in the normal sense. In addition, he also argues that there is no possibility of proving the Gˆ ateaux differentiability by using only the lower semicontinuity and 9  Chapter 1. Introduction to Density Functional Theory convexity of the functional. However, he has not excluded the possibility of proving it correct by other ways. This disputed issue must be clarified in the future studies. Recently different authors pointed out the nonuniqueness of external electrostatic and magnetic fields yielding a given many-electron ground-state [48, 49], which implies the nondifferentiability of the ground-state energy functional in the spin-density-functional theory (SDFT). These results put the whole framework of SDFT into question. G´ al defensed the theory by studying the constrained functional differentiation [50–55]. Luo [56] also proposed a constructive definition of constrained functional derivative in DFT. The locality of the density functional derivatives has also been a controversial problem. The “locality hypothesis” states that the density functional derivative can be in the form of a multiplicative local function. This hypothesis was challenged by Nesbet [57–60] who found that the functional derivative of the noninteracting kinetic energy functional can be orbital-dependent, namely δTs [ρ] ψi (r) = (εi − υ(r))ψi (r), (1.33) δρ(r) where εi is the energy eigenvalue of orbital ψi (r) and υ(r) is the KS effective potential. But Lindgren and Salomonson [43, 61] pointed out that the derivative in Eq. (1.33) must be evaluated within the normalized density domain. And once it is so, the orbital energy term in Eq. (1.33) disappears. They also extended the definition of noninteracting kinetic energy functional to the unnormalized density domain. Then the functional derivative of the new functional just equals the old derivative plus a system-independent constant and the locality hypothesis still holds. Holas and March [42, 62] also disagreed with Nesbet and proved the “locality hypothesis” to be a theorem. Liu and Ayers [63] also confirmed the existence and uniqueness of the functional derivative of Ts [ρ] from different ways.  10  Chapter 1. Introduction to Density Functional Theory  1.5 1.5.1  Key Concepts and Relations in DFT Exchange-Correlation Hole and Pair-Correlation Function  The two-particle density matrix for an N -particle system is defined as Γ2 (r1 σ1 r2 σ2 , r′1 σ1′ r′2 σ2′ ) = N (N − 1)  Ψ∗ (r1 σ1 r2 σ2 x3 x4 · · · xN )×  Ψ(r′1 σ1′ r′2 σ2′ x3 x4 · · · xN )dx3 dx4 · · · dxN , (1.34) where ri and σi denotes the spacial and the spin coordinates, respectively, and xi denotes a combined coordinate of space and spin. The two-particle density, which is the diagonal element of the density matrix above, is ρ2 (r1 σ1 , r2 σ2 ) = Γ2 (r1 σ1 r2 σ2 , r1 σ1 r2 σ2 ).  (1.35)  So the conditional probability of finding an electron at r2 with spin σ2 if there is another electron at r1 with spin σ1 is P (r2 σ2 |r1 σ1 ) =  ρ2 (r1 σ1 , r2 σ2 ) . ρ(r1 σ1 )  (1.36)  Here ρ(r1 σ1 ) is just the electron density. The XC hole function is defined by the equation hxc (r1 σ1 , r2 σ2 ) = P (r2 σ2 |r1 σ1 ) − ρ(r2 σ2 ).  (1.37)  It is easy to show that  σ2  hxc (r1 σ1 , r2 σ2 )dr2 = −1,  (1.38)  whose physical meaning is that this hole contains one electron. Since the two-particle density is symmetric, namely ρ2 (r1 σ1 , r2 σ2 ) = ρ2 (r2 σ2 , r1 σ1 ),  (1.39)  so according to the definition in Eq. (1.37) for the XC hole function we have hxc (r1 σ1 , r2 σ2 ) = hxc (r2 σ2 , r1 σ1 )  ρ(r2 σ2 ) . ρ(r1 σ1 )  (1.40)  The interaction energy between electrons is Eee =  Ψ  1 Ψ r12  =  1 2  ρ(r1 )ρ(r2 ) dr1 dr2 + Exc , r12  (1.41) 11  Chapter 1. Introduction to Density Functional Theory where r12 is the distance between two electrons. And the XC energy Exc can be written as Exc = =  1 2σ 1 2  ρ(r1 σ1 )hxc (r1 σ1 , r2 σ2 ) dr1 dr2 r12  1 σ2  h ρ(r1 σ1 )υxc (r1 σ1 )dr1 ,  (1.42)  σ1  h (r σ ) is the potential of the XC hole with the definition where υxc 1 1 h υxc (r1 σ1 ) = σ2  hxc (r1 σ1 , r2 σ2 ) dr2 . r12  (1.43)  Like the XC energy, we can also decompose the XC hole function into the exchange and correlation component, which is given by hxc (r1 σ1 , r2 σ2 ) = σ1 σ2  hx (r1 σ1 , r2 σ1 ) + σ1  hc (r1 σ1 , r2 σ2 ).  (1.44)  σ1 σ2  The exchange hole function can be defined exactly from the Hartree-Fock expression for the exchange energy, given by Ex = σ  1 2  ρ(r1 σ)hx (r1 σ, r2 σ) dr1 dr2 , r12  (1.45)  where the exchange hole function, given in terms of spin orbitals ψi (rσ), is 1 hx (r1 σ, r2 σ) = − ρ(r1 σ)  2  N  ψ ∗ (r1 σ)ψ(r2 σ) .  (1.46)  i  From the equation above it is easy to know hx (r1 σ, r2 σ)  0,  (1.47)  and σ  hx (r1 σ, r2 σ)dr2 = −1.  (1.48)  According to Eq. (1.38) one can deduce that hc (r1 σ1 , r2 σ2 )dr2 = 0,  (1.49)  σ1 σ2  12  Chapter 1. Introduction to Density Functional Theory but unlike hx (r1 σ, r2 σ), hc (r1 σ1 , r2 σ2 ) can be either positive or negative. We can also define the pair-correlation function as g(r1 σ1 , r2 σ2 ) =  ρ2 (r1 σ1 , r2 σ2 ) . ρ(r1 σ1 )ρ(r2 σ2 )  (1.50)  Obviously hxc (r1 σ1 , r2 σ2 ) = ρ(r2 σ2 )[g(r1 σ1 , r2 σ2 ) − 1].  (1.51)  So the interaction energy is Eee =  1 2σ  g(r1 σ1 , r2 σ2 ) dr1 dr2 . r12  (1.52)  [g(r1 σ1 , r2 σ2 ) − 1] dr1 dr2 . r12  (1.53)  ρ(r1 σ1 )ρ(r2 σ2 ) 1 σ2  And the XC energy is Exc =  1 2σ  ρ(r1 σ1 )ρ(r2 σ2 ) 1 σ2  Like the XC hole function, the pair-correlation function also has the property  σ2  1.5.2  ρ(r2 σ2 )[g(r1 σ1 , r2 σ2 ) − 1]dr2 = −1.  (1.54)  Scaling Relations  Although we do not know the exact expression of energy density functional, studying its properties under certain mathematical transformation might help to construct approximations for it. A scaling transformation on the spacial coordinates is defined as ri → λri ,  (1.55)  where λ is the scaling constant. To preserve the normalization of wavefunctions, the scaled wave-function should be Ψλ (rN ) = λ3N/2 Ψ(λrN ).  (1.56)  Here rN is a shorthand notation of the N -particle spacial coordinates. With this definition, the scaled density is ρλ (r) = λ3 ρ(λr).  (1.57)  13  Chapter 1. Introduction to Density Functional Theory For the kinetic and electron-electron interaction energy operators, it is easy to show that 1 T (rN ), λ2 1 Vee (rN ). λ  Tˆ(λrN ) = Vˆee (λrN ) =  (1.58) (1.59)  Straightforwardly, the scaling relation of the noninteracting kinetic energy functional and the classical Coulomb potential energy functional are Ts [ρλ ] = λ2 Ts [ρ],  (1.60)  J[ρλ ] = λJ[ρ].  (1.61)  The scaled one-particle density matrix is N ∗ ψλ,i (r)ψλ,i (r′ )  Γλ (r, r′ ) = i  N  = λ  3  ψi∗ (λr)ψi (λr′ ) i  3  = λ Γ(λr, λr′ ).  (1.62)  So the scaled exchange energy functional is |Γλ (r, r′ )|2 1 drdr′ 4 |r − r′ | λ6 |Γ(λr, λr′ )|2 = − drdr′ 4 |r − r′ | λ |Γ(λr, λr′ )|2 = − dλrdλr′ 4 |λr − λr′ | = λEx [ρ].  Ex [ρλ ] = −  (1.63)  The scaling relations can be used to derive some useful results. Suppose a local density functional can be written as F [ρ] =  f (ρ(r))dr,  (1.64)  where f is some function of ρ. If the scaling relation of F [ρ] reads F [ρλ ] = λn F [ρ],  (1.65) 14  Chapter 1. Introduction to Density Functional Theory where n is the scaling index, then λ−3  f (λ3 ρ(λr))dλr = λn  f (ρ(r))dr,  (1.66)  finally we can deduce that f (λ3 ρ) = λn+3 f (ρ).  (1.67)  The simplest solution for the functional equation [64] above is f (ρ) = Cρ1+n/3 ,  (1.68)  where C is some constant. So if F [ρ] = Ex [ρ], then n = 1 (see Eq. (1.63), according to Eq. (1.64) and Eq. (1.68) we get the same expression for the local density approximation (LDA) exchange energy functional in Eq. (1.10); and if F [ρ] = Ts [ρ], then n = 2 (see Eq. (1.60), in the same way we get the same expression for the Thomas-Fermi kinetic energy functional in Eq. (1.7). Unlike Ts [ρ], J[ρ] and Ex [ρ] which have simple scaling relations, the interacting kinetic energy functional T [ρ], the electron-electron interaction energy functional Vee [ρ], and the correlation energy functional Ec [ρ] have not any simple scaling relations. This is because that the scaled wave-function Ψλ (rN ) does not minimize the expectation value of Tˆ + Vˆee , but minimize the expectation value of Tˆ + λVˆee . This fact can be shown as the following: E[ρ] = min  Ψ→ρ  Ψ∗ (rN )[Tˆ(rN ) + Vˆee (rN )]Ψ(rN )drN  = min λ3N  Ψ∗ (λrN )[Tˆ(λrN ) + Vˆee (λrN )]Ψ(λrN )drN  = min λ−2  Ψ∗λ (rN )[Tˆ(rN ) + λVˆee (rN )]Ψλ (rN )drN  Ψ→ρ  Ψ→ρ  = λ−2 min Ψλ |Tˆ + λVˆee |Ψλ . Ψλ →ρλ  (1.69)  Obviously the quantity Tˆ + λVˆee scales like λ2 under wave-function scaling transformation, which make it impossible that the total energy functional, Tˆ + Vˆee , has a simple scaling relation. Since all the scaling relations of other components of the total energy functional are known simple, the only unknown scaling relation of Ec [ρ] can not be simple. A careful analysis reveals that the following scaling inequalities must hold [65]: T [ρλ ] > λT [ρ], Vee [ρλ ] < λVee [ρ], Ec [ρλ ] < λEc [ρ] (λ < 1),  (1.70)  T [ρλ ] < λT [ρ], Vee [ρλ ] > λVee [ρ], Ec [ρλ ] > λEc [ρ] (λ > 1).  (1.71) 15  Chapter 1. Introduction to Density Functional Theory Recently the approximate scaling relation of the kinetic-energy component of the correlation energy density functional for atoms are discussed by Liu et al. [66]. Calder´ın and Stott also derived the scaling relations for the KS potential [67]. For the case of scaling the spin densities separately in SDFT, please see Ref. [68].  1.5.3  Adiabatic Connection Formula  The adiabatic connection formula is also called coupling constant integration or coupling strength integration formula [69–73]. In the Kohn-Sham implementation of DFT, a noninteracting model system is used; while for the real system, full correlation effects must be considered. One can define a partially interacting system with the Hamiltonian ˆλ = − 1 H 2  N i  ∇2i + Vˆ (ρ, λ) + λVˆee ,  (1.72)  where 0 λ 1, λ is the coupling strength parameter. V (ρ, λ) is the one-body potential which corresponds the density ρ and has a parameter dependency on λ. When λ = 0, V (ρ, 0) = Vs (ρ) is just the Kohn-Sham effective potential; and when λ = 1, V (ρ, 1) = Vext (ρ) is just the external potential. Suppose the noninteracting Kohn-Sham reference system (λ = 0) connects to the fully interacting real system (λ = 1) smoothly and adiabatically by a continuum of partially interacting systems (0 < λ < 1), all of which correspond to the density of the real system. Then Exc [ρ] = FHK [ρ] − Ts [ρ] − J[ρ] =  ˆ + λVˆee |Ψλmin Ψλmin |H  λ=1  ˆ + λVˆee |Ψλmin − Ψλmin |H 1  = 0  λ=0  − J[ρ]  d ˆ + λVˆee |Ψλmin dλ − J[ρ]. Ψλ |H dλ min  (1.73)  With the Hellmann-Feynman theorem the equation above can be simplified to 1 Ψλmin |Vˆee |Ψλmin dλ − J[ρ]. (1.74) Exc [ρ] = 0  16  Chapter 1. Introduction to Density Functional Theory One can regain Eq. (1.42) with a coupling constant integration average for the XC hole function as 1  ¯ xc (r1 σ1 , r2 σ2 ) = h 0  hλxc (r1 σ1 , r2 σ2 )dλ,  (1.75)  where hλxc (r1 σ1 , r2 σ2 ) is the XC hole function of the partially interacting system. So the adiabatic connection formula for the XC energy is Exc = =  1 2σ 1 2σ  1 σ2  0  ¯ xc (r1 σ1 , r2 σ2 ) ρ(r1 σ1 )h dr1 dr2 r12  1 σ2  1  =  ρ(r1 σ1 )hλxc (r1 σ1 , r2 σ2 ) dr1 dr2 dλ r12  λ Uxc dλ.  (1.76)  λ is the potential energy of exchange-correlation at the intermediate Here Uxc coupling strength λ. Eq. (1.76) stimulates us to approximate the XC energy by 1  Exc = 0  λ λ=0 λ=1 Uxc dλ ≈ aUxc + (1 − a)Uxc ,  (1.77)  where a is some fractional number to mix the potential energies of the noninteracting with those of the fully interacting systems. This idea leads to the famous hybrid functional family [74, 75]. The simplest hybrid functional can be written as hyb Exc = aExexact + (1 − a)ExGGA + EcGGA .  (1.78)  Here Exexact is calculated by the Hartree-Fock type expression for exchange energy (see Ref. [23]). Hybrid functionals are perhaps the most successful density functionals in daily quantum chemical calculations and have gained much attention even from nontheorists.  1.5.4  Bounds on Exact Functionals  An important technique in developing new functionals is to construct approximate expressions which obey rigorous mathematical bounds of the exact functional. So these bounds become the standard to calibrate the current density functionals. Lieb and Thirring [31] proposed that Ts [ρ]  TTF [ρ],  (1.79) 17  Chapter 1. Introduction to Density Functional Theory where Ts is the noninteracting kinetic energy and TTF is the Thomas-Fermi kinetic energy (see Eq. (1.7)). For the kinetic energy density, there is also an inequality [76, 77] which reads τ τvW , (1.80) where τvW is the energy density of the von Weizs¨acker functional. If all occupied KS orbitals in one system are divided into to two sets A and B; ρA and ρB are electron densities generated from the orbitals in A and B, respectively, the following inequality holds [78]: Ts [ρA + ρB ]  Ts [ρA ] + Ts [ρB ].  (1.81)  A very beautiful lower bound for the XC energy was obtained by Lieb and Oxford [79]. It reads Exc [ρ]  −C  ρ4/3 (r)dr,  (1.82)  which C 1.68. Later Perdew [80] found a lower bound for C, giving 1.43 C 1.68. An slightly improved lower bound presented by Chan and Handy [81] is C 1.6358. Recently numerical evaluations of the tightness of the Lieb-Oxford bound have been done by Odashima and Capelle [82]. For a special discussion on the bounds on one-dimensional exchange energies, please see Ref. [83].  1.5.5  Functional Derivative Discontinuity of Density Functionals  In their influential paper [84] published almost 3 decades ago, Perdew et al. (PPLB) extended DFT to systems with a fractional number of electrons based on the zero-temperature grand canonical ensemble theory. PPLB considered that at zero temperature the ground-state of a system with a fractional number of electrons is a thermal mixture of two pure states with two closest integer numbers of electrons which the fractional number nests between. So the ground-state energy can be written in the form EN +ω = (1 − ω)EN + ωEN +1 ,  (1.83)  where N is some integer and 0 < ω < 1. According to this extension of DFT, the ground-state energy function E(N ) varies linearly between any two adjacent integers. The Janak theorem [85] tells us that ∂E = µ = εHOMO , ∂N  (1.84) 18  Chapter 1. Introduction to Density Functional Theory where εHOMO is the orbital energy of the highest partially occupied KS orbital and µ is the chemical potential of the system. This result combined with the discussion on Eq. (1.83) above leads to the well-quoted conclusion εHOMO =  −I (Z − 1 < N < Z) , −A (Z < N < Z + 1)  (1.85)  where I and A are the first ionization energy and the electron affinity of the Z-electron system, respectively. Normally, I = A, so PPLB found the so-called “functional derivative discontinuity”(FDD) in DFT. FDD is very important in the development of DFT. Any continuum XC functional, say the LDA or GGA functionals, can not represent FDD. They at the best describe FDD in an average sense. Usually good approximate functionals can give an εHOMO which is close to the nagetive of Mulliken’s electronegativity [86] − 21 (I + A). The failure of producing FDD accounts for the some wellknown difficulties of DFT, such as the incorrect dissociation behavior of some radicals and the bad description of band gaps of solids. Like the other problems on the functional derivatives in DFT, FDD is not a question free of disagreements. Perdew and Kurth [87] indicate that FDD arises partly from the XC energy functional, and entirely so if the number of electrons does not fall on the boundary of an electronic shell or subshell . Qian and Sahni [88] showed that FDD is due to the correlationkinetic component of the XC energy via quantal density functional theory. While Zahariev and Wang [44] argued that FDD totally origins from the noninteracting kinetic energy functional and there is no FDD for the XC energy functional of a system with an integer number of electrons. Very recently Sagvolden and Perdew [89] presented a new proof for the FDD of XC energy functional and also showed some numerical evidences of FDD from some model density. Another major conclusion of the PPLB paper, which is that for a system with an integer number of electrons it is true that εHOMO (N ) = −I,  (1.86)  where Z −1 < N < Z. Despite the many proofs in the literature [19, 84, 90], the relation above was doubted [91] and then debated [92–94]. There was also a new proof provided by Harbola later [95]. But it seems that none of these proofs end the debating. Luo [56] pointed out that the pivot of the problem is the definition of functional derivative. He showed by construction that the restricted functional derivative of the energy functional can only be determined up to an additive constant. This constant depends on the way 19  Chapter 1. Introduction to Density Functional Theory of defining the restricted functional derivative. If unrestricted functional derivative is considered, one must extend the definition of energy functional to the Fock space. And also the unrestricted functional derivative contains another additive constant which depends on the way of how to extend the definition of the energy functional. So in all, the relation in Eq. (1.86) might be right, but it is only one of the many reasonable choices. FDD is not some property of density functional which is in a sense of pure math, actually it appears for some physical considerations. We have already mentioned that FDD is one of the most important properties of the exact XC energy functional. And it governs the performance of the approximate XC functionals under many situations. Despite of the subtleness and intricacy of this problem, all the arguments above are not something about metaphysics. They can be tested and thus must have “right” or “wrong” judgements. So all previous works should be re-examined carefully and a deeper analysis on this problem is needed in order to give a completely sound answer.  1.5.6  Asympotic Behavior of the Exchange-Correlation Potential  Most of the popular density functionals focus on a good description of the total energy or total energy difference of the system, but fail to determine the energy change as a response to an external electromagnetic field change. That is why most of the functionals can not precisely predict such response properties as vertical excitation energies, NMR shielding constants and static isotropic polarizabilities. All these response properties are related to the structure of the XC potential which is defined in Eq. (1.20). Moreover, most of the popular functionals we used can generate accurate enough XC potentials in the range closed to the nuclei. But in the long range, those generated XC potentials deviate from the exact XC potential too much—normally they decay exponentially, much faster than the exact potential should do. Almbladh and von Barth studied the asymptotic structure of Vxc [96]. For the case when both the N - and (N − 1)-electron atomic systems are not degenerate, the asymptotic expression of Vxc (r) is 1 α Vxc (r) ∼ − − 4 + · · · , r → ∞, (1.87) r 2r where α is the static ground-state polarizability of the (N − 1)-electron system. If there is any orbital degeneracy in the N - and (N − 1)-electron system, the asymptotic structure would be α 1 QN −1 − QN − 4 + · · · , r → ∞, Vxc (r) ∼ − + 3 r r 2r  (1.88) 20  Chapter 1. Introduction to Density Functional Theory where QN −1 and QN are the quadrupole moments of the N - and (N − 1)electron systems, respectively. Qian and Sahni [97, 98] add O(1/r5 ) terms to both cases above, namely α 8κ0 χ 1 + ··· , Vxc (r) ∼ − − 4 + r 2r 5r5 1 QN −1 − QN α − 4+ Vxc (r) ∼ − + 3 r r 2r 8κ0 χ RN −1 − RN + + · · · , r → ∞, 5r5 r5  (1.89)  where κ20 /2 is the ionization energy; and RN −1 and RN are the hexadecapole moments of the N - and (N − 1)-electron systems, respectively; χ is an expectation value of the (N − 1)-electron system (see Ref. [98] for the detailed definition). They also pointed out [99] the asymptotic expressions for the exchange and correlation potentials should be 1 Vx (r) ∼ − + · · · , r α 8κ0 χ Vc (r) ∼ − 4 + + · · · , r → ∞, 2r 5r5  (1.90)  respectively. Obviously if the popular functionals are forced to produce potentials which have the correct asymptotic behavior explained above, better computational results for the response properties will be obtained. That is why researchers have cared for the asymptotic behavior of XC potentials for years. One natural idea is to fit the approximate XC potentials to the exact ones directly [100–105]. An alternative to deal with the same problem is the asymptotic correction (AC) method. In this method, people use the functional derivatives of the popular density functionals to model the XC potential in the energy-significant core region, while use the asymptotic form of the exact XC potential in the long-range. The simplest asymptotic correction form of the XC potential is the Fermi-Amaldi (FA) model, namely [106] 1 1 ρ(r′ ) FA Vxc (r) = − VH [r] = − dr′ , (1.91) N N |r − r′ |  where VH is the Hartree potential because of the classical Coulombic interaction and N is the number of the electrons in the system. The FA model potential is simple to implement and it is the functional derivative of the approximate XC energy functional − N1 J[ρ]. For most of the other AC potentials, they are not functional derivative of any popular XC energy 21  Chapter 1. Introduction to Density Functional Theory functional. FA model is also exact for the one-electron system. So the FA model has been widely used as the zero-order approximation to the XC potential in many calculations, for examples, Ref. [29, 30]. But the FA model oversimplifies the XC hole (it only gives a uniform hole function as −1/N ) and is not size-consistent. There are ways to fix this problem [107]. For modifications and extensions of the FA model, please see Ref. [108–111]. For a thorough discussion on the FA model, please see Ref. [112]. Since Perdew et al. [84] have shown the “functional derivative discontinuity” of the XC energy functional, the exact XC potential must jump by some system-dependent constant as the number of electrons goes through an integer. This means that the XC potential might not vanish asymptotically [84, 113, 114]. So the asymptotic expression for the XC potential should be 1 Vxc (r) ∼ − + C∞ . r  (1.92)  After some asymptotic analysis on the KS equations one can derive the generalized Koopmans’ theorem C∞ = I + εHOMO ,  (1.93)  where I is the ionization energy and εHOMO is the highest occupied KS orbital energy. There are a lot of AC schemes based on this theory [115–123]. Since any potential from a continuum functional can not represent such a discontinuity, researchers often divide the whole space into two or three regions. For the region which is close to the nuclei, the conventional approximate XC potentials are used; while for the long-range, the correction asymptotic expression of the XC potential plus the constant in Eq. (1.93) is used. Finally a switch or connection function is used at the turning point or in the middle region. Baer et al. [124, 125] also suggested a XC energy functional which can produce a XC potential with the correct asymptotic behavior. Della Sala and G¨ oling found that the KS exchange potential of finite systems can not approach its asymptotic limits simultaneously [126, 127]. Actually it approaches a nonzero constant along a nodal surface of the HOMO if it approaches zero elsewhere. Joubert [128] proved that the correlation potential also has a nonuniform asymptotic behavior and this behavior exactly cancels the nonuniform asymptotic behavior of the exchange potential. Like the debate on the functional derivative discontinuity of the XC energy functional, there are also some disagreements on the asymptotic behavior of the XC potential. Tozer and Handy [129] stated that if generalized gradient approximation (GGA) functionals are used, the asymptotic of the 22  Chapter 1. Introduction to Density Functional Theory XC potential must be some positive number. With a constructive approach, Pino [130] showed that the XC potential must decay to zero asymptotically, otherwise a shift in the electronic energy which is proportional to the number of electrons will appear. He also warned that any AC model which is involved with non-zero asymptotics for the XC potentials might lead to unphysical results. Another family of methods related to the AC methods is the long-range correction (LC) method which means combining the short-range part of exchange functional with the long-range part of the Hartree-Fock exchange integral. Many LC schemes [131–136] have been proposed. They are based the decomposition of the electron repulsion operator into the short- and long-range parts 1 1 − erf(µr12 ) erf(µr12 ) = + , (1.94) r12 r12 r12 where r12 is the distance between two electrons, µ is the parameter that determines the proportion between the two ranges and erf(µr12 ) is the standard error function. An LC density functional carries the general form LC Exc = Exsr + Exlr,HF + Ec ,  (1.95)  where Exsr denotes the short-range part of exchange energy evaluated with density functionals , Exlr-HF denotes the long-range part the exchange energy evaluated with the Hartree-Fock type integrals and Ec is the correlation energy functional. It was reported that LC methods can overcome the major problems of time-dependent density functional theory (TDDFT) [134] and is remarkably accurate for a broad range of molecular properties [135, 136]. The whole family of the methods is promising.  1.6  Kinetic Energy Functional  We have discussed the Thomas-Fermi (TTF ) and von Weizs¨acker (TvW ) kinetic energy functional in the previous section (see Eq. (1.7) and (1.11)). TTF is exact for the uniform electron gas and TvW is exact for the one-electron system. A natural idea on the improvement of both models might be a hybrid combination of both models, which lead to the TF(D)-λW model (see Eq. (1.12)). In addition, a second-order gradient expansion of the density of the nonuniform electron gas also produce a TF(D)-λW model with λ = 1/9 [18, 137, 138]. But this kind of model can only offer limited improvements [139] since both TTF and TvW incorporate the whole or some part of the  23  Chapter 1. Introduction to Density Functional Theory noninteracting kinetic energy [140]. In principle gradient expansion approximations (GEAs) can be achieved up to arbitrary high orders [141, 142]. The noninteracting kinetic energy functional can be written as Ts [ρ] =  {τ0 (ρ) + τ2 (ρ) + τ4 (ρ) + τ6 (ρ) + · · · }dr,  (1.96)  where τ0 (ρ) = c0 ρ5/3 , |∇ρ|2 , τ2 (ρ) = c2 ρ τ4 (ρ) = c4 ρ1/3  (1.97) (1.98) ∇2 ρ ρ  2  −  9 ∇2 ρ ∇ρ 8 ρ ρ  2  +  1 ∇ρ 3 ρ  4  .  (1.99)  ci , i = 0, 2, 4 are constants (see Ref. [141, 142]), τ0 and τ2 are TF and vW-type terms, respectively. The expression of τ6 is extremely complicated [142] and τ6 or other higher order terms actually diverge in the long range of atomic or molecular systems [142–144]. To cure the divergence problem, a local truncation scheme was proposed by Pearson and Gordon [145] which reads imax −1 1 TsPG [ρ] = τ2i + τ2imax dr, (1.100) 2 i=0  where τ2imax is the smallest terms in the expansion series in Eq. (1.96). Not restricting to the GEAs, a lot of GGAs have been also proposed. For instance, DePristo and Kress [146] suggested the following kinetic energy density functional with the Pad´e approximation [147], τDK = τ0  1 + 0.95x + a2 x2 + a3 x3 + 9b3 x4 , 1 − 0.05x + b2 x2 + b3 x3  (1.101)  where x = τ2 /τ0 , a2 , a3 , b2 , b3 are fitted parameters. For a comparative study on most of the existed GGA kinetic energy functionals, please see Ref. [148]. Very recently Perdew and Constantin [149] suggested a metaGGA (kinetic-energy-density-involved) kinetic energy functional which can recover the fourth-order GEA functional in the slowly varying density limit and the vW functional in the rapidly varying density limit. Another family of kinetic energy functionals [150–164] are based on the linear-response (LR) theory [165]. The response function χ is defined as χ(r − r′ ) =  δρ(r) , δυ(r′ )  (1.102) 24  Chapter 1. Introduction to Density Functional Theory where υ can be some potential, say the effective, external, Hartree or XC potential, which produces the corresponding response function, respectively. In the first paper of modern DFT [6], Hohenberg and Kohn had already pointed out that the second functional derivative of the total energy functional gives the total response function, that is KS δυeff (r) 1 δ 2 (Etot [ρ] − Ts [ρ]) δTs [ρ(r)] = = =− . ′ ′ ′ ′′ χ(r − r ) δρ(r ) δρ(r )δρ(r ) δρ(r′ )δρ(r′′ )  (1.103)  The last equal sign holds because δEδρtot should be a constant (the chemical potential of the system). For a UEG this relation can be shown in the momentum space as F  δTs [ρ(r)] δρ(r′ )δρ(r′′ )  ρ0  =−  1 χ ˜Lind (q)  =  π2 FLind (η), kF  (1.104)  where F denotes the Fourier transform and the Lindhard function [166] FLind has the expression FLind (η) =  1+η 1 1 − η2 + ln 2 4η 1−η  −1  .  (1.105)  Here η = q/2kF is the scaled momentum and kF is the Fermi wave vector. A general form for nonlocal functionals can be Tnl [ρ] =  f (ρ(r), ρ(r′ ), r, r′ )drdr′ ,  (1.106)  where f is some function which fulfills the dimensionality requirement. In this understanding, the Thomas-Fermi kinetic energy functional can be rewritten as TTF [ρ] = CTF  ρ5/3−β (r)δ(r − r′ )ρβ (r′ )drdr′ ,  (1.107)  where CTF is the TF constant in Eq. (1.7) and β is a parameter. If a kernel is introduced into the double integral, a general form for a TF-type nonlocal kinetic energy functional [160, 163] would be presented as TF Tnl [ρ] = CTF  ρ5/3−β (r)Ω(ζ(r, r′ ), |r − r′ |)ρβ (r′ )drdr′ ,  (1.108)  25  Chapter 1. Introduction to Density Functional Theory where ζ(r, r′ ) is a scaling factor. Here the kernel Ω is designed to make the relation in Eq. (1.104) hold. The von Weizs¨acker functional can also be written in following forms: I TvW [ρ] =  1 2  II TvW [ρ] = −  1 2  ∇ψ(r) · ∇ψ(r)dr  (1.109)  ψ(r)∇2 ψ(r)dr.  (1.110)  Inserting the integral kernel as we did above, two general forms for the vW-type nonlocal kinetic energy functionals [164] vW,I Tnl [ρ] =  1 2  vW,II Tnl [ρ] = −  1 2  ∇ψ(r) · ∇ψ(r′ )Ω(ζ(r, r′ ), |r − r′ |)dr  (1.111)  ψ(r)∇2 ψ(r′ )Ω(ζ(r, r′ ), |r − r′ |)dr.  (1.112)  are obtained. We have the freedom to choose the scaling factors in Eqs. above. Usually there are three cases: • The kernel can be a constant Fermi wave vector kF0 (r) which comes from a reference uniform electron density. This is called a densityindependent kernel. • The kernel can be a local Fermi wave vector kF (r). This is called a density-dependent kernel. • The kernel can be a two-body Fermi wave vector which has the expression 1/γ kFγ (r) + kFγ (r′ ) ζ(r, r′ ) = , (1.113) 2 where γ is some parameter. This is called a double density-dependent kernel. Recently Blanc and Canc´es proved that the density-independent functionals are not bounded from below [167] so that all calculations employing these models might be meaningless. The LR-theory-based kinetic energy functionals are mostly implemented in the orbital-free density functional theory [168, 169] (OF-DFT) and widely used in the studies of the metallic and extended systems.  26  Chapter 1. Introduction to Density Functional Theory  1.7  Exchange-Correlation Energy Functional  The Kohn-Sham approach is the dominant method in modern DFT implementations, which makes the exchange-correlation energy functional defined in Eq. (1.14) the core of current DFT researches. After decades there are a lot of popular XC functionals which are suitable for using under all kinds of situations. Here we only provide a brief summary on these XC functionals. In the year of 2000, Perdew [170] presented his dream about that the DFT users are climbing a “Jacob’s Ladder” to gain more and more accuracy. This ladder of XC functionals can be illustrated in the following: (chemical accuracy) HEAVEN rung5 fully nonlocal functional rung4 hybrid functional rung3 meta-GGA functional rung2 GGA functional rung1 LDA functional EARTH (Hartree theory) We would like do add the explanations and remarks as below: (1) The earth is the Hartree model, in which no XC effect is considered. This is the roughest individual-particle image of electrons. (2) Rung1 is the local density approximation (LDA) functional. The LDA version for exchange energy functional can be found in Eq. (1.10) and the most popular LDA correlation energy functional is the VWN functional [171]. The Xα method proposed in the early years by Slater [172] can also be considered as an empirical LDA exchange energy functional with one parameter. In LDA functionals, XC effects are accounted as if the system is a UEG. So one should not expect LDA functions to give precise results for some real chemical systems. (3) The functionals on rung2 are those which depend explicitly on the electron density gradients. They are semi-local functionals since they only depend on the gradient locally. They are not just gradient expansions starting from some simple models. Like the kinetic energy functionals, simple gradient expansions (GEA functionals) do not provide much improvements and diverge in the long-range of the finite systems. So some general forms containing density gradients other than the gradient expansions are invented. That is why these models are called general gra27  Chapter 1. Introduction to Density Functional Theory dient approximations (GGA). One of the most popular GGA exchange functional is the Becke88 functional [173], whose form is ExB88 [ρσ ] = ExLDA [ρσ ] − b  ρ4/3 σ  x2σ dr, 1 + 6bxσ sinh−1 xσ  (1.114)  where ExLDA is the Dirac exchange functional in Eq. (1.10), the reduced density gradient xσ is a dimensionless quantity for the spin σ with the definition |∇ρ| xσ = 4/3 , (1.115) ρ and b = 0.0042. For the GGA correlation functionals, one example is LYP functional [174, 175]. Its expression for closed-shell systems is ρ 11 − bρ2 |∇ρ|2 + −1/3 24 1 + dρ 3 5 7 (3π 2 )2/3 ρ8/3 + |∇ρ|2 − δ ωρ2 dr, b 10 12 72 EcLYP [ρ] = −a  (1.116)  where −1/3  ω =  e−cρ , ρ11/3 (1 + dρ−1/3 )  δ = cρ−1/3 +  dρ−1/3 . 1 + dρ−1/3  The parameters a = 0.04918, b = 0.132, c = 0.2533 and d = 0.349. GGA functionals usually have some improvements comparing to LDA functionals. For example, the combination of the B88 and LYP functional makes the BLYP functional, which can predict molecular geometries comparable to the Hartree-Fock results; and can also produce vibrational frequencies as accurate as the MP2 results. But the selfinteraction existed in BLYP functional will make it underestimate the reaction barrier seriously. (4) Rung3 contains the functionals which explicitly depend on the kinetic energy density of the system. The kinetic energy density is occ  τ= i  |∇ψi |2  (1.117)  It was reported that this introduction of τ -dependency significantly improves the calculated results [176]. 28  Chapter 1. Introduction to Density Functional Theory (5) We have touched the hybrid functionals on rung4 in Sec. 1.5.3. It is well-known that the exact Hartree-Fock exchange integral will completely eliminate the self-interaction effect. So Becke [74, 75] proposed a XC functional with a fractional mixing of the HF exact exchange energy. This family of functionals show surprising accuracy in daily computational chemistry and have become the most popular functionals. One of the famous example is B3LYP functional [177], which has the form B3LYP Exc = (1 − a0 )ExLDA + EcVWN + a0 ExHF  +ax (ExB88 − ExLDA ) + ac (EcLYP − EcVWN ),  (1.118)  where a0 , ax and ac are parameters. The B3LYP functional is so successful in predicting the geometries, spectroscopic and thermochemical properties of small molecules [178] that many experimental chemists think B3LYP as the synonym of DFT. (6) On rung5 there are some fully nonlocal functionals. Although the GGA and hybrid functionals are very successful in dealing many problems, a correct description of Rydberg states of atoms or dispersion forces between systems at large separation requires fully nonlocal XC functionals. There are a lot of orbital functionals being introduced [179–191]. Unlike the meta-GGA and hybrid functionals, they depend not only the occupied KS orbitals, but also the unoccupied ones. Strictly speaking, all the functionals on rung3–5 are only implicit functionals of density. To get the XC potential one must use the optimized potential method [192, 193] (OPM). There are highly efficient methods available [194, 195]. But usually a direct construction of the XC potential using the non-selfconsistent KS orbitals is sufficient in the routinely DFT calculations. For a comprehensive discussion on orbital-dependent functionals, please see Ref. [196]. Another family of nonlocal XC functionals employ the average density approximation [197–199] (ADA) and weighted density approximation [197–200] (WDA). According to Eq. (1.51) and (1.76) we have Exc =  ρ(r)ρ(r′ ) [¯ gxc (r, r′ , ρ) − 1]drdr′ , |r − r′ |  (1.119)  where g¯xc is the coupling constant integration averaged pair-correlation function. In the LDA model, the XC hole function is modeled as ′ UEG hLDA xc (r, r ) = ρ(r)[gxc (r) − 1],  (1.120) 29  Chapter 1. Introduction to Density Functional Theory UEG is the pair-correlation function of a UEG. Comparing LDA where gxc hole function with the exact one in Eq. (1.51), one might find that the differences are using the local density ρ(r) to replace the nonlocal UEG density ρ(r′ ), and using gxc instead of the exact one. In the ADA ansatz, one use an averaged local density ρ¯(r) to replace ρ(r′ ) in Eq. (1.51); and the UEG pair-correlation function is still used, but with the averaged density. So the ADA XC hole function is  ′ UEG hADA ¯(r)[˜ gxc (|r − r′ |, ρ¯(r)) − 1]. xc (r, r ) = ρ  (1.121)  The averaged density is determined through a recursive integral relation ρ¯(r) =  W [|r − r′ |, ρ¯(r)]ρ(r′ )dr′ ,  (1.122)  where W is the weight function. One can also define a function G[|r − r′ |, ρ(r)] = gxc [|r − r′ |, ρ(r)] − 1.  (1.123)  Another weighted density ρ˜(r) can be calculated from the sum rule ρ(r′ )GWDA [|r − r′ |, ρ˜(r)]dr′ = −1,  (1.124)  and the WDA XC hole function is ′ ′ WDA hWDA [|r − r′ |, ρ˜(r)]dr′ . xc (r, r ) = ρ(r )G  (1.125)  Many model WDA G functions are available [201–204]. To apply ADA functionals one must solve the averaged density form Eq. (1.122), which is not easy. In addition, both ADA and WDA functionals require a double integration operation in Eq. (1.119). This is computational expensive. But the fast Fourier transform (FFT) algorithm will tremendously lower the computing cost, so nowadays ADA and WDA functionals are widely used to treat periodic systems combined with pseudopotentials. (7) The heaven is to achieve chemical accuracy, the dream of every theoretical or computational chemist. Of course the computational cost increase as one climbs Jacob’s ladder. To choose functionals on which rung is a problem of a compromise between the research needs and the computing capability. The most popular XC functionals are listed in Tab. 1.2: 30  Chapter 1. Introduction to Density Functional Theory Table 1.2: Popular XC Functionals  Name Dirac Xα VWN SIC P86 B88 LYP PW91 B3LYP Gill96 PBE HCTH VSXC OPTX TPSS Minnesota a  b c  1.8  Year 1930 1951 1980 1981 1986 1988 1988 1991 1994 1996 1996 1998 1998 2001 2003 2000–  Reference [12] [172, 205] [171] [206] [207] [173] [174] [80, 208–211] [74, 75, 177] [212, 213] [214, 215] [216] [176] [217] [218] [219–233]  Category LDA LDA LDA – GGA GGA GGA GGA hybrid GGA GGA GGA meta-GGA GGA meta-GGA Manyc  Remark Xa a X, empirical Ca a XC, SICb Ca Xa Ca XCa XCa Xa a XC, non-empirical XCa, semi-empirical XCa Xa a XC, non-empirical XCa  “X” means exchange-only functional; “C” means correlation-only functional and “XC” means exchange-correlation functional. “SIC” means self-interaction correction. “Many” means GGA, meta-GGA, hybrid and hybrid meta-GGA all.  Applications of DFT  Nowadays DFT has become the favorite son of not only the computational chemists, but all chemists, condensed matter physicists and material scientists. A strong proof of this statement is that according to the statistics by CAS2 [234], DFT papers account for a large part of the most-cited papers since 1999. This is not a surprise since DFT is a single-particle theory which considers correlation effect. The single-particle nature makes DFT equations much simpler than those in the formidable many-body theories. And the including of correlation effect let DFT completely beat the HF method, which is also a single-particle theory. Usually, HF algorithms scales as O(N 4 ) 3 2  Chemical Abstract Society, a division of the American Chemical Society. N is the measure of the computing problem size, say the number of electrons of number of basis functions. 3  31  Chapter 1. Introduction to Density Functional Theory and all post-HF methods scale higher than O(N 5 ), while DFT only scales as O(N 3 ). This makes it one of the best theory to approach a linear-scaling [235]. Combined with pseudopotentials and plane-wave basis set, DFT is very suitable to deal with the solids. After the FFT algorithm was invented [236], it is easy to make the DFT calculations of solids linear-scaling. In addition, DFT has also been the best quantum method to study the dynamics of nuclei [237], since it is the most accurate quantum method which we can afford in a dynamic simulation. The relatively lower scaling factor of DFT attracts many researchers and they are now using it to calculate the properties of larger and larger systems, including the most important bio-macromolecules––proteins and DNAs. For a general performance review of density functionals, please see Ref. [238]. Despite the success and advantages mentioned above, DFT also has some drawbacks. The single-determinant nature of most the popular DFT implementations make them difficult to deal with open-shell systems. Or we can say, non-dynamic correlation is hard to be considered in the framework of DFT. And now most of XC functionals contain some empirical parameters which rely on numerical fittings on training sets. In some sense almost all DFT methods are semi-empirical methods. Numerical fittings do help improve the performance of DFT, but also decrease the transferability of the XC functionals. DFT also has some intrinsic difficult problems in its theory, for example, the self-interaction (SI) problem. It is SI which makes the XC potentials behave incorrectly in the long-range. And the consequence is the failure of predicting many response properties, especial the charge-transfer excitation energies. There are two branches in the studies of DFT, one is to make DFT faster; and the other is to make it more accurate. 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Schnol [241] had shown that eigenfunctions of an N particle Schr¨odinger operator vanish exponentially in absolute value if the potential is bounded from below and the corresponding eigenvalue is isolated and has finite multiplicity. Schmincke [242] obtained the same result for certain classes of unbounded potentials, from which Coulomb potentials are excluded. Patil and coworkers [243] had published a series of papers on the asymptotic behavior of wave functions of atomic and few-electron systems. Using a hyperspherical coordinate transform, Dawson [244] derived an exponential decaying asymptotic relation for atomic and molecular wave functions. Handy et al. [245] discussed the long-range exponential behavior of the Hartree-Fock orbitals; Handler et al. [246] worked out the prefactors for the Hartree-Fock orbitals. From a semiclassical analysis, Dzuba et al. [247] obtained a similar result on the long-range behavior of the Hartree-Fock orbitals. Ishida and Ohno [248] also studied the prefactors and remedied the 4  A version of this chapter has been submitted for publication. Y. A. Zhang and Y. A. Wang, Asymptotic Behavior of Finite-System Wave Functions and Physical Interpretation of Kohn-Sham Orbital Energies.  46  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions flaws of Handy’s proof [245]. Significant research efforts have been devoted to understand the long-range behavior of the Dyson orbitals and natural orbitals [249–266]. Similar to what Katriel and Davidson had done [249], Almbladh and von Barth [267] got the asymptotics of wave functions of finite systems and solids. Based on differential inequality techniques, researchers had derived many upper and lower bounds to the asymptotic wave functions and ground-state densities [268–276]. From the viewpoint of density functional theory (DFT) [277], the asymptotic behavior of the ground-state density had been investigated [278]. Starting from the asymptotic behavior of pair density [277–279], Ernzerhof et al. [280] reported some numerical studies of the asymptotic behavior of ground-state wave functions. In this chapter, we will study the asymptotics of wave functions with a general external potential that vanishes (or approaches some constant) at infinity. With the help of a comparison theorem for wave functions in the asymptotic region with different external potentials, we will employ the standard asymptotic analysis method [281] to estimate the long-range behavior of wave functions for a large set of external potentials, especially for those without analytical expressions. On the other hand, it has been well established that the energy of the highest occupied Kohn-Sham orbital (HOKSO) is exactly equal to the negative of the first vertical ionization potential [267, 277, 282–286]. But, this is true only for calculations if the exact exchange-correlation (XC) functional is used. In practise, the HOKSO energy will not be even close to the first vertical ionization potential if an approximate XC functional is employed [287, 288]. Although it is a traditional view [277] that there is no definite physical meaning for the Kohn-Sham (KS) orbital energies other than the HOKSO energy, researchers still have taken great efforts to interpret the KS orbital energies [288–299]. For example, Stowasser and Hoffmann [288] suggested a simple linear scaling relationship between the KS orbital energies and the Hartree-Fock orbital energies. Hamel et al. [291] concluded that the KS orbital energies calculated using the exact KS exchange potential can better approximate experimental vertical ionization potentials than do the Hartree-Fock orbital energies obtained via Koopmans’ theorem [277]. Vargas et al. proposed a Koopmans-like approximation scheme in the KS method and used the frontier KS orbitals to calculate many molecular properties [292]. Bartlett and coworkers [293] and Baerends and coworkers [294–297] suggested an exact or approximate mapping between the KS orbital energies and experimental vertical ionization potentials. Savin et al. [298, 299] showed that very accurate excitation energies can be obtained by taking the difference of the KS orbital energies, if the exact XC potential is 47  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions used in the calculation 5 . In order to infer the exact relationship between vertical ionization potentials and the KS orbital energies, we will reinvestigate the interesting question regarding the differences between the Dyson quasiparticle potential and the KS potential and between the Dyson orbitals and the KS orbitals hereafter.  2.2  Asymptotic Behavior of the Dyson Orbitals  In the following, we will closely follow the derivation presented by Katriel and Davidson [249] to derive the quasiparticle equations for the Dyson orbitals [300], whose asymptotic behavior will be then analyzed accordingly. The non-relativistic electronic Hamiltonian of an N -electron system is of this form: N ˆ N = TˆN + Vˆext H + VˆeeN , (2.1) where the kinetic energy, the external potential energy, and the electronelectron interaction operators are defined like usual 6 : N  TˆN =  N  tˆa = a=1  a=1  1 − ∇2a 2  ,  (2.2)  N N Vˆext =  vext (a) ,  (2.3)  a=1  and  N  VˆeeN = a<b  1 = |ra − rb |  N a<b  1 , rab  (2.4)  respectively. Here, rab is a short-hand notation for the distance between vectors ra and rb . The external potential vext (a) can be some attractive potential that vanishes at infinity, but we will restrict it to be the Coulombic 5  Although the numerical results support Savin et al.’s conclusion, their derivation in Ref. [298] has some flaws. For example, Eq. (3) of Ref. [298] is only true for fi0 . For other quasiparticle amplitudes, coupling terms will appear in the equations [249, 267]: the quasiparticle ` ´equations for different quasiparticle amplitudes differ from one another more than O 1/r5 . Hence, the derivation presented in Ref. [298] fails. 6 Throughout the text, {i, j, k, · · · } are dummy indexes for orbitals or wave functions (states), whereas {a, b} are electron indexes. Unless otherwise noted, all indexes are positive integers, e.g., i 1.  48  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions potential for now. In the next section, we will study the non-Coulombic systems in turn. Separating electron 1 from the remaining (N − 1) electrons, we can rewrite the Hamiltonian in Eq. (2.1) as ˆ ˆN = H ˆ N −1 + h(1) H + vˆee (1) ,  (2.5)  ˆ h(1) = tˆ1 + vext (1) ,  (2.6)  where and  N  vˆee (1) = b>1  1 . r1b  (2.7)  Given the eigenequations for the N -electron and (N − 1)-electron systems 7: ˆ N |ΨN = E N |ΨN , H (2.8) ˆ N −1 |ΨN −1 = E N −1 |ΨN −1 , H i i i  (2.9)  we can define a complete set of the Dyson orbitals {di (1)} [300] for the state |ΨN (not necessarily a ground state): −1 di (1) = ΨN |ΨN i  N −1  ,  (2.10)  where the subscript “N − 1” indicates integrating over the spin and spatial coordinates of the (N − 1) electrons. −1 ˆ N −1 form a complete set, we imSince all eigenfunctions {ΨN } of H i mediately obtain the Carlson-Keller expansion [301]: |ΨN (1, 2, . . . , N ) =  ∞  −1 di (1)|ΨN (2, 3, . . . , N ) . i  (2.11)  i=1  −1 Multiplying ΨN | to both sides of Eq. (2.8) from the left, we get i −1 ˆ N ΨN |H |ΨN i  N −1  −1 = E N ΨN |ΨN i  N −1  .  (2.12)  Substituting Eqs. (2.5)−(2.7), (2.9), and (2.10) into Eq. (2.12), we have N  ˆ h(1)d i (1) +  −1 ΨN i b>1  1 ΨN r1b  N −1  = −Ii di (1) ,  (2.13)  7 Throughout the text, all energy related quantities, including orbital energies and ionization potentials, are listed in the ascending order: Ei Ej , if i < j.  49  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions where the ith vertical ionization potential from the state |ΨN for the N electron system is 8 Ii = EiN −1 − E N . (2.14) −1 −1 Noticing that, for all b, ΨN |r1b |ΨN i simplify Eq. (2.13) to  N −1  are the same, we can further  N −1 ˆ + Ii di (1) = (N − 1)Wi (1) , h(1) + r1  (2.15)  di −1 1 ΨN − ΨN i r1 r12  (2.16)  where Wi (1) =  . N −1  Eq. (2.15) clearly indicates that each Dyson orbital di is coupled to all other Dyson orbitals through Wi in Eq. (2.16). Since we are only interested in the long-range behavior of the Dyson orbitals, we take electron 1 asymptotically away from all other electrons. As a result, we can take advantage of the well-known Laplace expansion for −1 r12 : ∞ 1 r2L = PL (cos θ12 ) , (2.17) r12 rL+1 L=0 1 where PL is the Lth Legendre’s polynomial, whose variable θ12 is the angle between vectors r1 and r2 . Consequently, the second term in Eq. (2.16) can be asymptotically expanded as −1 ΨN i  1 ΨN r12  N −1  ≃ =  ∞  1  rL+1 L=0 1 ∞ dj (1) r1L+1 L=0 j=1  −1 L ΨN |r2 PL (cos θ12 )|ΨN i  N −1  −1 L −1 ΨN |r2 PL (cos θ12 )|ΨN i j  N −1  ,  (2.18) where Eq. (2.11) has been taken into consideration. When L = 0, the corresponding term reduces to di /r1 , which exactly cancels the first term in 8 Throughout the text, all energy related quantities, including orbital energies and ionization potentials, are listed in the ascending order: Ei Ej , if i < j.  50  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions Eq. (2.16). Thus, the expression of Wi approaches its own asymptotic form: Wi (1) ≃ −  ∞  ∞  Aij (1) =  dj (1)Aij (1) ,  (2.19)  j=1  L>0  −1 L −1 ΨN |r2 PL (cos θ12 )|ΨN i j  N −1  .  (2.20)  Aij dj (1) ,  (2.21)  r1L+1  Now, Eq. (2.15) becomes N −1 ˆ h(1) + + Ii di (1) = (1 − N ) r1  ∞ j=1  which is a fully coupled differential equation: each di depends on the exact solutions of all {dj }. The exact solutions of Eq. (2.21) can be found through the iterative self-consistent field (SCF) procedure: making initial guesses for all {dj }, substituting such initial guesses to the right-hand side (RHS) of Eq. (2.21), solving for each di , using the newly obtained {dj } as initial guesses and repeating the above iterative process till full self-consistence is reached. Below, we will set up the SCF procedure accordingly. In the asymptotic region, the differences in the angular components of the Dyson orbitals can be safely ignored, and we will assume the system has a pseudo-spherical symmetry in the long range hereafter. We can then write the radial and angular components of a Dyson orbital as the following 9 : di (1) = Ri (r) ξi (θ, φ) σ ,  (2.22)  where Ri (r), ξi (θ, φ), and σ are radial, angular, and spin components, respectively. Afterwards, we select some direction (θ0 , φ0 ) and define a new (real) orbital which depends only on the radial variable: fi (r) = Ri (r)|ξi (θ0 , φ0 )| ,  (2.23)  from which we can define χi (r) = rfi (r) .  (2.24)  As a result, Eq. (2.21) turns into the equation for χi (r): − 9  1 d2 χi N −1 + Ii + vext (r) + χi = 2 2 dr r  ∞  Bij χj ,  (2.25)  j=1  Hereafter, we will drop the electron index 1 in r1 for simplicity.  51  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions where Bij = (1 − N )Aij .  (2.26)  To appreciate the asymptotic behavior of the Dyson orbital di , we need only to study the homogeneous equation [249]: −  1 d2 χ ˜i N −1 + Ii + vext (r) + χ ˜i = 0 . 2 2 dr r  Suppose the dominate asymptotic term of χ˜i is of this form  (2.27) 10 :  χ ˜i ∼ rβi e−αi r ,  (2.28)  the solutions for the fully coupled equation, Eq. (2.25), will have exponential asymptotic behavior too [249]. For a Coulomb system with vext = −Z/r (at large distance), it is straightforward to show that the solution in Eq. (2.28) has Z −N +1 , (2.29) αi = 2Ii , and βi = αi where Z is the total nuclear charge of the entire system. So we have χ˜i f˜i = ∼ rβi −1 e−αi r . r  (2.30)  Let Lmin be the smallest L for Aij = 0 in Eq. (2.21), we obviously have Mij = Lmin + 1 2. We can then derive the asymptotic version for Eq. (2.25): −  1 d2 χi N −1 + Ii + vext (r) + χi ≃ 2 2 dr r  ∞ j=1  Cij χj , rMij  (2.31)  where Cij is some constant, and Cij /rMij is the leading term in the expression of Eq. (2.26) for Bij . Now pay attention to the right-hand side (RHS) of Eq. (2.31). When i = j, Mii = 2, and the term Cij /r2 will produce a particular solution χpii =  ∞ n=0  10  Here, the notation f (x) ∼ g(x) means lim  c′in rn  x→∞  χ ˜i  (2.32)  f (x) = constant. g(x)  52  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions for the differential equation −  1 d2 χi N −1 Cii + Ii + vext (r) + χi = 2 χi , 2 dr2 r r  (2.33)  where χ ˜i is the homogenous solution for Eq. (2.33) and {c′in } are some constants. When i = j, the non-homogeneous term χj Cij /rMij contribution is governed by its own differential equation −  Cij N −1 1 d2 χi + Ii + vext (r) + χi = M χj , 2 2 dr r r ij  (2.34)  which will produce an approximate particular solution χpij =  Cij Dij χ ˜j = M χ ˜j , M ij (Ii − Ij )r r ij  (2.35)  upon substituting the homogenous solution χ ˜j for χj on the RHS of Eq. (2.34). Since Eq. (2.31) is linear, we readily apply the principle of superposition in the theory of ordinary equation [302]: any solution of Eq. (2.31) can be expressed as a sum of the homogenous solutions with undefined constant coefficients and the particular solutions. We thus propose to solve the coupled system in Eq. (2.31) iteratively. First, we substitute χj := χ ˜j into the RHS of Eq. (2.31) and obtain the expression for the first-round approximate solutions: [1] χi  =  ∞  ′[1]  n=0  Cin rn  [1]  ∞  χ ˜i +  Dij  j=i  rMij  χ ˜j ,  [1]  (2.36) [1]  where Dij = Cij /(Ii − Ij ). Then, we substitute χj := χj into the RHS of Eq. (2.31) and get the second-round approximation for the solutions: [2]  χi =  ∞ n=0  ′[2]  Cin rn  χ ˜i +  [2]  ∞  Dij  j=i  rMij  χ ˜j .  (2.37)  Continue this procedure until self-consistency is reached. We observe that the structure of the finally converged expression for χi is similar to those in Eqs. (2.36) and (2.37), only with varied coefficients: χi =  ∞ n=0  ′ Cin rn  χ ˜i +  ∞ j=i  Dij χ ˜j , rMij  (2.38) 53  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions which yields the expression for fi with the help of Eq. (2.24): lim di (1) = fi =  r→∞  ∞  ′ Cin rn  n=0  f˜i +  ∞ j=i  Dij ˜ fj . rMij  (2.39)  Of course, the self-consistent solutions given in Eq. (2.38) must be exact. Somewhat approximately, Appendix A offers an alternative way to obtain very similar results. Based on Eq. (2.39), we are ready to thoroughly prove an earlier statement made by Katriel and Davidson [249]. Theorem 1. All Dyson orbitals of a Coulomb system have the same exponential tail asymptotically. Proof. If i 2, rewrite the expression for fi in Eq. (2.39) as i−1  fi = j=1  Dij ˜ fj + rMij  ∞ n=0  ′ Cin rn  f˜i +  ∞  Dij ˜ fj . rMij j=i+1  (2.40)  Because of Eq. (2.30), the first sum of finite terms in the above equation behaves asymptotically as i−1 j=1  Dij ˜ f˜1 Di1 ˜ f ∼ . f ∼ 1 j rMi1 rMi1 rMij  (2.41)  Rewriting the third sum of infinite terms in Eq. (2.40) in the following way and then combining it into the second term 11 , we have ∞ n=0  ′ Cin rn  f˜i +  ∞  Dij ˜ f = Mij j r j=i+1  where ′′ Cim  =  ′ Cim  +  ′ Ci0 +  ∞ j=i+1  for m = Mij  ′ Ci1 + r  Dij  ∞ m=2  ′′  Cim rm  f˜i ,  f˜j , f˜i  (2.42)  (2.43)  2. Again, Eq. (2.30) guarantees that, for j > i, f˜j ∼ e−(αj −αi )r → 0 . f˜i  (2.44)  11  Here, we actually change the order of terms in a series and then re-sum it. According to Riemann’s Theorem [303], this might change the convergence property of the series. However, because we are dealing with well-behaved wave functions that can represent real electron densities, not some general functions, we thus assume our manipulation is valid.  54  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions This insures that the absolute magnitude of the summation term in Eq. (2.43) converges to some decreasing function with an upper bound Q0 : ∞  Dij  j=i+1  ∞  f˜j f˜i  Dij  j=i+1  f˜j < Q0 . f˜i  (2.45)  Consequently, we have ′ Ci0 +  ′ Ci1 + r  ∞ m=2  ′′  Cim rm  < =  ∞ n=0 ∞ n=0  ′ Cin + rn ′ Cin rn  +  ∞ m=2  Q0 rm  Q0 . r(r − 1)  (2.46)  It is well known that any converged Laurent-type series will not produce any exponential tail asymptotically [249, 304]; Eq. (2.39) further warrants the convergence of  ∞  n=0  ′ Cin n r .  Therefore, with the aid of Eq. (2.46), we know  that the RHS of Eq. (2.42) must also converges: ′ Ci0  C′ + i1 + r  ∞ m=2  ′′  Cim rm  ′ ˜ f˜i ∼ Ci0 fi ∼ f˜i .  (2.47)  Already Eq. (2.41) shows that the asymptotics of the sum of the second and third terms in Eq. (2.40) can be safely ignored when compared to the first term: f˜1 ≫ f˜i . (2.48) rMi1 As a result, we have f˜1 fi ∼ M . (2.49) r i1 If i = 1, fi only has the last two terms in Eq. (2.40). According to the above analysis, we can easily show that f1 ∼ f˜1 .  (2.50)  From Eqs. (2.49) and (2.50), we conclude that all Dyson orbitals have the same exponential tail. After we fully understand the asymptotic behavior of the Dyson orbitals, which form a complete set, it is then straightforward to analyze asymptotic behaviors of other types of orbitals as well as electron densities. 55  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions  2.3  Extenstion to Non-Coulombic Systems  In this section, we will see what could happen for a system with a nonCoulombic potential that vanishes (or approaches some constant) at infinity. In fact, we often deal with non-Coulombic potentials in our studies of physical problems. For example, we may apply an external electric field that is non-Coulombic in nature to an atomic or molecular system. Also, in the framework of DFT, the XC potential plays a central role but is nonCoulombic [277]. Specifically, we will assume the same form for the Hamiltonian operator defined in Eqs. (2.1)−(2.4), except that the external potential vext in Eq. (2.3) can be non-Coulombic. With this particular choice of the Hamiltonian operator, we can again carry out the same derivation as we have done in Section II till Eq. (2.27), where we have to re-evaluate the asymptotic form of the homogeneous solution χ ˜i for a non-Coulombic vext . In the following, to assist our further discussion, we first state two theorems (without proof) in the Sturm-Liouville theory of ordinary differential equations [305]. No-oscillation Theorem. Consider a Sturm-Liouville differential equation in the form d2 y − 2 + v(x) y = 0 . (2.51) dx If v(x) 0 for x ∈ [p, q], then any non-vannishing solution y has at most one zero in [p, q]. Comparison Theorem. Let v1 (x) and v2 (x) satisfy v1 (x)  v2 (x)  0 , for x  p.  (2.52)  Consider the two differential equations: d2 yn + vn (x)yn = 0 , n = 1, 2 . dx2 If the solutions of Eq. (2.53) are non-negative in [p, +∞) and obey −  yn (x) → 0 as x → +∞ , n = 1, 2 ,  (2.53)  (2.54)  then there exists a positive constant c such that y1 (x)  cy2 (x) , for x  p.  (2.55)  Now, let us conform the hydrogen-like equation, Eq. (2.27), under the influence of a general non-Coulombic external potential to Eq. (2.51) in the theorems above: d2 χ ˜i − + vi (r)χ ˜i = 0 , (2.56) dr2 56  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions where  N −1 . (2.57) r The no-oscillation theorem permits us to choose some interval in which the solution of Eq. (2.56) is single-signed. Noticing that Ii > 0 and vext (r) vanishes at infinity, we can always choose some Ra so that vi (r) is positive when r > Ra . According to the no-oscillation theorem, we know that the solution χ ˜i has either no zero or only one zero in [Ra , +∞). If χ ˜i has no zero, it is single-signed in [Ra , +∞). If χ ˜i has only one zero at Rb in [Ra , +∞), we can definitely say that χ ˜i is single-signed in [Rb , +∞). In both cases, can we find some R = max{Ra , Rb }, so that when r > R, χ ˜i is single-signed, positive or negative. For convenience, we can always choose χ ˜i to be a positive function in [R, +∞), because changing the sign of a wave function does not change its physical nature. The comparison theorem provides us a simple way to find the upper and lower bounds of orbitals under the influence of a general external potential. Several authors suggested some similar theorems [306, 307] in their studies of asymptotic behaviors of bound eigenfunctions [308]. We will carry out similar analysis below. Since vi → 2Ii = αi2 as r → ∞, Eq. (2.56) has an asymptotic form vi (r) = 2 Ii + vext (r) +  −  d2 χ ˜i + αi2 χ ˜i = 0 , dr2  (2.58)  whose solution has an exponential tail like e−αi r . For convenience, we rewrite vi (r) = αi2 + ui (r) ,  (2.59)  where ui (r) vanishes at infinity. In Eq. (2.57), if vext decays faster than a Coulombic potential −Z/r, the contribution of vext can then be safely ignored so that the same derivation in Section II can be completely reproduced only with a minor change in the definition of βi in Eq. (2.29): βi =  1−N . αi  (2.60)  We thus do not need to consider this case further. It is certainly possible that the leading decaying tail in vext decays slower than a Coulombic tail. Although we might not have an analytic expression for vext , we can bracket ui (r) with two known expressions: Zp rp  ui (r)  Zq , 0<q<p<1, rq  (2.61) 57  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions where Zp and Zq are some constants. We can then estimate the asymptotic behavior of χ ˜i with the help of the comparison theorem. Suppose we know Ii for the real system, we can introduce two companion systems whose equations are d2 χ ˜p − + vp (r)χ ˜p = 0 , (2.62) dr2 and d2 χ ˜q − + vq (r)χ ˜q = 0 , (2.63) dr2 where Zp Zq vp (r) = αi2 + p , and vq (r) = αi2 + q . (2.64) r r We can make their eigenvalues identical to αi2 , because we can adjust the values of parameters Zp and Zq (see Appendix B for the detailed reasoning). Since we know from Eqs. (2.61) and (2.64) that vp (r)  vi (r)  vq (r) ,  (2.65)  we can easily apply the comparison theorem to Eqs. (2.56), (2.62), and (2.63) and obtain the inequality cq χ ˜q  χ ˜i  cp χ ˜p .  (2.66)  Using the method of dominant balance [281], we can use the asymptotical forms of χ ˜p and χ ˜q (see Appendix C for derivation) to bracket f˜i : e−λq r cq r where λp =  1−q  −αi r  f˜i  e−λp r cp r  1−p  e−αi r ,  (2.67)  Zp Zq , and λq = . 2(1 − p)αi 2(1 − q)αi  (2.68)  e  Of course, if the tail of the total potential vi (r) has an explicit analytical expression, we can usually find the asymptotic behavior of f˜i by using the method of dominant balance [281]. When the tail of the total potential has no analytical expression, we can simply bracket it with two other potentials whose expressions are known. In some cases, it might be possible to bracket ui (r) with other types of functions than those appear in Eq. (2.61). Then, the non-oscillating and comparison theorems will result different upper- and lower-bound pre-exponential functions for f˜i in Eq. (2.67), but with the same exponential tail e−αi r . Nonetheless, with Eqs. (2.66) and (2.67) in 58  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions place of Eqs. (2.28) and (2.30), respectively, the same derivation in Section II can be continued all the way to the end. Because any constant shift in the external potential does not affect the orbitals or wave functions, the final result can be summarized in a more general theorem than Theorem 1. Theorem 2. All Dyson orbitals of a system with a Coulombic interelectron interaction potential and an external potential (either Coulombic or non-Coulombic) that vanishes or approaches some constant at infinity have the same exponential tail asymptotically. On the other hand, the KS theory in DFT [277] is a special case of a much bigger set of systems that are governed by some local total external potential (either Coulombic or non-Coulombic) but without any inter-electron twobody interaction potential. In other words, the Hamiltonian operator only has the first two terms on the RHS of Eq. (2.1). Consequently, there is no coupling between different Dyson orbitals: they are governed completely by the homogeneous equations, like Eqs. (2.27) and (2.56). We can then simply repeat the same arguments in this Section and gather the conclusion in the following theorem. Theorem 3. For a system without an inter-electron interaction potential but with a local total potential (either Coulombic or non-Coulombic) that vanishes or approaches some constant at infinity, the asymptotic exponential tails of different Dyson orbitals are dictated by distinct vertical ionization potentials.  2.4  Interpretation of the Kohn-Sham Orbital Energies  In the standard KS theory [277], a real, fully interacting Coulomb system is mapped onto a non-interacting system. The spatial KS orbitals {ψi } are the solutions of the eigenequations: tˆ + vext + vxc + vJ ψi (r) = εi ψi (r) ,  (2.69)  where vext is a Coulombic external potential, vxc captures the XC effects, vJ is the traditional Hartree potential, and {εi } are the KS orbital energies 12 . The electron density is then a sum of individual occupied orbital contributions: H  ρ(r) = i=1  oi |ψi (r)|2 ,  (2.70)  12 Throughout the text, all energy related quantities, including orbital energies and ionization potentials, are listed in the ascending order: Ei Ej , if i < j.  59  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions where {oi } are orbital occupation numbers and the upper summation limit “H” is the orbital index of the HOKSO. Similar to Eq. (2.23), the real radial KS orbital can be defined as gi (r) = RiKS (r)|ξiKS (θ0 , φ0 )| ,  (2.71)  such that it satisfies the asymptotic KS equation: −  1 d2 ϕi Z −N +1 + −εi + vKS (r) − ϕi = 0 , 2 2 dr r  (2.72)  where ϕi = rgi (r), and the Coulombic tail has been removed from the local KS potential vKS . This asymptotic KS equation looks very similar to the homogeneous Dyson equation, Eq. (2.27), and can be conformed into the same structure as Eq. (2.56). Hereafter, we will invoke Theorem 3 for the non-interacting KS system and Theorem 1 for the real system in our asymptotic analysis to interpret the physical meaning of the KS orbital energies {εi }. From Eqs. (2.10), (2.22), (2.23), and (2.70), we can express the same (radial) density in two different sums in the asymptotic region: ∞  H  oi gi2 (r) = N  ρ(r) =  fi2 (r) ,  (2.73)  i=1  i=1  where the second sum has an infinite number of terms and the first one is a finite sum. Substituting Eq. (2.39) into Eq. (2.73), after some rearrangement of the terms, we convert the RHS into summations in terms of {f˜i }:   ∞ ∞  H ˜ ˜ fi fj  [w]ij M [s]i f˜i2 + oi gi2 = N , (2.74)  r ij  i=1  i=1  j=i  where [s]i and [w]ij are abbreviated notations of two Laurent series: [s]i =  ∞ k=0  sik , rk  (2.75)  wkij , rk  (2.76)  and [w]ij =  ∞ k=0  60  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions ′ } and {D }. with coefficients {sik } and {wkij }, some simple functions of {Cin ij The RHS of Eq. (2.74) is a sum of converged Laurent series multiplied by exponential functions, whose asymptotic behavior is characterized by the slowest decaying tail13 [249, 304]:   ∞ ∞  ˜i f˜j  f [w]ij M ∼ f˜12 . (2.77) [s] f˜2 + N  i i r ij  i=1  j=i  Similarly, gH (r) is the only surviving orbital on the left-hand side (LHS) of Eq. (2.74) asymptotically, H  i=1  oi gi2 ∼ gH2 .  (2.78)  Then, Eqs. (2.29), (2.30), (2.49), (2.50), (2.74), (2.77), and (2.78) guarantee that gH (r) ∼ f1 (r) ∼ f˜1 (r) ∼ rβ1 −1 e−α1 r , (2.79)  Now, turn to Eq. (2.72), if vKS (r) decays slower than the Coulombic tail, the analysis presented in Section III will destroy the balance shown in Eq. (2.79). We thus have to conclude that vKS (r) must decay faster than the Coulombic tail. Since all KS orbitals obey the same differential equation (with the same vKS ), their asymptotic leading terms should have the same structure: gi (r) ∼ rβi −1 e−αi r , (2.80) where αi and βi are determined by their corresponding orbital energies εi : √ Z −N +1 αi = −2εi , and βi = . (2.81) αi  Even though Eq. (2.81) is virtually identical to Eq. (2.29), we cannot jump to claim something like [293–297]14 : 13  εH+1−i = −Ii ,  (2.82)  One might argue that the leading term of an infinite series does not necessarily represent the asymptotic behavior of the series from a pure mathematical point of view. This issue had been raised by Ahlrichs [251] and answered by Parr and Levy [252] and by Kartriel and Davidson [249]. On the other hand, it is well known that any converged Laurent-type series will not produce any exponential tail asymptotically [249, 304]. We thus can use the slowest decaying tail in the asymptotic analysis. If this were untrue, the validity of Eq. (2.83) would have been in doubt. Practically, only a finite number of wave functions or basis functions can be used to carry out quantum mechanical calculations. Under such a condition, the difficulty posed by Ahlrichs [251] does not exist. 14 Throughout the text, all energy related quantities, including orbital energies and ionization potentials, are listed in the ascending order: Ei Ej , if i < j.  61  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions right away. However, according to Eqs. (2.30), (2.80), and (2.81), both sides of Eq. (2.79) must have identical exponential tail, yielding this famous result [267, 277, 282–286]: εH = −I1 . (2.83)  We name the above asymptotic analysis “exponential tails matching,” which can be propagated further after we have just matched the tails of 2 (r) and f˜2 . Removing the contribution of the HOKSO to the density gN 1 from both sides of Eq. (2.74), we get   ∞ ∞  H−1 ˜ ˜ fi fj  [w]ij M [s]i f˜i2 + oi gi2 = N − oH gH2 . (2.84)  r ij  i=1  i=1  j=i  Although we do not exactly know which terms will be left on the RHS, we will argue that the asymptotic leading term of the remaining terms on the RHS of Eq. (2.84) cannot be a cross term involving f˜i f˜j , whose asymptotic leading term can be shown to be [w]ij  f˜i f˜j ∼ rβij e−αij r , rMij  (2.85)  where αij = (αi + αj ) , and βij = (βi + βj − Mij )  (2.86)  for i = j. Obviously, αij and βij can never satisfy Eq. (2.81), which is obeyed only by {αi , βi } coming from some KS orbital. Hence, all the cross terms can be excluded from the process of exponential tails matching, H−1  oi gi2  =N  i=1  ∞  [s]i f˜i2 ,  (2.87)  i=k 2  and only some surviving square term (not necessarily f˜22 ) in Eq. (2.84) can match the leading term on the LHS, gH2 −1 : gH2 −1 ∼ f˜k2 , k  2.  (2.88)  Similar to Eq. (2.83), we will have εH−1 = −Ik , k  2,  (2.89)  which is to say that all those slower decaying cross terms must be completely summed into gH2 :   αij < αk k−1  ˜ ˜ fi fj  [w]ij M [s]i f˜i2 + gH2 ∼ . (2.90)  r ij  i=1  j=i  62  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions We continue the procedure of the exponential tails matching until we exhaust all occupied KS orbitals. Our conclusion becomes clear. Theorem 4. Every Kohn-Sham orbital energy exactly equals the negative of some vertical ionization potential: εH+1−i = −Ik ,  1  i  k,  (2.91)  where both orbital energies and vertical ionization potentials are arranged in ascending order 15 . In practise, however, we might be able to do better than what Theorem 4 tells us. To see this, we first simplify the RHS of Eq. (2.74) after removing all those cross terms from exponential tails matching: H  oi gi2 = N i=1  ∞  [s]i f˜i2 .  (2.92)  i=1  On the RHS of the above equation, we can usually partition the entire set of {f˜i } according to their relative weights (magnitudes) in the summation:  2 , 1 i lvalence ;  Primary : 1 < N [s]i f˜i2 (2.93)  2 ˜ Secondary : 0 < N [s]i fi ≪ 1 , i > lvalence .  Spectroscopically, members of the primary set correspond to the major (sharp) peaks in the photoelectron spectrum (PES), whereas members of the secondary set are mainly in the shake-up (correlation) domain [309]. Most of the major peaks locate in the lower energy or valence region of the PES; the shake-up features are normally much weaker in oscillation strength and are lying in the higher energy (core electron) area [309]. In the language of configuration interaction, the primary peaks are characterized by singleelectron hole states, whereas the shake-up features are composed of many n-particle-(n + 1)-hole configurations [309]. When we perform the exponential tails matching procedure for all occupied KS orbitals, one by one from the HOKSO to the core KS orbitals, individual members of the primary set in Eq. (2.93) will take turns to be summed into pertinent KS orbitals. Because of the large relative magni2 tudes of the primary {f˜i }, some significant amount of residual f˜i+1 term 2 ˜ will be left after the complete fi term along with some small contributions from higher-lying faster-decaying {f˜k |k > i} have already been removed by 15 Throughout the text, all energy related quantities, including orbital energies and ionization potentials, are listed in the ascending order: Ei Ej , if i < j.  63  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions 2 the gi2 term on the LHS of Eq. (2.92). The residual f˜i+1 term will then be 2 summed into and removed by gi+1 , if the exponential tails matching is taken to the next iteration. As a result, there will most likely be an exact line-up between the orbitals on both sides of Eq. (2.92) in the valence region:  gH+1−i ∼ f˜i , 1  i  lvalence ,  (2.94)  which immediately spells out Eq. (2.82). Unfortunately, this simply mapping will be destroyed in the core electron region of the PES where a single board feature may envelop many shake-up peaks, among which we cannot know exactly which residual f˜k2 term will survive the tails matching process a priori. Accordingly, Eq. (2.91) becomes true. Despite the above arguments are not fully built upon rigorous mathematical deduction, we now have a much succinct physical understanding. Conjecture. Every Kohn-Sham orbital energy exactly equals the negative of some vertical ionization potential 16 :   Valence Region : εH+1−i = −Ii , 1 i lvalence ; (2.95)  Core Region : εH+1−i = −Ik , k i > lvalence .  On the other hand, based on some approximate theoretical arguments, Baerends and coworkers suggested that orbital energies got from the exact KS potential is a good approximation to the vertical ionization potentials [294–297], namely, Eq. (2.82) or the first half of Eq. (2.95). Subject to an adiabatic approximation, Bartlett et al. supported the above assessment in the framework of time-dependent DFT [293]. It is also not easy to analyze how accurate Eq. (2.82) really is in general, given some drastic approximations invoked in the theoretical reasoning leading to it [293–297]. As a result, there is very limited means to improve its accuracy. It is quite encouraging, however, that existing numerical evidence [293–297] only partially embraces Eq. (2.82), but fully reinforces Eq. (2.95). It is absolutely necessary that extensive numerical tests must be performed to verify the validity of our conjecture. Nonetheless, Theorem 4 (or its more refined conjecture) is a universal statement, which supersedes Eq. (2.82).  2.5  Conclusions  In pursuing the quest of interpreting the physical meaning of the occupied Kohn-Sham orbital energies, we have gone through rigorous mathematical 16 Throughout the text, all energy related quantities, including orbital energies and ionization potentials, are listed in the ascending order: Ei Ej , if i < j.  64  Chapter 2. Asymptotic Behavior of Finite-System Wave Functions analysis of asymptotic behaviors of wave functions of finite systems whose external potentials vanish or approach some constant at infinity. During this journey, we have proven four theorems and proposed one conjecture. 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Dissertation (Indiana University, Bloomington, IN, USA, 1995).  70  Chapter 3  Conventional and Unconventional Density Variations in Density Functional Theory 17  3.1  Background  Published nearly 20 years ago, the “unconventional density variation” (UDV) paper of Perdew and Levy [310] is quite interesting, because it is inconsistent to the functional differentiability theorem by Lieb [311–316]. Lieb’s Theorem 3.10 [313] guarantees such UDV’s from any ground-state (GS) densities {n0 (r)} to be included, as long as all densities {n(r)} considered belong to the convex set JN ≡ {n(r) | n(r) ≥ 0, n(r) = N, n(r) ∈ H1 (R3 )}, a subset of the Banach space Y ≡ L3 ∩ L1 , where R3 denotes the three-dimensional coordinate real space, H1 (R3 ) = {g | g ∈ L2 (R3 ), ∇g ∈ L2 (R3 )} is a Sobolev space, and Lb (R3 ) is a Banach space with a norm 1/b f b = |f (r)|b < +∞, 0 < b < +∞, respectively. It was further showed by Lieb [313] that the external potentials {v(r)}, corresponding to the GS densities {n0 (r)} ⊂ JN ⊂ Y, belong to the dual space of Y, which again is a Banach space Y ∗ ≡ L3/2 + L∞ ≡ {v(r) | v(r) = v3/2 (r) + v∞ (r), v3/2 (r) ∈ L3/2 , v∞ (r) ∈ L∞ }. L∞ is the Banach space of bounded functions, with the norm f ∞ = ess sup |f (r)| < ∞, where the essential supremum is the smallest upper bound of |f (r)| almost everywhere [317]. In particular, the GS densities of some external potentials in Y ∗ are everywhere positive due to the Unique Continuation Theorem [313, 318]. Normally, a family of density variations (DV’s) about a GS density n0 (r) for a system with a fixed number of electrons in Hilbert space can be defined 17  A version of this chapter has been submitted for publication. Y. A. Zhang and Y. A. Wang, “Are the Unconventional Density Variations Really Unconventional?”.  71  Chapter 3. Density Variations in DFT as δn(r) = n(r) − n0 (r) = yf (r) ,  (3.1)  18  where y is a small positive DV parameter (y ≪ 1) and f (r) is continuous and differentiable with a null normalization f (r) = 0. In the entire set of DV’s {δn(r)}, the UDV’s {δnu (r)} were defined as those DV’s violate the following condition [310]: |f (r)|  n0 (r)  < M < +∞ ,  (3.2)  for a finite positive number M 19 . It was further argued [310] that energy variations of {δnu (r)} from the GS energy will be of order y, not the conventional y 2 behavior, δEv [n0 , δnu ] = Ev [n0 (r) + δnu (r)] − Ev [n0 (r)] ∝ O(y) .  (3.3)  A noninteracting two-electron Hydrogen anion was used as an illustration [310] to confirm the above conclusion and to justify the “derivative discontinuity of the energy” [319–323]. If this assessment [310] were correct, the entire theoretical foundation of modern density-functional theory (DFT) [311–316, 319–327] would have to be reshaped. In the following, we will show that such an assessment [310] is incorrect. (For a direct comparison, we will adopt a very similar notational system to that of Ref. [310].)  3.2  The Model System  For a noninteracting two-electron Hydrogen anion, its exact GS energy density functional is known to be a sum of the von Weizs¨acker (vW) functional [325, 326] and the nuclear-electron attraction energy density functional: Ev [n(r)] = TvW [n(r)] + Vne [n(r)] 1 n(r) − ∇2 n(r) − = 2 =  |∇n(r)|2 8n(r)  −  n(r) r  .  n(r) r (3.4)  18  Without losing generality, y will only take very small numerical values (y → 0+) unless otherwise noted hereafter. 19 Hereafter, capital italic Roman characters, such as M , P , and Q, will be used to denote arbitrary finite positive numbers unless otherwise noted.  72  Ground-State Energy Functional E v [n] (a. u.)  Chapter 3. Density Variations in DFT -0.2  -0.4  conventional variation unconventional variation  -0.6  -0.8  -1.0 0.0  0.2  0.4  0.6  0.8  1.0  Density Parameter y Figure 3.1: Exact reproduction of Fig. 1 of Ref. [310] for two noninteracting electrons bound to the Hydrogen-atom Coulomb potential v(r) = −1/r. The detailed definitions of the conventional and unconventional density variations are shown in Eqs. (9) and (10). The energy unit is in Hartrees.  The Euler-Lagrange equation of the above explicit energy density functional is conveniently written as an eigenvalue equation [310]: 1 1 − ∇2 − 2 r  n(r) = µ n(r) .  (3.5)  In Ref. [310], Eq. (3.5) was equated to the Schr¨odinger equation for the Hydrogen atom: 1 1 − ∇2 − ψi (r) = ǫi ψi (r) , (3.6) 2 r whose first two solutions, ψ1s (r) = π −1/2 e−r  and ψ2s (r) = (8π)−1/2 1 −  r −r/2 e , 2  (3.7)  were claimed to be identical to the first two solutions of Eq. (3.5). This association is obviously flawed [328]: ψ2s (r) is not a solution of Eq. (3.5), 73  Chapter 3. Density Variations in DFT 1.5  1+Ev[nu(r,y)] 1.65  1.0  Energy (10  -12  a. u.)  64.06 y  0.5  0.0 0.0  1.0  2.0  3.0  4.0  5.0  -9  Density Parameter y (10 ) Figure 3.2: The fine detail of the “unconventional density variation” energy curve of Fig. 1 for very small values of the density variation parameter y (from 0 to 5.0 × 10−9 ). The dotted line is the fitted curve, whereas the solid line is the exact energy curve (shifted upward by 1 Hartree for a better view).  because n(r) has to be nonnegative everywhere, but ψ2s (r) is not so. Wang and Carter [325] have discussed this issue related to the n(r)-formulation of Eq. (3.5) and used the correct ϕ(r)-formulation: 1 1 − ∇2 − 2 r  ϕ(r) = µϕ(r) ,  (3.8)  where ϕ2 (r) = n(r). More generally, one should use the complex-ϕ(r)formulation normally employed for inhomogeneous Bose-condensed fluids [329], in which |ϕ(r)|2 = n(r). From the two solutions in Eq. (3.7), two different DV paths were then introduced [310]: the UDV path, nu (r, y) = (1 − y) n1s (r) + y n2s (r) ,  (3.9)  and the conventional density variation (CDV) path, nc (r, y) = (1 − y)A (1 − αr)2 n1s (r) + yBe(1−β)r n2s (r) ,  (3.10) 74  Chapter 3. Density Variations in DFT 3.0  2.5  1+Ev[n1s(r,y,Z=0.25)] 1.26  2.0  Energy (10  -12  a. u.)  1.60 y  1.5  1.0  0.5  0.0 0.0  1.0  2.0  3.0  4.0  5.0  -10  Density Parameter y (10 ) Figure 3.3: The fine detail of the energy density functional curve of Eq. (3.11) for Z = 0.25 and very small values of the density variation parameter y (from 0 to 5.0 × 10−10 ). The dotted line is the fitted curve, whereas the solid line is the exact energy curve (shifted upward by 1 Hartree for a better view).  where α = y(2y − 1)/2, β = (2y − 1)(y − 1) + 1, A = (1 − 3α + 3α2 )−1 , B = β 5 /(β 2 − 3β + 3), n1s (r) = 2 |ψ1s (r)|2 , and n2s (r) = 2 |ψ2s (r)|2 , respectively. In Fig. 3.1, we have used Mathematica [330] and reproduced the same Fig. 1 of Ref. [310], but we have not observed the energy change behavior as stated in Eq. (3.3) for the UDV path [310]. The UDV energy curve of Fig. 3.1 for very small values of y is zoomed in to very fine details in Fig. 3.2, which clearly illustrates that this UDV curve has a zero slope when y goes to zero, even though in Fig. 3.1, on a larger scale, the same UDV curve appears to vary linearly with small values of y. In fact, the exact energy curve can be accurately fitted with a nonlinear power function: . Ev [nu (r, y)] = −1 + 64.06 y 1.65 , which immediately invalidates the general correctness of Eq. (3.3).  75  Chapter 3. Density Variations in DFT 7.0  1+Ev[n1s(r,y,Z=0.5)]  6.0  1.83  Energy (10  -12  a. u.)  2.02 y 5.0  4.0  3.0  2.0  1.0  0.0 0.0  1.0  2.0  3.0  4.0  5.0  -7  Density Parameter y (10 ) Figure 3.4: The fine detail of the energy density functional curve of Eq. (3.11) for Z = 0.5 and very small values of the density variation parameter y (from 0 to 5.0 × 10−7 ). The dotted line is the fitted curve, whereas the solid line is the exact energy curve (shifted upward by 1 Hartree for a better view).  3.3  Numerical Tests in Hilbert Space  To further examine the validity of the UDV condition in Eq. (3.2), we test the following family of DV’s: n1s (r, y, Z) = (1 − y) n1s (r) + y n1s (r, Z) ,  (3.11)  where n1s (r, Z) = 2 |ψ1s (r, Z)|2 is the GS density of a noninteracting 2electron Hydrogen-like atom with nuclear charge Z (Z > 0) and GS wave function, 1/2 −Zr ψ1s (r, Z) = Z 3 /π e . (3.12) Through controlling the value of Z, we can introduce either an UDV’s or a CDV’s in Eq. (3.11). For example, one can obtain the following expressions from Eqs. (3.1),  76  Chapter 3. Density Variations in DFT 7.0  1+Ev[n1s(r,y,Z=0.67)]  6.0  2.00  Energy (10  -10  a. u.)  0.25 y 5.0  4.0  3.0  2.0  1.0  0.0 0.0  1.0  2.0  3.0  4.0  5.0  -5  Density Parameter y (10 ) Figure 3.5: The fine detail of the energy density functional curve of Eq. (3.11) for Z = 0.67 and very small values of the density variation parameter y (from 0 to 5.0 × 10−5 ). The dotted line is the fitted curve, whereas the solid line is the exact energy curve (shifted upward by 1 Hartree for a better view).  (3.2), (3.7), and (3.11): δn1s (r, y, Z) = n1s (r, y, Z) − n1s (r)  = y [n1s (r, Z) − n1s (r)] = y f1s (r, Z) ,  |f1s (r, Z)| n1s (r)  = =  (3.13)  |n1s (r, Z) − n1s (r)| n1s (r)  −r  2/π e  Z 3 e−2(Z−1)r − 1 ,  (3.14)  and δn1s (r, y, Z) n1s (r, Z) =y − 1 = y Z 3 e−2(Z−1)r − 1 . n1s (r) n1s (r) (i) When Z > 1, n1s (r, Z) decays faster than n1s (r) and  (3.15)  n1s (r). Eq. (3.14) 77  Chapter 3. Density Variations in DFT goes to zero and Eq. (3.15) goes to y asymptotically, lim  r→∞  |f1s (r, Z)| n1s (r)  = 0,  lim  r→∞  δn1s (r, y, Z) =y . n1s (r)  This DV path is conventional. Similarly, (ii) when 1 > Z has a CDV path: lim  r→∞  |f1s (r, Z)|  (iii) When lim  r→∞  n1s (r)  1 2  < M < +∞ and  (3.16) 1 2,  δn1s (r, y, Z) →∞. r→∞ n1s (r) lim  one again  (3.17)  > Z > 0, one instead has an UDV path:  |f1s (r, Z)| n1s (r)  → +∞ and  δn1s (r, y, Z) →∞. r→∞ n1s (r) lim  (3.18)  Figs. 3.3––3.5 exhibit the energy functional curves as functions of Z for very small values of y. It is clear that for all cases, the energy density functional does not vary linearly with respect to y: Ev [n1s (r, y, Z)] ≈ −1 + a y b ,  (3.19)  where the exponent b is greater than 1. On the other hand, it is intriguing to study the consequences if Z = 0, even though it is formally not allowed for number-conserving densities within JN . Nonetheless, it can be thought to be a general case in Fock space [327], in which the densities belong to the convex set J ≡ {n(r) | n(r) ∈ JN , N ∈ R+ } ≡ {n(r) | n(r) 0, n(r) ∈ L1 , ∇ n(r) ∈ L2} ⊂ Y, where R+ denotes the space of nonnegative real numbers. It is then straightforward to deduce the following expressions for Z = 0 from Eqs. (3.11) to (3.15): n1s (r, y, Z = 0) = (1 − y) n1s (r) ,  (3.20)  n1s (r, Z = 0) = 0 ,  (3.21)  δn1s (r, y, Z = 0) = n1s (r, y, Z = 0) − n1s (r)  = −yn1s (r) = y f1s (r, Z = 0) ,  |f1s (r, Z = 0)| n1s (r)  = =  (3.22)  |−n1s (r)| n1s (r)  n1s (r) =  2/π e−r ,  (3.23) 78  Chapter 3. Density Variations in DFT and  −yn1s (r) δn1s (r, y, Z = 0) = = −y . n1s (r) n1s (r)  (3.24)  Obviously, when Z = 0, we obtain a CDV path, |f1s (r, Z = 0)|  |δn1s (r, y, Z = 0)| = y < M < +∞ , r→∞ n1s (r) n1s (r) (3.25) on which the energy density functional varies linearly with respect to y: lim  = 0 and  Ev [n1s (r, y, Z = 0)] ≈ −1 + y .  (3.26)  Figs. 3.3––3.5 and the above analysis indicate that the energy density functional has different power-law dependence on y as Z passes through the UDV region to the CDV region:  CDV : b = 1 , if Z = 0 ;    UDV : b → 1+ , if Z → 0+ ; Ev [n1s (r, y, Z)] = −1 + a y b 1 ≥ Z > 0 ; UDV : 2 > b > 1 , if   2  CDV : b = 2 , if Z > 12 . (3.27)  3.4  Theoretical Analysis  The above result is in direct contradiction to the claims of Perdew and Levy [310]. After a careful reading of the argument as presented in section IV on Page 6268 of Ref. [310], we learned that Eq. (33) of Ref. [310] is incorrect. To expose the mistake, let us reason along the lines of Ref. [310] as follows. For a given GS density n0 (r), there is a GS wave function Ψ0 (r1 σ1 , . . . , rN σN ) with spatial and spin coordinates {ri σi } uniquely mapped to it: N  n0 (r) =  Ψ0 (r1 σ1 , . . . , rN σN ) i=1  = N |Ψ0 |2  N −1  ,  δ(r − ri ) Ψ0 (r1 σ1 , . . . , rN σN ) (3.28)  where the subscript “N − 1” indicates the integration of |Ψ0 |2 to be carried out only for (N − 1) spatial coordinates and for all N spin coordinates. The  79  Chapter 3. Density Variations in DFT constrained search will associate a normalized, antisymmetric wave-function variation yχ(r1 σ1 , · · · , rN σN ) with δn(r) = yf (r), such that f (r) = N (Ψ∗0 χ + χ∗ Ψ0 ) = N  (Ψ∗0 χ  ∗  + χ Ψ0 )  N −1 N −1  + yN |χ|2  + y nχ (r) ,  N −1  (3.29)  where nχ (r) is the density of wave function χ(r1 σ1 , · · · , rN σN ). It should be emphasized that the second term on the right-hand side (RHS) of Eq. (3.29) must not be ignored, because yχ(r1 σ1 , · · · , rN σN ) might be bigger in magnitude than Ψ0 (r1 σ1 , . . . , rN σN ) for some regions in space. With Eq. (3.29) in place of the erroneous Eq. (33) of Ref. [310], one will derive the correct version of Eq. (39) of Ref. [310]: |f (r)|  n0 (r)  2 nχ (r) +  y nχ (r) n0 (r)  ,  (3.30)  whose second term on the RHS can be unbound for UDV’s in the asymptotic region in space for a finite y, lim  r→∞  y nχ (r) n0 (r)  → +∞ .  (3.31)  Consequently, there should be no condition like Eq. (3.2) imposed on any DV, and UDV’s are not abnormal. We can rigorously prove our argument based on the analysis of the Gˆ ateaux functional derivative [311–316]. Formally speaking, the variation of a density functional H[n(r)] at a GS density n0 (r) due to a DV δn(r) can be expressed as δH[n0 , δn] = H[n0 + δn] − H[n0 ] = h([n0 ]; r) δn(r) + ξ[n0 , δn] ,  (3.32)  where h([n0 ]; r) is the Gˆ ateaux functional derivative of H[n(r)] at the GS density n0 (r): δH[n(r)] = h([n0 ]; r) , (3.33) δn(r) n=n0 if h([n0 ]; r) is a single-valued bounded function without any dependence of δn(r) and the residual satisfies lim  y→0  ξ[n0 , yδn] =0. y  (3.34)  80  Chapter 3. Density Variations in DFT Applying this formalism to the energy density functional of Eq. (3.4), we get δEv [n0 , δn] = Ev [n0 +δn]−Ev [n0 ] = ev ([n0 ]; r) δn(r) +ξvW [n0 , δn] , (3.35) where the Gˆ ateaux functional derivative of Ev [n(r)] at n0 (r) and the residual of the vW functional are ev ([n0 ]; r) =  δEv [n(r)] δn(r)  tvW ([n0 ]; r) = =  n=n0  1 δTvW [n(r)] =− + r δn(r)  δTvW [n(r)] δn(r) 1 8  ∇n0 (r) n0 (r)  n=n0 2  −2  =−  ,  (3.36)  n=n0  1 ∇2 n0 (r) 2 n0 (r)  ∇2n0 (r) n0 (r)  ,  (3.37)  and y2 ξvW [n0 , yδn] = 8  ∇n0 (r) 1 δn(r) − ∇δn(r) n(r, y) n0 (r)  2  ,  (3.38)  respectively, for arbitrary δn(r) with n0 (r) + yδn(r) = n(r, y) ∈ JN . Eqs. (3.36) and (3.37) evidently indicate that ev ([n0 ]; r) is independent of δn(r). It yet remains to be proven that Eq. (3.38) is at most O(y b ), where b > 1, so that Eq. (3.34) is satisfied for arbitrary δn(r) with n0 (r) + yδn(r) = n(r, y) ∈ JN . Because of the general inequality for two complex or vector entities p and q, |p − q|2 2 |p|2 + |q|2 , (3.39)  Eq. (3.38) also obeys the following inequality, ξvW [n0 , yδn]  y2 4  w(r)δn(r) n(r, y)  +  y2 4  where w(r) = δn(r)  ∇n0 (r) n0 (r)  |∇δn(r)|2 n(r, y)  ,  (3.40)  2  .  (3.41)  Due to the fact that n0 (r) is a GS density for a Coulomb potential v(r) = −1/r ∈ Y ∗ , one can immediately infer the following condition: ∇n0 (r) n0 (r)  2  = u(r) = u3/2 (r) + u∞ (r) ∈ Y ∗ ,  (3.42) 81  Chapter 3. Density Variations in DFT from Eqs. (3.36) and (3.37). Utilizing the famous H¨ older inequality in functional analysis [317, 318]: fg  f  1  c  · g  d  ,  (3.43)  for 1/c + 1/d = 1, f ∈ Lc , and g ∈ Ld , we can then straightforwardly conclude that for arbitrary δn(r) with n0 (r) + yδn(r) = n(r, y) ∈ JN , the integral of |w(r)| in Eq. (3.41) is also finite, w(r)  1  =  |w(r)| =  |δn(r)| ·  =  δn(r)u(r)  δn(r)  1  + δn(r)u3/2 (r)  1 · u∞ (r)  3  2  1  δn(r)u∞ (r)  δn(r)  ∇n0 (r) n0 (r)  · u3/2 (r)  1  ∞+ 3/2  < R < +∞ .  (3.44)  Here, we have used the facts that δn(r) ∈ Y ≡ L3 ∩ L1 and u(r) ∈ Y ∗ . Now, let us consider the following inequality, yw(r)δn(r) n(r, y)  = | s(r, y)w(r) |  |s(r, y)w(r)| < S(y)R ,  (3.45)  where the function s(r, y) is defined as s(r, y) =  yδn(r) , n(r, y)  (3.46)  and S(y), a function of y, is the upper bound of s(r, y) for all space of r, |s(r, y)|  S(y) < +∞ .  (3.47)  If |δn(r)|/n0 (r) is bounded everywhere by a finite constant W , we easily ascertain that S(y) is O(y), |s(r, y)|  S(y) =  yW ∝ O(y) < +∞ , 1 − yW  (3.48)  for y less than W −1 . If |δn(r)|/n0 (r) is unbounded asymptotically, we can partition the space into two regions: Region I where |δn(r)|/n0 (r) is bounded by X(y) and 82  Chapter 3. Density Variations in DFT Region II otherwise. As y approaches zero, we can choose X(y), such that y is less than X −1 (y) and lim yX(y) = 0 ,  (3.49)  y→0  and Region II is completely outside of a sphere of radius rb (y) δn(r) n0 (r)  > X(y) Region II  δn(r) n0 (r)  20 ,  .  (3.50)  r rb (y)  From the above equation, it is clear that as y goes to zero, both X(y) and rb (y), as functions of y, will diverge, 1 1 = lim =0. y→0 X(y) y→0 rb (y) lim  (3.51)  We then can split the integrals of Eq. (3.45) into two pieces, according to the space partition: | s(r, y)w(r) | =  s(r, y)w(r) |s(r, y)(r)|  <  + s(r, y)w(r)  Region II  + |s(r, y)w(r)|  Region II  Region I  Region I  yRX(y) + |s(r, y)w(r)| 1 − yX(y)  r>rb (y)  ,  (3.52)  where the subscripts outside of the integrals signify the integration domains of the integrals. At y → 0, the limit of the second term of the last expression of Eq. (3.52) is zero due to the upper bound shown in Eq. (3.45) and the vanishing integration volume, lim |s(r, y)w(r)|  y→0  r>rb (y)  = lim |s(r, y)w(r)| rb →∞  r>rb (y)  =0.  (3.53)  We thus conclude that if |δn(r)|/n0 (r) is bounded everywhere, |s(r, y)w(r)| is O(y) according to Eqs. (3.45) and (3.48); otherwise, |s(r, y)w(r)| is O(y k ) with k > 0 according to Eqs. (3.45), (3.49), (3.51) to (3.53). In other words, we have lim |s(r, y)w(r)| = 0 .  y→0  (3.54)  20 Of course, part of Region I might be outside of this sphere, but Region II is exclusively outside of this sphere.  83  Chapter 3. Density Variations in DFT Hence, the first term on the RHS of Eq. (3.40) is finite and O(y b ) with b > 1 as y → 0, y2 4  w(r)δn(r) n(r, y)  <  y |s(r, y)w(r)| ∝ O(y b ) < +∞ . 4  (3.55)  Moreover, for an arbitrary δn(r) with n(r, 1) = n0 (r) + δn(r) = n(r) ∈ JN , one can derive the following inequality based on Eq. (3.39), |∇δn(r)|2 = |∇n(r) − ∇n0 (r)|2  2 |∇n(r)|2 + |∇n0 (r)|2  .  (3.56)  Employing Eq. (3.56) for the second term on the RHS of Eq. (3.40), one gets y2 4  |∇δn(r)|2 n(r, y)  y2 2  |∇n(r)|2 n(r, y)  +  |∇n0 (r)|2 n(r, y)  y2 2  .  (3.57)  The first term on the RHS of Eq. (3.57) obviously satisfies the following inequality, y2 2  |∇n(r)|2 n(r, y)  =  y 2  |∇n(r)|2 yn(r) n(r) n(r, y)  yM (y) 2 <  |∇n(r)|2 n(r)  yM (y)P < +∞ , 2  (3.58)  where the finite upper bound P is from the integral involving n(r) simply because n(r) ∈ JN and the upper bound M (y) is a function of y, yn0 (r) + yδn(r) yn(r) = n(r, y) n0 (r) + yδn(r)  M (y) < +∞ .  (3.59)  Following the same arguments presented from Eq. (3.45) to Eq. (3.54), we can readily conclude that if [n(r)/n(r, y)] is bounded everywhere, y |∇n(r)|2/n(r, y) is O(y); otherwise, y |∇n(r)|2/n(r, y) is O(y k ) with k > 0 as y → 0. In general, we have lim  y→0  y |∇n(r)|2 n(r, y)  =0.  (3.60)  84  Chapter 3. Density Variations in DFT Thus, the first term on the RHS of Eq. (3.57) is finite and O(y b ) with b > 1 as y → 0, y 2 |∇n(r)|2 ∝ O(y b ) < +∞ . (3.61) 2 n(r, y) Similarly, one can work with the second term on the RHS of Eq. (3.57) and get another inequality, y2 2  |∇n0 (r)|2 n(r, y)  =  y 2  |∇n0 (r)|2 yn0 (r) n0 (r) n(r, y)  yN (y) 2 <  |∇n0 (r)|2 n0 (r)  yN (y)Q < +∞ , 2  (3.62)  where the finite upper bound Q is from the integral involving n0 (r) because n0 (r) ∈ JN and the upper bound N (y) is a function of y, yn(r) − yδn(r) yn0 (r) = n(r, y) n0 (r) + yδn(r)  M (y) + S(y) = N (y) < +∞ .  (3.63)  Following the same arguments presented from Eq. (3.45) to Eq. (3.54), we can also conclude that if [n0 (r)/n(r, y)] is bounded everywhere, y |∇n0 (r)|2/n(r, y) is O(y); otherwise, y |∇n0 (r)|2/n(r, y) is O(y k ) with k > 0 as y → 0. In general, we have lim  y→0  y |∇n0 (r)|2 n(r, y)  =0.  (3.64)  Thus, the second term on the RHS of Eq. (3.57) is finite and O(y b ) with b > 1 as y → 0, y 2 |∇n0 (r)|2 ∝ O(y b ) < +∞ . (3.65) 2 n(r, y) Combining Eqs. (3.57), (3.61), and (3.65), we gather that the second term on the RHS of Eq. (3.40) is finite and O(y b ) with b > 1 as y → 0, 0<  y2 4  |∇δn(r)|2 n(r, y)  ∝ O(y b ) < +∞ ,  (3.66)  for arbitrary δn(r) with n0 (r) + yδn(r) = n(r, y) ∈ JN . 85  Chapter 3. Density Variations in DFT Consequently, with the aid of Eqs. (3.40), (3.55), and (3.66), Eq. (3.34) is satisfied. As a result, the energy variation manifests an O(y b ) behavior (with b > 1), δEv [n0 , δn] = Ev [n0 + δn] − Ev [n0 ] =  ev ([n0 ]; r) δn(r) + ξvW [n0 , δn]  = ξvW [n0 , δn] ∝ O(y b ) ,  (3.67)  where the term involving the Gˆ ateaux functional derivative is zero due to the Euler-Lagrange equation at the GS density n0 (r), ev ([n0 ]; r) =  δEv [n(r)] δn(r)  =µ,  (3.68)  n0  and the null normalization δn(r) = 0. In Eq. (3.67), b = 2 if |δn(r)|/n0 (r) is bounded everywhere; otherwise, b > 1. To this end, had the conclusion of Ref. [310] been correct, the EulerLagrange equation at the GS density n0 (r), Eq. (3.68), would have been invalid in order to produce an O(y) behavior in Eq. (3.67) and the entire edifice of DFT [313, 319, 321, 324, 326, 327] would have been in danger as a result. Fortunately, it is not so as we just proved!  3.5  More Numerical Evidence  We continue to verify our theory through the following family of DV’s: n2s (r, y, Z) = (1 − y) n1s (r) + y n2s (r, Z) ,  (3.69)  where n2s (r, Z) = 2 |ψ2s (r, Z)|2 is the total 2s-orbital density of a noninteracting two-electron Hydrogen-like atom with nuclear charge Z (Z > 0) and 2s wave function: ψ2s (r, Z) = Z 3 /(32π)  1/2  (2 − Zr) e−Zr/2 .  (3.70)  We can achieve either an UDV or a CDV in Eq. (3.69) by controlling the value of Z. For example, one can straightforwardly write the following expressions from Eqs. (3.1), (3.2), (3.7), and (3.69): δn2s (r, y, Z) = n2s (r, y, Z) − n1s (r)  = y [n2s (r, Z) − n1s (r)] = y f2s (r, Z) ,  (3.71) 86  Chapter 3. Density Variations in DFT 6.0  5.0  1+Ev[n2s(r,y,Z=1.0)] 1.61  4.0  Energy (10  -10  a. u.)  33.86 y  3.0  2.0  1.0  0.0 0.0  1.0  2.0  3.0  4.0  5.0  -8  Density Parameter y (10 ) Figure 3.6: The fine detail of the energy density functional curve of Eq. (3.69) for Z = 1.0 and very small values of the density variation parameter y (from 0 to 5.0 × 10−8 ). The dotted line is the fitted curve, whereas the solid line is the exact energy curve (shifted upward by 1 Hartree for a better view).  |f2s (r, Z)| n1s (r)  = =  |n2s (r, Z) − n1s (r)| n1s (r)  2/π e−r  Z3 (2 − Zr)2 e−(Z−2)r − 1 , 32  (3.72)  and δn2s (r, y, Z) n1s (r)  n2s (r, Z) −1 n1s (r) Z3 = y (2 − Zr)2 e−(Z−2)r − 1 32  = y  .  (3.73)  87  Chapter 3. Density Variations in DFT 2.0  1+Ev[n2s(r,y,Z=1.34)] 1.98  60.17 y  Energy (10  -13  a. u.)  1.5  1.0  0.5  0.0 0.0  1.0  2.0  3.0  4.0  5.0  -8  Density Parameter y (10 ) Figure 3.7: The fine detail of the energy density functional curve of Eq. (3.69) for Z = 1.34 and very small values of the density variation parameter y (from 0 to 5.0 × 10−8 ). The dotted line is the fitted curve, whereas the solid line is the exact energy curve (shifted upward by 1 Hartree for a better view).  Accordingly, we readily establish the four DV regions, according to the values of Z:  (r,y,Z)  CDV : Z > 2, lim |f√2s (r,Z)| = 0, lim δn2s = 0;  n1s (r)  n1s (r) r→∞ r→∞      (r,y,Z)   → ∞; lim |f√2s (r,Z)| < M < +∞, lim δn2s  CDV : 2 Z > 1, r→∞ n1s (r) n1s (r) r→∞ (r,y,Z)   → ∞; UDV : 1 Z > 0, lim |f√2s (r,Z)| → +∞, lim δn2s  n1s (r)  n1s (r) r→∞ r→∞      (r,y,Z)|   CDV : Z = 0, lim |f√2s (r,Z)| = 0, |δn2s = y < M < +∞. n1s (r) r→∞  n1s (r)  (3.74)  88  Chapter 3. Density Variations in DFT and the power-law behavior of the energy density functional:  CDV : d = 1, if Z = 0;        UDV : d → 1+, if Z → 0+; d Ev [n2s (r, y, Z)] = −1 + c y UDV : 2 > d > 1, if 1 Z > 0;        CDV : d = 2, if Z > 1.  (3.75)  Figs. 3.6 and 3.7 unquestionably confirm the above conclusions and for all cases, the energy density functional does not vary linearly with respect to y, if Z > 0. More amusingly, when Z = 0, the energy density functional varies linearly with respect to y for a CDV path, which again contradicts the claims of Ref. [310].  3.6  Density Variations in Fock Space  Until now our discussions are restricted in Hilbert space, where the density variations do not change the normalization of the total electron density. It is interesting to extend our study to Fock space, in which the number of the electrons is not fixed. We want to study the following family of model densities: n2sF (r, y, Z) = 0.5 (1 − y) n1s (r) + y n2s (r, Z) , (3.76) with normalization n2sF (r, y, Z) = (1 + y). Figs. 3.8 and 3.9 display the curves of the energy functional for small values of y. When Z = 2.0, one has a CDV path, whereas Z = 0.67 represents an UDV path. In both cases, the energy behaves nonlinearly, lying above the straight line Ev = −0.5y, which is the exact GS energy curve for the fixed external potential of the model system in Fock space [321]. Such a nonlinear behavior is very different from that of a typical sublinear function (the dashed curve in Fig. 3.9), which goes below the true GS straight-line segment for very small values of y. This fact confirms our conclusion that the order of the nonlinear energy curve should be higher than one. Moreover, the insets of Fig. 3.8 and 3.9 unveil that the energy behaves almost linearly for very small values of y, and the slope of the energy curve converges to the chemical potential of the model system, which is −0.5, as y → 0. These two numerical tests lend further support to the thermodynamical interpretation of the functional derivative of the energy density functional in Fock space [321]. 89  Chapter 3. Density Variations in DFT 0.0 0.0 -0.1 -0.2 -0.3 -0.4  -2  Energy (10 a. u.)  -2.0  -0.5 0.0  -4.0  -6.0  -8.0 0.0  0.2  0.4  0.6  0.8  1.0  0.5+Ev[n2sF(r,y,Z=2.0)] -0.50 y 4.0  8.0  12.0  16.0  20.0  -2  Density Parameter y (10 ) Figure 3.8: The energy density functional curve of Eq. (3.76) for Z = 2.0 and the density variation parameter y (from 0 to 0.2). The solid line is the exact energy curve (shifted upward by 0.5 Hartrees for a better view). The dotted straight line is the true ground-state energy curve, which has a slope of −0.5 (the theoretical chemical potential of the model system). The inset shows the convergence of the two curves for small values of y.  To further understand the behavior of the energy density functional variation in Fock space, we consider the following general scenario, using the similar notational system to that on Page 6268 of Ref. [310]. For an open system with its number of electrons N between two adjacent integers (J −1) and J, the GS density n(N ; r) and the GS energy Ev [n(N ; r)] are linear interpolations of the two adjacent corresponding entities with integer numbers of electrons [310, 321, 327]: n(N ; r) = (N − J + 1) n(J; r) + (J − N ) n(J − 1; r) ∈ J ,  (3.77)  and Ev [n(N ; r)] = (N − J + 1) Ev [n(J; r)] + (J − N ) Ev [n(J − 1; r)] , (3.78) respectively. 90  Chapter 3. Density Variations in DFT 0.0 0.0  -5.0 0.0  -5  Energy (10 a. u.)  -2.5  2.5  5.0  7.5  10.0  -0.5  0.5+Ev[n2sF(r,y,Z=0.67)] -0.50 y -3  0.5  -2×10 y -1.0 0.0  1.0  2.0  3.0  4.0  -5  Density Parameter y (10 ) Figure 3.9: The energy density functional curve of Eq. (3.76) for Z = 0.67 and the density variation parameter y (from 0 to 4.0 × 10−5 ). The solid line is the exact energy curve (shifted upward by 0.5 Hartrees for a better view). The dotted straight line is the true ground-state energy curve, which has a slope of −0.5 (the theoretical chemical potential of the model system). The dashed line shows a typical square root curve lying below the straight line for very small values of y. The prefactor of the square root function is chosen to make a reasonable comparison with the other two curves. The inset shows the convergence of the solid and dotted curves for very small values of y. Both the x and y coordinates of the inset should be multiplied by 10−11 .  Case A. If y = J − N → 0+, the energy variation of Eq. (3.78) and the DV of Eq. (3.77) are measured from the GS density n(J; r): δEv [n(J; r), y] = y{Ev [n(J − 1; r)] − Ev [n(J; r)]} ,  (3.79)  and δn(J; r, y) = n(N ; r) − n(J; r)  = y[n(J − 1; r) − n(J; r)]  = yf (J; r) ,  (3.80) 91  Chapter 3. Density Variations in DFT where f (J; r) = n(J − 1; r) − n(J; r). Eq. (3.80) results the following expressions: n(J − 1; r) |δn(J; r, y)| =y − 1 < M < +∞ , n(J; r) n(J; r)  (3.81)  and |f (J; r)|  n(J; r)  =  n(J − 1; r) n(J; r)  −  n(J; r) < P < +∞ ,  (3.82)  because the GS densities n(J − 1; r) and n(J; r) are everywhere positive [313, 318, 327] and have the exponential asymptotic limits [324, 326, 327]: lim n(J; r) ∼ e−2r  √  2IJ  r→∞  and lim n(J − 1; r) ∼ e−2r  √  ,  2IJ−1  r→∞  (3.83) ,  (3.84)  governed by their ionization potentials IJ and IJ−1 with IJ−1 IJ 0. In other words, if the DV is on the electron-deficient side, the DV will be a CDV and the corresponding energy variation is O(y) because of Eq. (3.79). This is the opposite to the statement on Page 6269 of Ref. [310]. Case B. If y = N − J + 1 → 0+, the energy variation of Eq. (3.78) and the DV of Eq. (3.77) are measured from the GS density n(J − 1; r): δEv [n(J − 1; r), y] = y{Ev [n(J; r)] − Ev [n(J − 1; r)]} ,  (3.85)  and δn(J − 1; r, y) = n(N ; r) − n(J − 1; r)  = y[n(J; r) − n(J − 1; r)] = yf (J − 1; r) , (3.86)  where f (J − 1; r) = n(J; r) − n(J − 1; r). Eq. (3.86) results the following expressions as r → ∞: δn(J − 1; r, y) n(J − 1; r)  n(J; r) −1 n(J − 1; r) √ √ √ ∼ e2 2 r( IJ−1 − IJ ) → ∞ ,  = y  (3.87)  and |f (J − 1; r)|  n(J − 1; r)  =  n(J; r)  − n(J − 1; r) n(J − 1; r) √ √ √ ∼ e− 2 r(2 IJ − IJ−1 ) ,  (3.88) 92  Chapter 3. Density Variations in DFT where the asymptotic limit of Eq. (3.88) depends on the sign of (4IJ −IJ−1 ). If 4IJ > IJ−1 , Eq. (3.88) will be square-integrable. Table 3.1 21 shows the values of (4IJ − IJ−1 ) for the ground states of atomic ions with 2 to 18 electrons, based on the highly accurate theoretical ionization potential data of Davidson and coworkers [331–333]. Apparently, most of the values of (4IJ − IJ−1 ) is positive and Eq. (3.88) is square-integrable. In fact, regardless of the square-integrability of Eq. (3.88), the energy variation is always O(y) for the DV’s on the electron-abundant side because of Eq. (3.85). Table 3.1: Values of (4IJ −IJ−1 ) (in Hartrees) for the ground states of atomic ions with 2 to 18 electrons, based on the highly accurate theoretical ionization potential data of Davidson and coworkers [331–333].  H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn  Z  J =2  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  −0.39 1.61 6.62 14.62 25.63 39.64 56.65 76.68 99.72 125.78 154.86 186.98 222.13 260.33 301.58 345.89 393.26 443.72 497.27 553.92 613.69 676.58 742.62 811.81 884.17  3  4  −0.90 −1.99 −0.11 −2.98 0.70 −3.96 2.30 −4.93 4.67 −5.90 7.79 −6.87 11.67 −7.83 16.30 −8.80 21.68 −9.76 27.82 −10.72 34.72 −11.68 42.37 −12.64 50.78 −13.59 59.95 −14.54 69.88 −15.49 80.58 −16.43 92.04 −17.37 104.27 −18.29 117.28 −19.22 131.05 −20.13 145.60 −21.04 160.94 −21.94 177.05 −22.83 193.95 Continued  5  −0.34 0.30 1.82 4.13 7.19 11.02 15.60 20.94 27.03 33.88 41.50 49.87 59.00 68.89 79.56 90.99 103.19 116.16 129.90 144.43 159.74 175.82 on next  6  7  8  9  −0.26 0.76 2.61 5.23 8.62 12.76 17.66 23.31 29.72 36.88 44.79 53.46 62.89 73.08 84.02 95.73 108.21 121.46 135.47 150.27 165.84 page  −0.23 1.05 3.14 6.01 9.65 14.04 19.18 25.08 31.72 39.11 47.24 56.12 65.74 76.10 87.19 99.02 111.58 124.88 138.90 153.66  −0.56 0.71 2.84 5.75 9.44 13.89 19.11 25.09 31.83 39.34 47.60 56.64 66.44 77.00 88.35 100.46 113.34 127.00 141.42  −0.29 1.28 3.69 6.90 10.87 15.61 21.12 27.37 34.39 42.15 50.67 59.94 69.97 80.74 92.27 104.55 117.57 131.35  21  Table 3.1 and the ionization potential data of Davidson and coworkers [331–333] can be downloaded from the web: http://www.chem.ubc.ca/faculty/wang/groupsite/papers/IPdiff.htm.  93  Chapter 3. Density Variations in DFT Table 3.1 – 959.71 −23.70 1038.46 −24.56 1120.44 −25.42  Continued from 211.64 192.71 230.12 210.37 249.40 228.84  previous page 182.21 169.15 199.35 185.38 217.30 202.33  Fe Co Ni  26 27 28  J=  10  11  12  13  14  15  F −0.14 Ne 1.67 Na 4.32 Mg 7.76 Al 11.99 Si 16.97 P 22.72 S 29.21 Cl 36.47 Ar 44.48 K 53.24 Ca 62.75 Sc 73.02 Ti 84.04 V 95.81 Cr 108.33 Mn121.60 Fe 135.64 Co 150.40 Ni 165.95  −0.79 −0.98 −0.73 −0.23 0.51 1.46 2.62 3.99 5.56 7.34 9.32 11.51 13.90 16.50 19.31 22.33 25.55 28.99 32.63  −0.11 0.57 1.72 3.26 5.17 7.44 10.05 13.01 16.31 19.95 23.94 28.27 32.94 37.96 43.32 49.03 55.08 61.46  −0.28 0.19 1.17 2.55 4.30 6.40 8.85 11.65 14.80 18.29 22.12 26.29 30.82 35.68 40.89 46.44 52.35  −0.15 0.60 1.80 3.38 5.33 7.63 10.29 13.29 16.63 20.31 24.34 28.71 33.42 38.47 43.87 49.61  −0.10 0.82 2.15 3.86 5.94 8.37 11.14 14.26 17.72 21.51 25.64 30.10 34.89 40.02 45.47  156.62 172.57 189.30  145.87 161.16 177.18  16  17  18  −0.28 0.67 2.05 3.81 5.95 8.45 11.30 14.50 18.05 21.94 26.19 30.78 35.71 40.99  −0.07 1.03 2.57 4.48 6.76 9.40 12.39 15.73 19.41 23.43 27.79 32.49 37.52  0.05 1.30 2.97 5.02 7.43 10.20 13.32 16.78 20.59 24.74 29.23 34.06  Summarizing both cases, we can safely say that the linear-order behavior of the energy variation for open systems in Fock space has nothing to do with the square-integrability of |f (r)|/ n0 (r), regardless of whether the DV is an UDV or a CDV. Therefore, the UDV arguments of Ref. [310] cannot provide any ground for the cause of the “derivative discontinuity of the energy” [321].  3.7  Conclusions  Numerical evidence and theoretical arguments suggest that the “unconventional density variations” are still in the allowed density variation domain of any ground-state density. In Hilbert space where the number of electrons of a system is fixed, the energy variation is of nonlinear order higher than linear order of the density variation about the ground-state density. In 94  Chapter 3. Density Variations in DFT Fock space where the number of electrons of a system is flexible, the energy variation behaves linearly with respect to the density variation about the ground-state density.  95  Bibliography  Bibliography [310] J. P. Perdew and M. Levy, Phys. Rev. B 31, 6264 (1985). [311] H. Englisch and R. Englisch, Physica 121A, 253 (1983). [312] H. Englisch and R. Englisch, Phys. Stat. Solidi. B 124, 373 (1984). [313] E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983). [314] R. van Leeuwen, Adv. Chem. Phys. 43, 25 (2003). [315] E. H. Lieb, in Density Functional Methods in Physics, edited by R. M. Dreizler and J. da Providˆencia (Springer, New York: Plenum, 1985), p. 31. [316] H. Englisch and R. Englisch, Phys. Stat. Solidi. B 123, 71 (1983). [317] G. de Barra, Introduction to Measure Theory (Van Nostrand Reinhold, London, 1974). [318] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators (Academic Press, San Diego, 1978). [319] J. P. Perdew, in Density Functional Methods in Physics, edited by R. M. Dreizler and J. da Providˆencia (Springer, New York: Plenum, 1985), p. 265. [320] J. P. Perdew and M. Levy, Phys. Rev. B 56, 16021 (1997). [321] J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Phys. Rev. Lett. 49, 1691 (1982). [322] J. P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 (1983). [323] J. P. Perdew and M. Levy, in Many-Body Phenomena at Surfaces, edited by D. C. Langreth and H. Suhl (Academic, New York, 1984), p. 71. [324] R. M. Dreizler and E. K. U. Gross, Density Functional Theory: An Approach to the Quantum Many-Body Problem (Springer-Verlag, Berlin, 1990). [325] Y. A. Wang and E. A. Carter, in Theoretical Methods in Condensed Phase Chemistry, edited by S. D. Schwartz (Kluwer, Dordrecht, 2000), chap. 5, pp. 117–184. 96  Bibliography [326] R. G. Parr and W. Yang, Density-functional Theory of Atoms and Molecules (Oxford University Press, Inc., 200 Madison Avenue, New York, NY 10016, USA, 1989), 1st ed. [327] F. E. Zahariev and Y. A. Wang, Phys. Rev. A 70, 042503 (2004). [328] H. English, H. Fieseler, and A. Haufe, Phys. Rev. A 37, 4570 (1988). [329] A. Griffin, Can. J. Phys. 73, 755 (1995). [330] Wolfram Research, Inc., Mathematica, Version 5.2, Champaign, IL (2005). [331] E. R. Davidson, S. A. Hagstrom, S. J. Chakravorty, V. M. Umar, and C. F. Fischer, Phys. Rev. A 44, 7071 (1991). [332] S. J. Chakravorty, S. R. Gwaltney, E. R. Davidson, F. A. Parpia, and C. F. Fischer, Phys. Rev. A 47, 3649 (1993). [333] S. J. Chakravorty and E. R. Davidson, J. Phys. Chem. 100, 6167 (1996).  97  Chapter 4  Perturbative Total Energy Evaluation in SCF Iterations––Tests on Molecular Systems 22  4.1  Introduction  Most quantum mechanical calculations, for example, Hartree-Fock and density functional theory (DFT) [334–336] methods, are carried out iteratively. Although these methods have lower computational scalings than post-HartreeFock methods do, they are still too expensive to deal with systems with thousands of atoms routinely. Searching for algorithms with higher computational efficiency has become an active research area of contemporary theoretical chemistry [337]. Both Hartree-Fock and DFT methods utilize matrix diagonalization in the self-consistent field (SCF) procedure. The computational cost of matrix diagonalization usually scales as O(N 3 ), where N is a measure of the size of the system, i.e., the number of basis functions. If N becomes very large, a full Hartree-Fock or DFT calculation will become unaffordable. Additionally, it normally takes many iterations to fully converge the SCF cycle. This further increases the computational cost. So, reducing the number of matrix diagonalizations is an effective way to make calculations of large systems accessible. We will focus on this issue in this chapter. For a given density ρ(r) and external potential vext (r), the DFT total electronic energy is conveniently decomposed into four parts: E[ρ] = Ts [ρ] + Vext [ρ] + EH [ρ] + Exc [ρ],  (4.1)  22 A version of this chapter has been submitted for publication. Y. A. Zhang and Y. A. Wang, Perturbative Total Energy Evaluation in SCF Iterations––Tests on Molecular Systems.  98  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations where Ts is the kinetic energy, Vext = ρ(r)vext (r) is the external potential energy, EH [ρ] is the Hartree energy, and Exc [ρ] is the exchange-correlation (XC) energy. In each iteration of solving the Kohn-Sham (KS) equations, one uses an input density ρin (r) to construct the Hartree (vH ) and XC (vxc ) potentials and obtains the output orbitals {ψiout } and their corresponding orbital energies {εout i }: 1 out − ∇2 + vext (r) + vH [ρin ](r) + vxc [ρin ](r) ψiout = εout i ψi . 2  (4.2)  The output density ρout (r) can then be obtained from the output orbitals: occ  ρout (r) = i  fiout |ψiout |2 ,  (4.3)  where fiout is the occupation number of ψiout . Multiplying the left-hand side of Eq. (4.2) with the complex conjugate of ψiout , integrating over the entire space, and summing over all occupied orbitals, one arrives at occ  fiout εout = Ts [ρout ]+Vext [ρout ]+ ρout (r){vH [ρin ](r)+vxc [ρin ](r)} . (4.4) i i  We can then write an approximate DFT total electronic energy during the iterative process as occ  E  HKS  in  [ρ , ρ  out  ] = i  fiout εout − ρout (r){vH [ρin ](r) + vxc [ρin ](r)} + i  EH [ρout ] + Exc [ρout ].  (4.5)  This is the well-known Hohenberg-Kohn-Sham (HKS) functional [338], an upper bound of the exact ground-state KS total electronic energy [339]. Alternatively, Harris approximated the total electronic energy based on some consideration from perturbation theory [340]: occ  E  Harris  in  [ρ , ρ  out  ]= i  in in in in fiout εout i − ρ (r)vxc [ρ ](r) −EH [ρ ]+Exc [ρ ]. (4.6)  The Harris functional can be obtained by replacing all ρout in Eq. (4.5) with ρin , although fiout and εout must come from solving the KS equations. i Unlike the HKS functional, the Harris functional is neither an upper bound nor a lower bound of the exact ground-state energy in general [341–343]. 99  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations Practically, both the HKS and Harris functionals are often about the same quality if self-consistency is not achieved; sometimes the Harris functional yields better results than the HKS functional does due to error cancelation [344]. When the converged electron density ρKS (r) is obtained at the end of the SCF cycle, both the HKS and Harris functionals yield the exact KS total electronic energy E KS [ρKS ]: occ KS  KS KS KS KS fiKS εKS i − ρ (r)vxc [ρ ](r) − EH [ρ ] + Exc [ρ ],  KS  E [ρ ] = i  (4.7)  where fiKS and εKS i are the occupation number and the energy eigenvalue of the converged ith KS orbital, respectively.  4.2  Perturbation Expansions of Total Electronic Energy Functionals  We can expand Eqs. (4.5) and (4.6) around the converged electron density ρKS (r). After some algebraic manipulations, the final results (up to second order in density difference) are as the following [344]: E HKS [ρin , ρout ] = E KS [ρKS ] + {ρout (r1 ) − ρin (r1 )} E  Harris  in  [ρ , ρ  out  C(r1 , r2 ){ρout (r2 ) − ρKS (r2 )} , KS  KS  ] = E [ρ ] + {ρ in  out  (4.8)  in  (r1 ) − ρ (r1 )}  C(r1 , r2 ){ρ (r2 ) − ρKS (r2 )} ,  (4.9)  where C(r1 , r2 ) =  δvxc [ρ(r1 )] 1 + |r1 − r2 | δρ(r2 )  1 2  .  (4.10)  ρin  Based on this appreciation, we recently proposed the corrected HKS (cHKS) and corrected Harris (cHarris) total electronic energy functionals [345]: E cHKS [ρin , ρout ] = E HKS [ρin , ρout ] + {ρout (r1 ) − ρin (r1 )} E  cHarris  in  [ρ , ρ  out  C(r1 , r2 ){ρb (r2 ) − ρout (r2 )} ,  ] = E  Harris  in  [ρ , ρ  out  b  ] + {ρ  out  in  (r1 ) − ρ (r1 )}  C(r1 , r2 ){ρ (r2 ) − ρ (r2 )} ,  (4.11)  in  (4.12)  where ρb stands for some electron density “better” than ρin and ρout . Obviously, the performances of these two functionals are dictated by the choice of ρb . 100  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations In practice, the functional derivative term in C(r1 , r2 ) might be difficult to compute since the form of many XC functionals are quite complicated. To avoid this problem, the following finite-difference approximation can then be adopted [345, 346]: {ρout (r1 ) − ρin (r1 )}  δvxc [ρ(r2 )] δρ(r1 )  ρin  r1  ≈ vxc [ρout (r2 )] − vxc [ρin (r2 )],  (4.13) where the integration on the left-hand side is carried out on the variable r1 . Consequently, we can simplify Eqs. (4.11) and (4.12) as E cHKS [ρin , ρout ] = E HKS [ρin , ρout ] + 1 b out in (ρ − ρout )(veff − veff ) r1 ,r2 , 2 E cHarris [ρin , ρout ] = E Harris [ρin , ρout ] + 1 b out in (ρ − ρin )(veff − veff ) r1 ,r2 , 2  (4.14)  (4.15)  where the KS effective potential veff = vext + vH + vxc . In fact, the cHKS and cHarris models are very general and can be employed in other quantum chemistry methods. Specifically, we consider extension of the cHKS and cHarris concepts to the Hartree-Fock method. For simplicity, we only study the closed-shell case with spinless density matrices. The occupation numbers fi are either 2 or 0 and are the same for ψiout and ψiHF in the following derivation. The ordinary Hartree-Fock equation is written as   occ ∗ ψj (r2 )ψj (r2 ) − 1 ∇2 + vext (r1 ) + 2 dr2  ψi (r1 )− 2 |r1 − r2 | j  occ j  ψj∗ (r2 )ψi (r2 ) dr2 ψj (r1 ) = |r1 − r2 |  εHF i ψi (r1 ), (4.16)  where ψi and εHF are the ith Hartree-Fock orbital and its corresponding i orbital energy, respectively. We can construct the Hartree-Fock density matrix P HF (r1 , r2 ) from the Hartree-Fock obitals: occ  ψi∗ (r1 )ψi (r2 ),  P HF (r1 , r2 ) = 2  (4.17)  i  101  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations whose diagonal elements define the Hartree-Fock density ρHF (r). The HartreeFock total electronic energy reads occ  EHF = 2 i  HF HF εHF i − EH − EHFX ,  (4.18)  HF ) energies are where the Hartree-Fock Hartree (EHHF ) and exchange (EHFX defined as  EHHF = HF EHFX =  1 HF ρ (r1 )vHHF (r1 ) r1 , 2 1 HF HF P (r1 , r2 )vHFX (r1 , r2 ) 2  (4.19) r1 ,r2 ,  (4.20)  with ρHF (r2 ) dr2 , |r1 − r2 | 1 P HF (r2 , r1 ) HF vHFX (r1 , r2 ) = − . 2 |r1 − r2 | vHHF (r1 ) =  (4.21) (4.22)  Analogously, one can define the Hartree-Fock versions of the HKS and Harris total electronic energy functionals: occ  E  HKS HF  = 2 i out  εout − ρout vHin i  r1  in − P out vHFX  r1 ,r2  +  out EH + EHFX ,  (4.23)  occ Harris EHF = 2  i  in εout − EHin − EHFX , i  (4.24)  where the superscripts “in” and “out” denote the input and output quantities, respectively. The difference between the HKS and Harris functionals can be readily shown to be HKS Harris EHF − EHF =  =  1 out in (P out − P in )(vHFX − vHFX ) r1 ,r2 + 2 1 out (ρ − ρin )(vHout − vHin ) r1 2 1 out in (P out − P in )(vee − vee ) r1 ,r2 , 2  (4.25)  where vee (r1 , r2 ) = vH (r1 )δ(r2 − r1 ) + vHFX (r1 , r2 ).  (4.26) 102  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations Also, the difference between the Harris and Hartree-Fock total electronic energy functionals can be grouped into three parts: occ Harris EHF − EHF = 2  i  in HF (εout − εHF i i ) − (EH − EH ) − 2 ○ 1 ○  1 in in HF P vHFX − P HF vHFX 2  r1 ,r2 ,  (4.27)  3 ○  which will be analyzed part by part in turn. 1 in Eq. (4.27) with orbital perturbation theory. First, we evaluate part ○ The Hartree-Fock equation for ψiHF is 1 HF HF − ∇2 + vext (r1 ) + veff (r1 ) ψiHF = εHF i ψi , 2  (4.28)  in − v HF ) where veff (r1 ) is the Hartree-Fock effective potential. Let δveff = (veff eff be the perturbation in the potential of the system described by Eq. (4.28). Then, to second order in orbital change, we have occ  εout i  HF  HF  − εi =  j  HF  ψj |δveff |ψj  + j=i  | ψiHF |δveff |ψjHF |2 , HF εHF i − εj  (4.29)  and consequently, occ HF (εout i −εi )  i  1 HF in ) −vee = P HF (vee 2  occ vir r1 ,r2 + i  t  | ψiHF |δveff |ψtHF |2 , (4.30) HF εHF i − εt  where in the second term of the right-hand side the summation index t runs over all the virtual orbitals. Similarly, applying this orbital perturbation theory to the Hartree-Fock equation for ψiout , 1 in out − ∇2 + vext (r1 ) + veff (r1 ) ψiout = εout i ψi , 2  (4.31)  we have occ i  out (εHF i − εi ) =  1 out HF in P (vee − vee ) 2  occ vir r1 ,r2  + i  t  | ψiout |δveff |ψtout |2 , εout − εout t i (4.32) 103  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations HF in ). After working where the perturbation in the potential is −δveff = (veff −veff out the Taylor expansion of the second-order term of Eq. (4.32) at the Hartree-Fock quantities, we find that its difference from the second-order term of Eq. (4.30) only enters in third order. Hence, we can cancel the second-order terms of Eqs. (4.30) and (4.32) and derive a much simpler yet 1 accurate expansion for part ○:  1 = ○  1 in HF (P out + P HF )(vee − vee ) 2  r1 ,r2  + O(δ 3 ).  (4.33)  2 and ○ 3 can be exactly exIt is straightforward to show that parts ○ panded as 2 = ○ = =  1 in in (ρ vH − ρHF vHHF ) r1 2 1 in in 1 (ρ vH − ρHF vHHF ) r1 + (ρHF vHin − ρin vHHF ) 2 2 1 in HF in HF (ρ + ρ )(vH − vH ) r1 , 2  r1  (4.34)  and 3 = ○ = =  1 in HF (P in vHFX − P HF vHFX ) r1 ,r2 2 1 1 in HF in HF (P in vHFX − P HF vHFX ) r1 ,r2 + (P HF vHFX − P in vHFX ) 2 2 1 in HF (P in + P HF )(vHFX − vHFX ) r1 ,r2 , 2  r1 ,r2  (4.35)  where the second terms after the second equal signs are zero after interchanging the two integration dummy variables, r1 and r2 : (ρHF vHin − ρin vHHF )  r1  = = 0,  ρHF (r1 )ρin (r2 ) dr1 dr2 − |r1 − r2 |  ρin (r1 )ρHF (r2 ) dr1 dr2 |r1 − r2 | (4.36)  and in HF (P HF vHFX − P in vHFX )  r1 ,r2  = −  1 2  − = 0.  P HF (r1 , r2 )P in (r2 , r1 ) dr1 dr2 |r1 − r2 | P in (r1 , r2 )P HF (r2 , r1 ) dr1 dr2 |r1 − r2 | (4.37) 104  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations Substituting Eqs. (4.33)–(4.35) into Eq. (4.27), we arrive at Harris EHF − EHF =  = = =  1 in HF (P out + P HF )(vee − vee ) 2 (ρin + ρHF )(vHin − vHHF ) r1 −  r1 ,r2 −  in HF (P in + P HF )(vHFX − vHFX ) r1 ,r2 1 in HF (P out − P in )(vee − vee ) r1 ,r2 2 1 out in (P in − P HF )(vee − vee ) r1 ,r2 2 1 Tr{(P in − P HF )(F out − F in )}, 2  (4.38)  where F out and F in are Fock matrices built from output and input HartreeFock orbitals, respectively. With the aid of Eqs. (4.25) and (4.38), we can easily deduce the difference between the HKS and Hartree-Fock total electronic energy functionals: HKS EHF − EHF =  =  1 out in (P out − P HF )(vee − vee ) r1 ,r2 2 1 Tr{(P out − P HF )(F out − F in )}. 2  (4.39)  Therefore, we can define the Hartree-Fock versions of the cHKS and cHarris functionals: 1 cHKS HKS EHF = EHF + Tr{(P b − P out )(F out − F in )}, 2 1 cHarris Harris EHF = EHF + Tr{(P b − P in )(F out − F in )}, 2  (4.40) (4.41)  where P b denotes some density matrix of “better” quality than P in and P out . Interestingly, Eqs. (4.14) and (4.15) look very similar to the above two equations. The preceding derivation supersedes Finnis’s analysis [344]: Our argument builds upon the general orbital perturbation theory and, as such, apparently can be applied to all levels of quantum chemistry theory. It is also necessary to mention the connection between our functionals and the energy direct inversion of the iterative subspace (EDIIS) method proposed by Kudin et al. [347]. The EDIIS energy functional is written as k  E  EDIIS  (P˜ ) = i=1  1 ci E0 (Pi ) − 2  k i,j=1  ci cj Tr{(Fi − Fj )(Pi − Pj )},  (4.42) 105  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations k where Pi,j are density matrices; P˜ = ki=1 ci Pi with ci 0 and i ci = 1; Fi,j are Fock matrices; and E0 (Pi ) is the Hartree-Fock energy evaluated with Pi . The quadratic term looks very similar to our second-order correction terms in Eqs. (4.40) and (4.41), so the EDIIS method can be viewed as finding an optimal reference zero-order total energy and the corresponding second-order correction term in a linear subspace of the density matrices. The EDIIS scheme arises from an optimization consideration; we propose our functionals based on perturbation theory. In the end, both approaches arrive at similar results.  4.3  Implementation and Computational Details  The NWChem 5.0 source code [348] was modified to run all calculations. The cHKS and cHarris total electronic energy functionals were evaluated in each DFT or Hartree-Fock SCF iteration. For the estimation of ρb or P b , we took advantage of the direct inversion of the iterative subspace (DIIS) method [349], which is readily available in the code. The 6-31G∗ basis set was used for all calculations. LDA exchange functional [350] and the VWN 5 correlation functional [351] were used for all DFT calculations. We tested our models on four molecular systems: HF, H2 O, CrC, and SiH4 (with one elongated Si−H bond). In DFT calculations, the H−F bond length was 0.920 ˚ A, the H−O bond length was 0.965 ˚ A, the H−O−H bond ◦ angle was 103.75 , the Cr−C bond length was 2.00 ˚ A, the regular Si−H bond length was 1.47 ˚ A, the elongated Si−H bond length was 4.00 ˚ A, and ◦ any H−Si−H bond angle was 109.47 . For Hartree-Fock calculations, the H−F bond length was 0.911 ˚ A, the structure of H2 O was the same as above, and the structure of SiH4 was similar except that the elongated Si−H bond length was 12.0 ˚ A. To guarantee the convergence, the criterion of the root mean square of the electron density difference was set to be 1 × 10−7 in all DFT calculations for HF, H2 O, and SiH4 , and 1×10−8 for CrC. For Hartree-Fock calculations, the convergence threshold of the norm of the orbital gradient was set to be 1 × 10−7 . In all calculations, the default initial guesses were adopted to start the SCF procedure, and DIIS began immediately upon becoming possible and was used throughout the SCF processes. Damping (with 70% of previous density) was employed in the first 20 SCF iterations in some DFT calculations for HF and H2 O to investigate the performance of our models. For the DFT calculations on SiH4 , a 50% damping factor was utilized. For the Hartree-Fock calculations on SiH4 , the damping factor was 40% and 10% 106  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations before and after the 100th iteration, respectively. In all calculations, level shifting was used (with a parameter of 0.5 a.u.) whenever the gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital was less than 0.05 Hartrees.  4.4  Results and Discussions  First we present the results for HF and H2 O in Figs. 4.1 and 4.2. Both show the same tendency: the cHKS and cHarris energies are virtually identical to each other throughout the entire iteration process. This behavior is reasonable because the difference between the cHKS and cHarris functionals is of third order in the density or density matrix variation (see the analysis in Section 4.2). The HKS and Harris energies are roughly of the same quality. The cHKS and cHarris models do improve upon the HKS and Harris energies that come from the current iteration, but they are not better than the HKS and Harris energies that come from the next iteration. Such a behavior is not surprising because the DIIS density or density matrix is designed to minimize the total energy gradient of the SCF process [349], but is not necessarily optimal for the total energy minimization. It is very difficult to propose a general scheme to achieve fast, smooth convergence for any arbitrary molecular system. Therefore, in most modern quantum chemistry packages, one has to rely on trial and error to choose the best convergence scheme among many options. For further testing of the performance of our models in DFT calculations with different convergence schemes, we intentionally slowed down the SCF convergence with damping (with 70% of previous density) on the same two molecules. This commonly used procedure is also known as linear mixing or Pratt mixing. The results are displayed in Fig. 4.3. It shows that the cHKS and cHarris energies are always together and so are the HKS and Harris energies. Amazingly, the cHKS and cHarris models converge much faster than the HKS and Harris energies do. Our corrected models converge to within 1 × 10−6 Hartrees only after five SCF iterations, but the HKS and Harris energies do not achieve this kind of accuracy until iteration 15. Of course, a damped DIIS density or density matrix is not optimal for minimizing the total energy gradient, so the HKS and Harris total energies converge very slowly in the above two cases shown in Fig. 4.3. Even with such “bad” input densities, the perturbative corrections in the cHKS and cHarris models still demonstrate significant improvement. However, when we perform the same tests on the same molecular sys107  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations 0  HF LDA  -2  -6  10  log |E-E  KS  |  -4  -8  HKS Harris -10  cHKS cHarris  -12  1  2  3  4  5  6  Iteration  0  H O LDA 2  -2  -6  10  log |E-E  KS  |  -4  -8  HKS Harris cHKS  -10  cHarris -12  1  2  3  4  5  6  7  Iteration  Figure 4.1: Convergence of the total energies (in Hartrees) of HF (top) and H2 O (bottom) molecules evaluated with the HKS (circles), the Harris (hollow triangles), the cHKS (squares), and the cHarris (hollow stars) functionals during the SCF iterations of Kohn-Sham calculations.  tems using the Hartree-Fock method, we do not observe a similar dramatic improvement. This is because for most molecular systems at their equilibrium geometries, the DIIS method (even with damping) already provides an 108  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations excellent convergence scheme and our correction models do not show much greater enhancement (see Fig. 4.2). On the other hand, if the molecular system is far away from its equilibrium geometry, the damped Hartree-Fock 0  HF Hartree-Fock  -2  -6  10  log |E-E  HF  |  -4  -8  HKS Harris cHKS  -10  cHarris  -12 1  2  3  4  5  6  Iteration 0  H O Hartree-Fock  -2  2  -6  10  log |E-E  HF  |  -4  -8  HKS Harris -10  cHKS cHarris  -12  1  2  3  4  5  6  7  8  Iteration  Figure 4.2: Convergence of the HF total energies of HF (top) and H2 O (bottom) molecules evaluated with the HKS (circles), the Harris (hollow triangles), the cHKS (squares), and the cHarris (hollow stars) functionals during the SCF iterations of Hartree-Fock calculations.  109  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations 0  HF LDA (damping)  -2  -6  10  log |E-E  KS  |  -4  -8  HKS Harris -10  cHKS cHarris  -12  0  4  8  12  16  Iteration 0  H O LDA (damping)  -2  2  -6  10  log |E-E  KS  |  -4  -8  HKS Harris -10  cHKS cHarris  -12  0  4  8  12  16  Iteration  Figure 4.3: Convergence of the total energies (in Hartrees) of HF (top) and H2 O (bottom) molecules evaluated with the HKS (circles), the Harris (hollow triangles), the cHKS (squares), and the cHarris (hollow stars) functionals during the SCF iterations of Kohn-Sham calculations with damping.  results might possess a similar pattern to the DFT counterparts. A polyatomic molecule, SiH4 with one elongated Si−H bond, is then employed to illustrate this scenario. Fig. 4.4 shows that when one Si−H bond is much 110  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations  0  -2  log10|E-EKS|  -4  HKS Harris cHarris cHKS  -6  -8  SiH4 with elongated Si-H bond  -10  LDA  -12 0  10  20  30  40  Iteration 0  -2  HKS Harris cHarris cHKS  log10|E-EHF|  -4  -6  -8  -10  SiH4 with elongated Si-H bond Hartree-Fock  -12 100  110  120  130  140  150  160  Iteration  Figure 4.4: Convergence of the total energy (in Hartree) of a SiH4 molecule with an elongated Si−H bond, evaluated with the HKS (circles), the Harris (hollow triangles), the cHarris (squares), and the cHKS (hollow stars) functionals during the SCF iterations of a Kohn-Sham calculation (top) and a Hartree-Fock calculation (bottom). For the Kohn-Sham calculation, the elongated Si−H bond length is 4 ˚ A and the damping factor is 50%. For the Hartree-Fock calculation (only shown the data after 100th iteration), the elongated Si−H bond length is 12 ˚ A, and the damping factor is 40% and 10% before and after the 100th iteration, respectively. The Hartree-Fock data of first 140 iterations behave very similarly.  111  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations elongated, both DFT and Hartree-Fock methods have great difficulty to achieve fast SCF convergence. It takes more than 30 and 140 iterations for the damped DIIS scheme to locate the fast descending path in the DFT and  singlet CrC LDA  -4  10  log |E-E  KS  |  -2  HKS70  -6  cHKS70 HKS0 cHKS0  -8 0  10  30  20  Iteration 0  triplet CrC  LDA  -4  10  log |E-E  KS  |  -2  HKS70  -6  cHKS70 HKS0 cHKS0  -8  0  10  20  30  40  Iteration  Figure 4.5: Convergence of the total energies (in Hartrees) of singlet (top) and triple (bottom) CrC molecules during the SCF iterations of Kohn-Sham calculations. “HKS70” and “cHKS70” denote the HKS and cHKS energies with 70% damping, respectively. “HKS0” and “cHKS0” denote the HKS and cHKS energies without damping, respectively.  112  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations Table 4.1: Convergence of the SCF process of DFT calculations on CrC  Molecular State  Damping  Energy Model  Iterationsa  Iterationsb  No  HKS cHKS  28 23  73  Yes  HKS cHKS  32 25  85  No  HKS cHKS  32 23  78  Yes  HKS cHKS  35 27  91  1 CrC  3 CrC  a b  Number of iterations to converge the total energy within 1 × 10−6 Hartrees. Number of iterations to converge the total energy within 1 × 10−12 Hartrees.  Hartree-Fock calculations, respectively. After the turning point, the cHKS and cHarris models bear a similar fast converging trait to what Fig. 4.3 exhibits. To further investigate the performance of our models in different chemical environment, we choose one notoriously difficult case for almost all conventional SCF convergence acceleration methods–––the CrC molecule [352– 357]. DFT calculations are performed for both singlet and triplet states of CrC without and with damping (with the same damping parameter as before). Because the cHKS model works slightly better than the cHarris model (see Fig. 4.3), we only show the cHKS results in Fig. 4.5. Unlike the cases for HF or H2 O, damping does not affect the convergence of the conventional SCF scheme strongly for CrC. This effect can be observed from the results in Table 4.1. In both cases, damping only slows the complete SCF process (below 1 × 10−12 Hartrees) by 12 iterations. The cHKS model always uses 5 to 9 fewer iterations than the HKS model does to converge the total energy below 1 × 10−6 Hartrees. Interestingly, we also find that calculations with different damping parameters pose similar convergence behavior. This supports our conclusion regarding the damping calculations for HF and H2 O: whatever input densities are given, perturbative corrections always deliver improvement, sometimes quite substantial. Except for the first several iterations, the cHKS model always produces an 113  Chapter 4. Perturbative Total Energy Evaluation in SCF Iterations energy higher than the converged one, which seems to suggest that the cHKS functional converges from above without much oscillation. This is different from the behavior of the Harris and cHarris models, whose convergence oscillates with iteration and is much less uniform.  4.5  Conclusion  In summary, we have generalized the cHKS and cHarris functionals to the Hartree-Fock method and have done case studies on several molecules. Numerical evidence shows that the performance of such perturbative correction models is greatly affected by the choice of the estimation of the density or density matrix. Even when the quality of such estimation is not good for those commonly implemented convergence schemes for DFT or Hartree-Fock methods, our perturbative correction models accelerate the convergence of total energy drastically. 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Maclagan and G. E. Scuseria, J. Chem. Phys. 106, 1491 (1997). [356] G. L. Gutsev, L. Andrews, and C. W. Bauschlicher, Jr., Theor. Chem. Acc. 109, 298 (2003). [357] A. Kalemos, T. H. Dunning, Jr., and A. Mavridis, J. Chem. Phys. 123, 014302 (2005).  116  Chapter 5  Theoretical Studies of the Tautomers of Pyridinethiones 23  5.1  Introduction  There has been continued interest in organosulfur compounds with thiocarbonyl groups because of their diverse chemistry [358],various biological applications [359, 360], and the tendency of the C=S bond to oxidize easily to form the corresponding thiols and disulfides [361]. Pyranthiones and pyridinethiones are two important classes of such organosulfur compounds [362]. Four years ago, pyranthiones were explored as ligands because of their trans-influencing ability [363]. A number of pyridinethiones have also been patented for therapeutic antioxidant properties [364] and effectiveness against carcinoma [365]. Pyridinethiones and their oxygen analogues, pyridinones, can tautomerize to the corresponding thiol/enol form (Fig. 5.1) [366, 367]. It has been shown, however, that the prevailing form of these compounds is the thione/ketone form [368]. For pyridinethiones, the S–H group in the thiol form can easily oxidize to bridge two molecules through a disulfide bond in polar solvents in air (Fig. 5.2) [367, 369]. Interestingly, Stoyanov et al. reported that pyridinethiones convert to the thiol tautomer when left in aqueous or alcoholic solutions for 24 hours even though the thione form is more dominant [367]. Evidence for this tautomerism was also observed recently in the syntheses of pyridinethiones studied by our group [362]. Monga and Orvig recently reported [362] the preparation of Hmppt and Hdppt (Fig. 5.3) by reacting thiomaltol with ammonia and methylamine, respectively. The resulting pyridinethiones were thoroughly characterized 23  A version of this chapter has been published. Y. A. Zhang, V. Monga, C. Orvig, and Y. A. Wang (2008) Theoretical Studies of the Tautomers of Pyridinethiones, J. Phys. Chem. A, 112:3231–3238.  117  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones X  XH OH  N  OH N  H  X=O; pyridinones X=S; pyridinethiones Figure 5.1: Tautomeric forms of pyridinones and pyridinethiones  by elemental analysis, electron impact mass spectrometry, melting point determination, as well as UV-Visible absorption, IR, and NMR spectra. X-ray crystal structures were also obtained for some of these compounds. Although the thiol and thione forms of the tautomer could not be separately isolated, some experimental characterization data suggested that the thione form is the dominant tautomer. A strong νC=S stretch in the IR spectra confirmed the thione form of the tautomer to be the solid-state chemical structure of the pyridinethiones. The solution-state characterization (via NMR and UV-Vis spectra) of these compounds also suggests the presence of the thione tautomeric forms. Nonetheless, the data collected could not be unambiguously assigned to one tautomeric form over the other. As mentioned before, when left in aqueous or alcoholic solutions, the monomeric (thiol) forms of these compounds were found to convert to their dimeric (disulfide) forms. Evidence for the formation of an S–S bridge between two molecules was obtained with IR and NMR spectra. Unfortunately, the S–H or N–H proton could not be assigned distinctly due to probable exchange of the proton between S and N in solution (Fig. 5.1). This process has been studied before, and the structures have been verified by comparing the experimental data with theoretical calculations for the IR and NMR data [367]. Hence, more extensive comparisons between the theoretical and experimental data should allow the assignment of the dominant tautomeric form of these pyridinethiones. Herein, we report ab initio and DFT calculations of Hmppt, its methyl analog, Hdppt, their thiol tautomers, and their dimeric forms. Our results reported here provide a comprehensive library of data for any future comparisons of these and other such similar compounds.  118  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones  S  S OH  N  CH3  N  H  SH  H3C  N  S  N  CH3  CH3  N  CH3  Hmppt-2  Hmppt-1'  HS OH  OH  H  Hmppt-1  HO  SH OH  S  S  HO  2H+ 2e  OH  H3C  N  N  CH3  H3C  Hmppt-d-1  S  Hdppt  H3C  N  N  CH3  CH3  CH3  Hmppt-d-2  Hdppt'  S  CH3  N H  N CH3 CH3  S OH  N H  OH  N CH3 CH3  S  O  S OH  HO  S  O  2H+ 2e  S  O H3C  O N CH3  N CH3 CH3  Hdppt-d  Figure 5.2: Proposed mechanism for the dimerization of the two pyridinethiones studied in this work: 3-Hydroxy-2-methyl-4(para)-pyridinethione (Hmppt) and 3-Hydroxy-1,2-dimethyl-4(para)-pyridinethione (Hdppt)  119  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones S  S  S OH  N  CH3  OH O  CH3  H  Hmppt  OH N  CH3  CH3  Thiomaltol  Hdppt  Figure 5.3: Synthesis of the pyridinethiones: (a) xs NH4 OH, H2 O/EtOH, 34 ℃, 36 h; (b) xs 40% MeNH2 , H2 O, 75 ℃, 12 h.  5.2  Details of Computational Methods  We have performed ab initio and DFT calculations to obtain the NMR, IR, and UV-Vis data for Hmppt, Hdppt, and their dimers. We optimized Hmppt-1 and Hmppt-2 in the gas phase at the B3LYP [370, 371]/6-31G(d,p) and B3LYP/6-311++G(2df,p) levels of theory, and found only negligible differences between two types of geometries. So we will use the B3LYP/631G(d,p) geometries for all species throughout this paper unless specially mentioned. Geometries of Hmppt-1 and Hmppt-2 obtained on different levels are included in the Supporting Information for reference. The total energies and the Gibbs free energies (at 298 K) of molecules of interest were computed at the B3LYP/6-31G(d,p) and MP2/6-31G(d,p) levels of theory. Partial charges were computed from the natural bond orbital (NBO) analysis [372]. The 13 C and 1 H NMR chemical shifts of all tautomers were predicted with the Gauge-independent Atomic Orbital (GIAO) [373] and the Continuous Set of Gauge Transformations (CSGT) [374] approaches at the HF/6-311++G(2df,p), MP2/6-311++G(2df,p), B3LYP/6311++G(2df,p) and PBEPBE/6-311++G(2df,p) levels of theory. The polarizable continuum model (PCM) [375] was employed to account for the solvation effects. All geometries were re-optimized at the B3LYP/6-31G(d,p) level in the solution environment. Vibrational frequencies were calculated at the HF, MP2, and B3LYP levels of theory with the basis set 6-31G(d,p). To improve the agreement between theory and experiment, we took the usual practice of employing scaling factors to bring our calculated frequencies closer to the existing experimental data [376]. Since the scaling factor for B3LYP/6-31G(d,p) calculations is not available, we instead used the scaling factor for B3LYP/6-31G(d), which should not bring any significant numerical error. Time-dependent density functional theory(TDDFT) [377] 120  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones with different density functionals (B3LYP and PBEPBE [378, 379]) was utilized to calculate the UV-Vis spectra. All calculations were performed with the quantum chemistry package of Gaussian 03 [380].  5.3  Results and Discussions  Both Hmppt and Hdppt can dimerize in solution. While in solution, it is very difficult to crystallize the Hmppt-monomer/dimer mixture, so X-ray data are available only for the Hdppt-dimer [362]. X-ray quality crystals for Hdppt-dimer were obtained by slow evaporation in ethyl acetate. The structure is shown in Fig. 5.4, and the selected bond lengths and bond angles are listed in Table 5.1. For comparison, we also provide the data of the optimized geometry from our DFT calculations.  Figure 5.4: ORTEP diagram of Hdppt-dimer (50% thermal ellipsoids), adapted from Ref. [362a] .  As can be seen in Table 5.1, our theoretical results agree quite well with the experimental X-ray data. Because the S–S bond is very flexible, we imposed C2 symmetry on the dimer to accelerate the calculation and to reduce the demand on computational resources (especially in the property calculations). In the end, we found that the optimized geometry without the C2 symmetry constraint is virtually identical to that obtained with the  121  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones constraint. Thus, it is legitimate for us to impose the same C2 symmetry constraint in additional calculations whenever possible. Table 5.1: Selected bond lengths (in ˚ A) and bond angles (in deg) of the Hdppt-dimer.  Bond S(1)-S(1∗ ) S(1)-C(1) O(1)-C(2) N(1)-C(3) N(1)-C(4) N(1)-C(7) C(1)-C(2) C(1)-C(5) C(2)-C(3) C(4)-C(5) C(3)-C(6)  a b  X-raya/Theoryb 2.0472(7)/2.1306 1.763(1)/1.771 1.278(2)/1.245 1.362(2)/1.355 1.351(2)/1.364 1.482(2)/1.472 1.424(2)/1.457 1.383(2)/1.390 1.427(2)/1.462 1.377(2)/1.385 1.489(2)/1.497  Bond Angles S(1)-S(1∗ )-C(1) S(1)-C(1)-C(2) S(1)-C(1)-C(5) C(3)-N(1)-C(4) C(3)-N(1)-C(7) C(4)-N(1)-C(7) N(1)-C(3)-C(2) N(1)-C(3)-C(6) N(1)-C(4)-C(5) O(1)-C(2)-C(1) O(1)-C(2)-C(3) C(2)-C(1)-C(5) C(1)-C(2)-C(3) C(1)-C(5)-C(4) C(2)-C(3)-C(6)  X-raya/Theoryb 104.22(5)/102.59 112.1(1)/119.8 126.0(1)/119.3 123.0(1)/122.8 119.4(1)/119.3 117.5(1)/117.9 119.9(1)/121.2 120.1(1)/120.7 119.9(1)/119.4 120.8(1)/124.4 123.4(1)/121.0 121.9(1)/120.6 115.8(1)/114.6 119.4(1)/121.3 120.0(1)/118.1  From Ref. [362a] with experimental uncertainty shown in the parentheses. Optimized geometry from B3LYP/6-31G(d,p) calculations.  We carried out NBO partial charge analysis for Hmppt-1 and other molecules shown in Fig. 5.2 (Hmppt-1’ is a resonance form of Hmppt1). The partial charge results are shown in Fig.s 5.5–5.7. For Hmppt-1 we found that both in vacuum and in solution (H2 O or DMSO), the N atom always carries a large negative charge (> −0.5), and the S atom always carries a relatively small negative charge (< −0.5). This might suggest that Hmppt-1’ poorly represents the actual electronic structure, since in Hmppt1’ the N atom should carry less negative charge than usual and the S atom should carry more negative charge. Moreover, the bond length of C1–C2 (see the numbering in Fig.s 5.5–5.7) in Hmppt-1 is calculated to be 1.43˚ A, ˚ which is much longer than the bond length of C2=C3 (1.37A), indicating a double bond structure between C2 and C3. Thus, it is safe to conclude that Hmppt-1’ does not represent the dominant form of the real electronic structure. We also found a similar situation in the charge and structural 122  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones analyses of Hdppt and Hdppt’: Hdppt’ is not a good representation of the actual electronic structure of Hdppt. Hdppt-dimer might be somewhat different, since it can only exist in the formally charged form (see Fig. 5.2). Its N atom has a charge c.a. −0.3, smaller than those in Hmppt-1 and Hmppt-2 and their dimers, whereas its O atom carries a larger negative charge (> −0.7). These results, compared to charge distributions of the Hmppt dimers, support the N-formal-positivecharged structure of the Hdppt-dimer, in agreement with the X-ray crystal structure data. It was also observed that different solvents affect the charge distributions slightly. The dipole moments of each species and total/free energy differences between them were also studied with small and large basis sets, both in vacuum and in solution (see Tab.s 5.2 and 5.3 and Appendix D for details). We found that basis set affects dipole moments only slightly. Dipole moments calculated from the small basis set are included in Supporting Information for reference. Data shown in Tab. 5.3 clearly indicate that Hmppt-1 is more stable than is Hmppt-2 in DMSO or H2 O, although the MP2 calculations predict that Hmppt-2 is just slightly more stable than is Hmppt-1 in vacuum. Such irregularity should not pose any real problem because we already have coherent results (from different methods) for the relative stability in DMSO or H2 O, based on which we can compare our theoretical results with experimental data. As expected from those positive energy differences in Tab. 5.3, the more polar thione form of Hmppt is favored in the polar solution environment. In addition, the polar solvents even increase the energy difference between Hmppt-2 and Hmppt-1. Table 5.2: Theoretical predictions of the dipole moments (in Debye) of all tautomers in different media (vacuum, DMSO, or H2 O), calculated at the B3LYP/6311++G(2df,p)//B3LYP/6-31G(d,p) level of theory.  Tautomer Hmppt-1 Hmppt-2 Hmppt-d-1 Hmppt-d-2  Vacuum 7.53 2.18 1.67 10.06  DMSO 12.64 3.54 3.62 18.84  H2 O 12.74 3.72 3.83 19.38  The dimers show another story. In the tautomerism of Hmppt, the proton migrates between the N and S atoms, but the proton migrates between 123  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones  (-0.214/-0.442/-0.449) S  (0.268/0.280/0.280) 10 H  1C  5C (0.299/ 0.291/ 0.291)  2C  (-0.289/ -0.292/ -0.292)  (-0.529/ 3C -0.507/ -0.507) (0.002/ 0.018/ N 0.018)  (0.244/ 0.279/ 0.280) 11 H  (-0.704/ -0.712/ -0.714) O  (-0.220/ -0.196/ -0.196)  8C (0.153/ 0.194/ 0.195)  13 H (0.499/0.516/0.521)  CH3  12 H (0.440/0.504/0.502)  (a) Hmppt-1 16 H (0.157/0.216/0.212) S(-0.028/-0.070/-0.057) (-0.241/ -0.230/ -0.227) 1C  (0.255/0.271/0.271) 10 H (-0.265/ -0.270/ -0.277)  (0.229/0.243/0.242) 11 H  2C  3C (-0.010/ -0.021/ -0.019)  (-0.692/-0.710/-0.712) O 5C (0.302/ 0.301/ 0.297)  12 H (0.501/0.526/0.533)  (0.190/0.184/0.184) 8C N (-0.440/ -0.480/ -0.486)  CH3  (b) Hmppt-2  Figure 5.5: NBO charge anlysis of pyridinethiones and their dimers in Vacuum/DMSO/H2 O (part I).  124  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones  (0.118/0.121/0.121) S (-0.275/ -0.268/ -0.268) 2C  (0.259/0.273/0.275) 15 H  14C  (-0.259/ -0.259/ -0.260)  3C (0.315/ 0.317/ 0.318)  5H (0.504/0.535/0.539)  (0.190/0.187/0.186) 6C  12C (-0.011/ -0.020/ -0.021)  (0.232/0.245/0.245) 13 H  (-0.685/-0.703/-0.705) O  N (-0.433/ -0.469/ -0.471)  CH3  (a) Hmppt-d-1 (left part)  (0.188/0.115/0.110) S (-0.266/ -0.278/ -0.280) 2C  (0.262/0.279/0.280) 14 H (-0.263/ -0.259/ -0.262)  (0.240/0.272/0.273) 12 H  13C  11C (-0.074/ -0.052/ -0.050)  (-0.644/-0.722/-0.725) O 3C (0.379/ 0.372/ 0.372)  (-0.479/ -0.476/ -0.471)  (0.180/0.199/0.202) 5C  N  CH3  29 H (0.438/0.498/0.500)  (b) Hmppt-d-2 (left part)  Figure 5.6: NBO charge anlysis of pyridinethiones and their dimers in Vacuum/DMSO/H2 O (part II).  125  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones  (-0.223/-0.430/-0.437) S  (0.267/0.282/0.282) 11 H  1C  (-0.286/ -0.287/ -0.287)  (0.239/ 0.274/ 0.275) 12 H  (-0.707/ -0.714/ -0.716) O  (-0.220/ -0.195/ -0.195)  5C (0.305/ 0.298/ 0.298)  2C  (-0.348/ 3C -0.323/ -0.322) (0.025/ 0.018/ N 0.025)  8C (0.155/ 0.194/ 0.195)  13 H (0.499/0.515/0.519)  CH3  CH3  (a) Hdppt  (0.118/0.119/0.116) S (-0.264/ -0.273/ -0,275) 1C  (0.262/0.281/0.282) 21 H (-0.265/ -0.256/ -0.256)  (0.234/0.268/0.269) 22 H  2C  3C (-0.065/ -0.045/ -0.044)  (-0.650/-0.723/-0.726) O 5C (0.386/ 0.379/ 0.378) (0.174/0.198/0.200) 8C  N(-0.385/ -0.294/ -0.293)  CH3  CH3  (b) Hdppt-d (left part)  Figure 5.7: NBO charge anlysis of pyridinethiones and their dimers in Vacuum/DMSO/H2 O (part III).  126  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones Table 5.3: Total energy and Gibbs free energy (at 298 K) differences with the zero-point energy correction (in kcal/mol) of all tautomers in different media (vacuum, DMSO, or H2 O), calculated at the B3LYP/6-31G(d,p) and MP2/631G(d,p)//B3LYP/6-31G(d,p) levels of theory.  Energy Difference EHmppt-2 −EHmppt-1 EHmppt-d-2 −EHmppt-d-1  Method B3LYP MP2 B3LYP MP2  Vacuum ∆E ∆G 4.0 4.1 −0.3 −0.9 26.6 26.4 27.8 27.6  DMSO ∆E ∆G 12.5 12.3 7.9 7.7 5.2 5.3 8.0 8.0  H2 O ∆E ∆G 12.6 12.1 7.9 7.6 4.4 4.0 7.5 7.1  the N and O atoms in the Hmppt-dimer. Hmppt-d-2 has a structure with a formal discrete positive/negative charge distribution that usually has an higher energy in vacuum, and its dipole moment is very large (∼ 10 Debyes), compared to the small dipole moment of Hmppt-d-1 (only ∼ 1.7 Debyes in vacuum). Although Hmppt-d-1 is favored in terms of total energy, the energy difference between Hmppt-d-2 and Hmppt-d-1 is smaller than the corresponding values of the Hmppt monomeric tautomers in solutions. This is, of course, because polar solvents can stabilize the more polar Hmppt-d-2. It is interesting to consider how the S and H atoms are bonded in these molecules, so we carried out vibrational frequency calculations and compared the results with the experimental data. The final results are shown in Tab. 5.4. Experimental IR data were collected from samples in solid phase, but a broad OH peak for H2O was seen in the Hmppt-dimer IR spectrum because the sample was not dry. In addition, O–H groups in Hmppt or in H2 O can form H-bonds with other O–H and N–H groups, which makes the IR spectrum above 2700 cm−1 very complicated. Hence, it is very difficult to match the calculated results with the experimental data precisely. Theoretically we found the vibrational frequencies of O–H and N–H bonds are 3055 ∼ 3192 cm−1 and 3329 ∼ 3499 cm−1 , respectively, but in direct contradiction to previous experimental assignments [362]: 3427 cm−1 for O–H and 3061 cm−1 for N–H vibrations. Even after we increased the basis set from 6-31G(d,p) to 6-311++G(2df,p) and reoptimized the geometries at B3LYP/6-311++G(2df,p), we still got 3179 cm−1 for O–H and 3484 cm−1 for N–H vibrations. Since all different theoretical methods consistently predict the vibrational frequencies of O–H and N–H bonds, further experimental studies should be carried out to resolve this discrepancy fully. 127  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones Table 5.4: Theoretical IR peak positions (in cm−1 ) and intensities of the important peaks (in parentheses) calculated at the HF/6-31G(d,p), MP2/6-31G(d,p), and B3LYP/6-31G(d,p) levels of theory, compared with the experimental results.  Scaling factorc  HF  MP2  B3LYP  0.8992  0.9370  0.9614  Expt.a Assignmentb  3055(171.5)  3161(98.0)  3192(160.4)  3329(182.5) 1605 Hmppt-1 1442 1189 1136(468.5) 869 804  3455(177.0) 1598 1456 1250 1154(119.1) 858 805  3499(119.5) 1607 1414 1220 1162(104.9) 874 826  3061 2932 2821 3427 1590 1445 1216 1174 886 782  3356(128.1) 2437(5.9) 1577 1470 Hmppt-2 1226 1157(12.9) 850 767  3488(104.2) 2507(4.1) 1542 452 1218 1124(53.5) 840 788  3527(104.2) 2553(9.1) 1561 1431 1208 1134(55.9) 894 805  3427 1590 1445 1216 1174 886 782  3370(133.7) 1574 Hmppt1535 d-1 1226 849 798  3505(103.3) 1534 1218 839 789  3544(126.6) 1556 1545 1203 851 803  3285(165.7)  3398(115.9)  3437(93.6)  3508 1630 1510 1215 833 816 3417 2806 2731 1630 1510 1215  Hmpptd-2  1587 1518 1218  1655 1608 1530 1565 1227 1232 Continued on next page  νNH (νOH ) νOH (νNH ) Ring and C–N–C bands νC=S Ring and C–N–C bands (νOH ) (νSH ) Ring and C–N–C bands (νC–H ) Ring and C–N–C bands νH2 O (νOH ) Ring and C–N–C bands  (νNH )  Ring and C–N–C bands 128  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones Table 5.4 – Continued from previous page Expt.a Assignmentb HF MP2 B3LYP 833 856 840 858 816 817 802 818 a b c  See the supporting information of Ref. [362] (with a ±4 cm−1 uncertainty). See Ref. [362]. New assignments from this work are shown in parentheses. See Ref. [376].  For the carbon–sulfur bond, the C=S peak is observed in Hmppt but not in Hmppt-dimer, which strongly suggests the formation of the S–S bond. Due to the reason of symmetry, we were unable to see the vibration of the S–S bond stretching in the IR spectra. It is clear that the calculated C=S stretching frequencies of Hmppt-1 agree better with the experimental data than do those of Hmppt-2 (the corresponding absolute average errors are 24 and 36 cm−1 , respectively), which favors Hmppt-1 in the tautomerism. According to our calculation, the absence of the S–H peaks around 2437 ∼ 2553 cm−1 in the experiment also disfavors the existence of Hmppt-2 in the sample. The GIAO results of the 13 C and 1 H NMR chemical shifts of all tautomers are compared with the experimental data in Tab.s 5.5–5.8. Because the CSGT results are basically consistent to the GIAO results, we have collected the CSGT results in Tab.s D.6–D.9 in Appendix D. From the average errors of the calculations, we found that the 13 C NMR chemical shifts of Hmppt-1 match better with the experimental data than do those of Hmppt2, and the same is true for those of Hmppt-d-1 and Hmppt-d-2. These results support the conclusion that Hmppt-1 and Hmppt-d-1 are favored in solution, specifically in DMSO. From the calculated 1H NMR results, we can see that Hmppt has an active H (from N–H) with the chemical shift ranging from 9.2 to 10.7 ppm. For Hmppt-2, we found the chemical shift for H (from S–H) ranges from 4.8 to 5.3 ppm. The experimental value is 12.84 ppm, much closer to the theoretical prediction of Hmppt-1. This fact thus strongly suggests that the Hmppt-1 form is favored. The solvent effects for the NMR properties of both tautomers were not considerable, since the average errors of the chemical shifts predicted in the gas and solution phases are roughly the same. In addition, for different levels of theory (ab initio and DFT) and methods computing NMR parameters (GIAO and CSGT), there are no significant differences in the results. Similarly, the calculated 13 C NMR chemical shifts also indicate that Hmppt-d-1 is favored (see Tab.s 5.7 and 5.8). Moreover, we notice that the theoretical chemical shifts for the active H (from O–H) in Hmppt-d-1 and 129  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones  Table 5.5: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-1 in DMSO, calculated with the GIAO method with the 6-311++G(2df,p) basis set at the HF, MP2, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C1a C2 C3 C5 C8 δb H10 H11 H12 H13 a b  c  HF 194.051 129.948 145.099 154.951 145.135 13.4 8.111 8.399 10.721 8.137  MP2 170.256 137.288 128.846 163.986 128.136 5.3 8.723 7.733 10.742 8.026  B3LYP 181.479 134.273 134.222 161.364 135.268 8.9 8.002 7.767 10.013 8.162  PBEPBE 172.171 131.980 128.607 158.655 129.426 3.8 7.938 7.598 9.763 8.181  Expt.c 169.86 125.22 128.34 151.76 126.85 7.29 7.50 12.84 8.64  See Fig. 5.5 for the numbering of the atoms. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  for the active H (from N–H) in Hmppt-d-2 are 6.9 ∼ 7.4 and 10.5 ∼ 12.2 ppm, respectively, whereas the experimental value is about 9.8 ppm, roughly between these two ranges. This indicates that both tautomeric forms contribute to the NMR spectrum of the Hmppt dimer and that Hmppt-d-1 is only slightly more favored than Hmppt-d-2. UV-Vis absorption spectra of Hmppts and its dimers calculated with TDDFT on different levels are compared with the available experimental data in Tab. 5.9. For Hmppt-1 and Hmppt-2, we found calculations with the large basis set, 6-311++G(2df,p), give better results than do those with the small basis set, 6-31G(d,p), but they show the same tendency. So, results of the monomers from the small basis set are shown only in Appendix D for comparison. Because of the sizes of the dimers, only small basis set results are available. Calculations of the monomers suggest that such small basis set results are still acceptable as a compromise between performance and accuracy. As different density functionals are concerned, we found that PBEPBE gives similar results as B3LYP does, so we only list the B3LYP results in Tab. 5.9 and keep the PBEPBE results in Appendix D for refer130  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones  Table 5.6: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-2 in DMSO, calculated with the GIAO method with the 6-311++G(2df,p) basis set at the HF, MP2, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C1a C2 C3 C5 C8 δb H10 H11 H12 H16 a b  c  HF 133.187 134.987 150.782 158.595 159.701 21.7 8.045 8.651 6.917 5.340  MP2 124.479 134.989 143.139 160.767 149.767 20.4 8.053 8.414 6.875 5.246  B3LYP 131.071 136.307 147.697 161.260 155.974 21.6 7.874 8.434 6.875 4.903  PBEPBE 127.011 133.869 144.199 158.135 151.737 19.7 7.832 8.407 6.922 4.881  Expt.c 169.86 125.22 128.34 151.76 126.85 7.29 7.50 8.64 12.84  See Fig. 5.5 for the numbering of the atoms. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  ence. The maximum UV-Vis absorption wavelength, λmax , of Hmppt was determined experimentally to be 326 nm. Calculations for Hmppt-1 yield a relatively accurate estimation of these absorption peak positions (see Tab. 5.9), whereas calculations for Hmppt-2 do not support the existence of such a peak in the proximity of 326 nm. This strongly disfavors Hmppt-2 in the tautomerism. We can also reach a similar conclusion for the Hmppt-dimers. Our theoretical prediction (246 nm) matches the experimental λmax value 238 nm well for Hmppt-d-1, while Hmppt-d-2 has only very weak peaks around 238 nm (at 234 nm), which certainly cannot be considered as candidates for λmax . On the other hand, Hmppt-d-2 does show very strong peaks above 400 nm in our calculations (444 nm, not shown in Tab. 5.9), which were not observed in our previous experiments. Comparison of the calculations in the gas phase and in H2 O shows the solvation effects are negligible. Hence, the characteristics of the corresponding UV-Vis spectra strongly infer that the structure of Hmppt-d-1 represents the actual structure better than does Hmppt-d-2. 131  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones  Table 5.7: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-d-1 in DMSO, calculated with the GIAO method with the 6-311++G(2df,p) basis set at the HF, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C2a C3 C6 C12 C14 δb H5 H13 H15 a b  c  5.4  HF 137.520 158.745 160.864 149.800 135.285 11.5 7.320 8.588 7.391  B3LYP 137.894 159.879 157.534 147.231 135.957 10.8 7.306 8.382 7.248  PBEPBE 133.841 156.344 153.541 143.949 133.257 7.3 7.396 8.371 7.210  Expt.c 133.16 148.99 145.69 139.25 117.58 9.8 7.87 7.10  See Fig. 5.6 for the numbering of the atoms. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  Conclusions  We have presented here a comprehensive theoretical study on the tautomers of Hmppt and its dimer in the vacuum and in polar solutions (DMSO and H2 O). Comparisons were also made with Hdppt and its dimeric form to substantiate our conclusions about the dominant tautomeric form observed in our previous experiments. Calculations at a range of levels of theory consistently support a thione form preference, although the degrees of the preferences may vary. This study can be used as a reference for investigating other similar molecules.  132  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones  Table 5.8: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-d-2 in DMSO, calculated with the GIAO method with the 6-311++G(2df,p) basis set at the HF, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C2a C3 C5 C11 C13 δb H12 H14 H29 a b  c  HF 144.950 178.877 159.927 122.985 138.287 18.6 7.637 8.010 12.192  B3LYP 149.495 174.873 153.375 120.534 136.011 17.4 7.318 7.764 11.256  PBEPBE 146.440 170.277 147.689 116.410 133.203 15.0 7.158 7.757 10.943  Expt.c 133.16 148.99 145.69 139.25 117.58 7.87 7.10 9.8  See Fig. 5.6 for the numbering of the atoms. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  Table 5.9: Theoretical UV-Vis absorption positions (in nm) and oscillator strengths (in parentheses) for single excitations of thiol and thione tautomers of Hmppt and Hmppt-dimer in vacuum and H2 O, calculated at B3LYP levels, compared with experimental data obtained in H2 O. Transitions with large CI coefficients are shown below each peak position.  Tautomers  Hmppt-1c  Vacuum 206 (0.2072) Hb − 3 →Lb 0.29 H−2 →L+2 0.53 H−1 →L+6 −0.14 H−1 →L+7 −0.21  H2 O 206 (0.2621) H−3 →L −0.23 H−2 →L+1 0.56 H−1 →L+4 −0.14 H→L+6 0.19 H→L+20 0.11 252 (0.0975) 249 (0.1454) H−2 →L 0.63 H−3 →L+1 0.13 H−3 →L+2 −0.11 H−2 →L 0.61 H→L+1 0.19 281 (0.0037) 280 (0.0001) Continued on next page  Assignmenta 210 (4.13)  245 (3.92)  276 (3.43)  133  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones Table 5.9 – Continued from previous page Vacuum H2 O Assignmenta H−1 →L+1 0.70 H−2 →L −0.22 H→L+1 0.64 325 (0.3166) 319 (0.3693) 326 (4.21) H−2 →L+2 −0.12 H−2 →L+1 0.11 H→L 0.60 H→L 0.63 199 (0.2005) 208 (0.0583) →L 0.16 H−3 →L 0.41 H−3 H−3 →L+1 −0.27 H−2 →L+1 −0.14 H−2 →L+2 −0.15 H−1 →L+1 0.37 H−2 →L+3 0.11 H−1 →L+2 0.42 0.12 H−1 →L+3 0.15 H→L+1 H→L+1 −0.11 H→L+4 −0.27 H→L+2 0.25 H−1 →L+5 0.18 208 (0.0038) H−2 →L+1 0.27 Hmppt-2c H−1 →L+1 0.55 H−1 →L+2 0.26 H−1 →L+3 −0.12 239 (0.0013) 245 (0.0290) H−3 →L −0.11 H−3 →L 0.11 H−2 →L 0.68 H−2 →L 0.49 H−1 →L −0.42 H→L+1 0.10 269 (0.0223) 266 (0.1388) H→ L+1 0.60 H−3 →L+1 0.12 H→L 0.30 H−2 →L −0.10 H−1 →L+1 0.16 H→L 0.63 206 (0.0747) 206 (0.0426) H−7 →L+1 0.19 H−7 →L+1 0.13 201 (4.27) H−5 →L+3 0.10 H−5 →L+3 0.10 H−3 →L+3 0.10 H−1 →L+2 0.24 −0.20 H−1 →L+2 0.30 H→L+3 H→L+3 −0.25 H→L+4 0.53 H→L+4 0.42 Hmppt-d-1d H→L+6 −0.11 Continued on next page Tautomers  134  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones Table 5.9 – Continued from previous page Vacuum H2 O Assignmenta 206 (0.0264) 206 (0.0279) H−5 →L+1 −0.37 H−5 →L+1 −0.36 H−1 →L+3 −0.21 H−1 →L+3 −0.14 H−1 →L+4 0.49 H−1 →L+4 0.55 H→L+2 0.13 245 (0.1157) 246 (0.1393) 238 (4.43) H−7 →L+2 −0.11 H−7 →L+2 −0.12 H−5 →L+4 0.11 H−1 →L+1 0.66 H−1 →L+1 0.66 270 (0.2546) 264 (0.2938) 270 (3.71) H−5 →L 0.67 H−5 →L 0.67 323 (0.0002) 323 (0.0029) H−5 →L 0.10 H−4 →L −0.13 336 (4.32) H−2 →L 0.68 H−3 → L 0.65 H→L 0.12 364 (0.1262) 354 (0.1581) H→L 0.68 H−3 →L −0.11 H→L 0.67 201 (0.2213) 203 (0.3136) H−10 →L −0.14 H−10 →L −0.12 H−9 →L 0.11 H−7 →L 0.11 H−7 →L 0.19 H−6 →L+1 0.49 H−6 →L+1 0.33 H−5 →L+3 −0.10 H−5 →L+4 −0.13 H−4 →L+4 −0.24 H−4 →L+3 0.13 H−3 →L+3 −0.16 H−3 →L+3 0.11 H−2 →L+4 −0.11 H−3 →L+5 −0.24 H−1 →L+4 −0.17 H−2 →L+6 0.32 H→L+3 0.13 Hmppt-d-2d H−1 →L+3 −0.12 H→L+4 0.13 220 (0.1801) 221 (0.1271) Continued on next page Tautomers  135  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones  Tautomers  a  Table 5.9 – Continued from previous page Vacuum H2 O Assignmenta →L 0.11 H−7 →L+1 −0.12 H−6 H−6 →L 0.23 H−5 →L+2 0.62 H−5 →L+2 −0.29 H−3 →L+4 −0.14 H−4 →L+4 −0.22 H−1 →L+3 0.11 H−3 →L+4 0.32 H−2 →L+3 −0.26 H−1 →L+4 −0.11 H−1 →L+6 0.18 H→L+3 0.12 H→L+5 −0.12 228 (0.1235) H−7 →L+1 −0.15 H−6 →L 0.21 H−5 →L+2 −0.23 H−1 →L+3 0.49 H→L+5 −0.21 234 (0.0777) 234 (0.0257) H−6 →L 0.48 H−4 →L+1 0.12 H−5 →L+2 0.19 H−4 →L+3 −0.10 H−4 →L+4 −0.13 H−3 →L+2 0.42 H−3 →L+4 −0.34 H−3 →L+4 0.36 H−2 →L+3 0.14 H−2 →L+3 −0.35 H→L+3 0.18 271 (0.0002) 269 (0.0008) H−5 →L+1 0.65 H−2 →L+2 −0.12 H−1 →L+3 0.16 H−1 →L+2 0.67 H→L+4 −0.12 364 (0.1134) 318 (0.0844) H−3 →L+1 0.14 H−5 →L −0.19 H−1 →L+1 0.65 H−1 →L+1 0.63 344 (0.0018) 337 (0.0005) H−4 →L 0.29 H−4 →L 0.62 H−3 →L 0.21 H−3 →L+1 0.18 H−2 →L+1 −0.15 H→L+1 0.24 H−1 →L −0.14 H→L+1 0.47  See the supporting information of Ref. [362a].  136  Chapter 5. Theoretical Studies of the Tautomers of Pyridinethiones b  “H” and “L” denote the HOMO and the LUMO, respectively. “H−m” and “L+n” denote the mth orbital below the HOMO and the nth orbital above the LUMO, respectively. c Calculate at B3LYP/6-311++G(2df,p)// B3LYP/6-31G(d,p) level of theory. d Calculate at B3LYP/6-31G(d,p) level of theory.  137  Bibliography  Bibliography [358] P. Metzner, D. R. Hogg, W. Walter, and J. Voss, in Organic Compounds of Sulphur, Selenium and Tellurium, edited by D. R. Hogg (The Chemical Society, London, 1977), vol. 4, p. 125 and 1979, vol. 5, p. 118. [359] (a) E. Block, Reactions of Organosulfur Compounds (Academic, New York, 1978), p. 26; (b) R. J. Cremlyn, An Introduction to Organosulfur Chemistry (Wiley, New York, 1996) chapts. 10 and 11. [360] G. A. Maw, in Sulfur in Organic and Inorganic Chemistry, edited by A. Senning (Marcel Dekker, New York, 1972), vol. 2, p. 113. [361] K. Steliou and M. Mrani, J. Am. Chem. Soc. 104, 3104 (1982) and references therein. [362] (a) V. Monga, B. O. Patrick, and C. Orvig, Inorg. Chem. 44, 2666 (2005). (b) V. Monga, K. H. Thompson, V. G. Yuen, V. Sharma, B. O. Patrick, J. H. McNeill, and C. Orvig, Inorg. Chem. 44, 2678 (2005). [363] J. Lewis, D. Puerta, and S. Cohen, Inorg. Chem. 42, 7455 (2003). [364] G. Tilbrook, R. Hider, and M. Moridani, Patent WO 98/25905 (1998). [365] N. Hoyoku, K. Yukihiro, S. Toshihiko, Y. Susumu, M. Hiromichi, and I. Yoichi, U.S. Patent 5093505 (1992). [366] A. Albert, Heterocyclic Chemistry an Introduction (Athlone, London, 1959). [367] S. Stoyanov, I. Petkov, L. Antonov, and T. Stoyanova, Can. J. Chem. 68, 1482 (1990). [368] H. Besso, K. Imafuku and H. Matsumura, Bull. Chem. Soc. Jpn. 50, 3295 (1977) and references therein. [369] J. D. Roberts and M. C. Caserio, Basic Principles of Organic Chemistry (W. A. Benjamin, New York, 1964). [370] A. D. Becke, Phys. Rev. A 38, 3098 (1988). [371] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988).  138  Bibliography [372] F. Weinhold, in Encyclopedia of Computational Chemistry, edited by P. V. R. Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer III, and P. R. Schreiner (Wiley, Chichester, UK, 1998), vol. 3, pp. 1792–1811. [373] (a) F. London, J. Phys. Radium. (b) R. McWeeny, Phys. Rev. 126, 1028 (1962). (c) R. Ditchfield, Mol. Phys. 27, 789 (1974). (d) J. L. Dodds, R. McWeeny, and A. Sadlej, J. Mol. Phys. 41, 1419 (1980). (e) K. Wolinski, J. F. Hilton, and P. Pulay, J. Am. Chem. Soc. 112, 8251 (1990). [374] (a) T. A. Keith and R. F. W. Bader, Chem. Phys. Lett. 194, 1 (1992). (b) T. A. Keith and R. F. W. Bader, Chem. Phys. Lett. 210, 223 (1993). (c) J. R. Cheeseman, M. J. Frisch, G. W. Trucks, and T. A. J. Keith, Chem. Phys. 104, 5497 (1996). [375] (a) M. T. Canc´es, B. Mennucci, and J. Tomasi, J. Chem. Phys. 107, 3032 (1997). (b) M. Cossi, V. Barone, B. Mennucci, and J. Tomasi, Chem. Phys. Lett. 286, 253 (1998). (c) B. Mennucci and J. Tomasi, J. Chem. Phys. 106, 5151 (1997). [376] A. Scott and L. Radom, J. Phys. Chem. 100, 16502 (1996). [377] (a) E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984). (b) E. K. U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1985). [378] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). [379] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 78, 1396 (1997). [380] Gaussian 03, Revision C.02, M. J. Frisch et al., Gaussian, Inc., Wallingford CT, 2004.  139  Chapter 6  Conclusions and Perspective The aim of this thesis is to explore the rigorous mathematical foundation, algorithm development, and the chemical applications of DFT. A solid mathematical foundation of DFT is a long-time dream of many mathematical physicists and theoretical chemists. Although Lieb has published a profound mathematical analysis on DFT for Coulomb systems [381], his analysis has not gained much attention from theoretical chemists. It is mainly because theoretical chemists are unfamiliar with the necessary mathematical knowledge and the language of chemists and mathematicians are different. Some researchers might think discussions on these kinds of problems are formal and have no actual influences on practical applications, but this is not the case. The long-range behavior of approximate XC potentials determines their performance with many response properties. The existence or not of “functional derivative discontinuity” when the system has an integer number of electrons affects the correct asymptotics of the XC potential. One must determine whether a constant shift should be added to the XC potential when the number of electrons goes through an integer number. This problem is related to the definition of functional derivative and chemical potential. As we have presented in the previous chapters, all the points above are controversial. Moreover, researchers even disagree with each other on the differentiability of the energy density functional! Like calculus in its early years, DFT is also a giant standing on his feet of clay. More formal work needs to be done and many questions need to be clarified to set up a complete and rigorous mathematical foundation of DFT. Iterative SCF cycles are ubiquitous in quantum chemistry calculations nowadays. So accelerating the convergence of such SCF cycles becomes a very important topic in the method development of quantum chemistry. After decades of efforts, there are robust schemes such as the level shifting [382] and DIIS [383] available. Since DFT focuses on electron density, acceleration of the total energy convergence by comparison of the electron density and other density related quantities in different SCF cycles is possible. Our group has done a series of works on this topic [384] and found it a promising direction to make DFT or even any general SCF-based quantum chemistry 140  Chapter 6. Conclusions and Perspective method much faster. This timesaver will not only extend our understanding of electronic structures to those of larger systems but also make the AIMD simulations on long time scales feasible. Modern chemistry and physics have become more and more intertwined so that the electronic structure becomes the starting point to understand a chemical phenomenon precisely. The final goal of any method development is solving practical problems. DFT is simple, easy to apply and of moderate computing requirements, this is why it is so popular among computational chemists. In the coming years, we will continue to have great interest in using DFT as a powerful tool to explore organic and biochemical reaction mechanisms, to predict spectroscopy properties of molecules, to design novel materials, and to simulate important processes taking place in the physical or biological world.  141  Bibliography  Bibliography [381] E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983). [382] V. R. Saunders and I. H. Hillier, Int. J. Quantum Chem. 7, 699 (1973). [383] P. Pulay, Chem. Phys. Lett. 73, 393 (1980). [384] B. Zhou and Y. A. Wang, J. Chem. Phys. 128, 084101 (2008).  142  Appendix A  Another Way to Solve the Coupled System To exactly solve the coupled system of Eq. (2.31) is not easy, since there are infinite number of coupling terms on the right-hand side of Eq. (2.31). Methods in the literature to attack such problems work for systems only with several coupled equations [385–389]. Since a complete decoupling of Eq. (2.31) might not be feasible, we now seek an approximate decoupling scheme. First, we make all orbitals of {χi } positive in the long range. We then re-index the set of {χi } to form a new set of orbitals {κi }, such that κj > κk for j < k asymptotically. The coupled equation for κ1 is −  κ N −1 1 d2 κ1 C11 κ + I + v (r) + κ1 + κ = ext 1 1 2 dr2 r r2  ∞  κ C1j  j=1  rM1j  κ  κj ,  (A.1)  κ , C κ , and M κ are constants associated with κ and have where I1κ , C11 1 1j 1j similar definitions as those in Eq. (2.31). Now, in the above equation, if all coupled orbitals have exponential tails in the long range, the left-hand side of Eq. (A.1) must have the same exponential tail as κ1 does. Thus, on the right-hand side, we can safely omit any term that decays faster than κ1 does in the long range. This leads to an approximate decoupled equation for κ1 :  −  κ 1 d2 κ1 N −1 C11 κ + I + v (r) + κ1 . κ ≃ ext 1 1 2 dr2 r r2  (A.2)  The solution of this equation is simply κ1 =  ∞ n=0  h1n rn  κ ˜1 ,  (A.3)  where h1n is some constant and κ ˜ 1 is the homogeneous solution of Eq. (A.1). 143  Appendix A. Another Way to Solve the Coupled System Then, we focus on the second slowest-decaying orbital κ2 . This time, we omit any term that decays faster than κ2 does in the long range on the right-hand side of the equation for κ2 , similar to Eq. (A.1). We get −  κ κ 1 d2 κ2 N −1 C12 C22 κ + I + v (r) + κ2 . κ ≃ κ + κ ext 2 1 2 2 dr2 r r2 rM12  (A.4)  Substituting Eq. (A.3) into Eq. (A.4) and invoking the same discussion below Eq. (2.34), we can write the solution for κ2 as: κ2 = [L ]2 κ ˜2 +  [L ]12 ˜1 , κ κ rM12  (A.5)  where [L ]2 and [L ]12 denote some Laurent-type series of r with coefficients Lk1 and Lk12 , respectively, similar to those two shorthand notations defined in Eqs. (2.75) and (2.76). In the same manner, we have the solution for any general orbital κi (i = 1): i−1 [L ]ij κi = [L ]i κ ˜i + ˜j , (A.6) κ κ Mij j=1 r which is very similar to Eq. (2.38). The only difference is that the numerators of the coupling coefficients are some Laurent series in Eq. (A.6) instead of some constants in Eq. (2.38). If fiκ = κi /r, we then have i−1  fiκ = [L ]i f˜iκ + j=1  [L ]ij ˜κ κ fj . rMij  (A.7)  With Eq. (A.7) in place of Eq. (2.39), we can produce a similar proof for Theorem 1 in Section II. For the exponential tail matching procedure in Section IV, we still have a summation of square terms and cross terms, virtually identical to the right-hand side of Eq. (2.74), after we express the electron density in terms of a summation of squares of all Dyson orbitals. Therefore, all the forthcoming analysis stays the same.  144  Bibliography  Bibliography [385] X. C. Cao, J. Phys. A: Math. Gen. 14, 1069 (1981). [386] X. C. Cao, J. Phys. A: Math. Gen. 15, 2727 (1982). [387] M. Humi, J. Phys. A: Math. Gen. 18, 1085 (1985). [388] X. C. Cao, J. Phys. A: Math. Gen. 25, 3749 (1992). [389] X. C. Cao, J. Phys. A: Math. Gen. 21, 617 (1988).  145  Appendix B  Coordinate Scaling Transform of the Eigenequation Consider the following eigenvalue problem: d2 y Z − my = ǫy , 2 dr r  (B.1)  where Z and m are constants. Apply a scaling transformation on the variable r, x = kr , (B.2) where k is a positive constant. Substituting Eq. (B.2) into Eq. (B.1), we have d2 y Zk m−2 ǫ − y = 2y . (B.3) 2 m dx x k Comparing Eq. (B.3) with Eq. (B.1), we can easily conclude that if we scale Z to Zk m−2 , the eigenvalue η is scaled by a factor of k −2 . So if ǫ > 0, we can always adjust Z to match the corresponding eigenvalue ǫ to any positive value.  146  Appendix C  Method of Dominant Balance Solution to Find the Asymptotic Solutions to a Differential Equation We want to identify the asymptotics of y(r) when r → ∞ in the following differential equation: d2 y Z − ǫ2 + m 2 dr r  y=0,  (C.1)  where ǫ > 0, Z, and m (0 < m < 1) are some constants. Without losing generality, let us define y(r) = eS(r) . (C.2) Substituting Eq. (C.2) into Eq. (C.1), we have d2 S + dr2  dS dr  2  = ǫ2 +  Z . rm  (C.3)  We already know that to first order S(r) ∼ −ǫr ,  (C.4)  which ensures that the second-order derivative term on the left-hand side of Eq. (C.3) can be safely ignored at large distance, d2 S ≪ dr2  dS dr  2  .  (C.5)  Eq. (C.3) then delivers Z Z Z2 dS ∼ − ǫ2 + m = −ǫ 1 + 2 m − 4 2m + · · · dr r 2ǫ r 8ǫ r  .  (C.6) 147  Appendix C. Method of Dominant Balance Solution Truncating the last expression above at first order and then integrating over r, we obtain S(r) ∼ −ǫr − λr1−m , (C.7) where λ=  Z . 2ǫ(1 − m)  (C.8)  Finally, the expression of y(r) asymptotically approaches y(r) ∼ e−λr  1−m  e−ǫr .  (C.9)  148  Appendix D  Supporting Information of Chapter 5 S  13 H 0.991  1.687 118.8  125.4  10 H 1.081  118.4  1.443  1.421  5C  2C  119.7  1.362 123.3 1.081  116.3  O  1.342  119.6  1.371  3C  11H  104.5  1C  119.2  8C  1.356  119.1  1.373  N  124.3  117.6  1.495  CH3  118.0  1.008  12 H  (a) Hmppt-1 geometry with 6-311++G(2df,p) S  13 H 0.993  1.696 118.8  125.5  10 H 1.083  1.424  5C  1.368 123.5  11H  O  1.344  119.4  1.378  3C  116.3  118.4  1.448  2C  119.7  1.083  104.1  1C  119.1  8C  1.360  119.2  1.377  N  123.9  118.0  1.498  117.9  CH3  1.010  12 H  (b) Hmppt-1 geometry with 6-31G(d,p)  Figure D.1: Geometries of Hmppt-1 dimers from calculations with small and large basis sets.  149  Appendix D. Supporting Information of Chapter 5 16 H 1.349 S 97.4  12 H 0.972  1.794 119.4  121.9  10 H 1.082  5C  2C 1.384  120.5 1.085  122.9  1.394  1.392  121.1  O  1.353  117.9  1.409  3C  8C 119.6  116.4 1.336  11H  108.1  1C  120.5  N  120.1  118.7 1.329  1.500  CH3  (a) Hmppt-2 geometry with 6-311++G(2df,p), 16H-S-1C-5C=91.0 degree. 16 H 1.352 S  12 H  97.6  0.975  1.803 119.2  122.0  10 H 1.085  1C  120.4 1.398  121.2  1.087  5C  2C  8C 119.2  1.342  O  1.355  117.8  1.415  3C  116.3  11H  123.3  1.401  1.390 120.4  107.6  N  119.9  118.5 1.334  1.505  CH3  (b) Hmppt-2 geometry with 6-31G(d,p), 16H-S-1C-5C=93.4 degree.  Figure D.2: Geometries of Hmppt-2 dimers from calculations with small and large basis sets. Table D.1: Theoretical predictions of the dipole moments (in Debye) of all tautomers in different media (vacuum, DMSO, or H2 O), calculated at the B3LYP/631G(d,p) level of theory.  Tautomer Hmppt-1 Hmppt-2 Hmppt-d-1 Hmppt-d-2  Vacuum 7.67 2.13 1.39 10.08  DMSO 12.57 3.32 3.11 17.93  H2 O 12.67 3.46 3.34 18.4  150  Appendix D. Supporting Information of Chapter 5 Table D.2: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-1 in vacuum, calculated with the GIAO and CSGT method with the 6-311++G(2df,p) basis set at the HF, MP2, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C1a C2 C3 C5 C8 δc H10 H11 H12 H13 a b c d  e  HF 214.387 /214.014b 126.838 /126.683 138.707 /139.099 154.849 /154.928 131.719 /131.652 12.9/12.9 7.685 /7.152 7.331 /6.882 7.231 /6.665 8.994 /8.179  MP2d 180.614 132.049 121.954 162.859 119.228 8.5 7.851 6.802 7.558 8.869  B3LYP 192.353 /191.529b 131.453 /131.22 127.467 /127.800 161.498 /161.176 124.196 /123.934 8.4/8.1 7.557 /7.052 6.827 /6.353 7.045 /6.464 8.906 /7.992  PBEPBE 180.870 /180.135b 129.254 /129.250 121.935 /122.435 158.561 /158.374 119.142 /118.988 7.2/6.9 7.496 /7.007 6.715 /6.236 6.981 /6.398 8.898 /7.976  Expt.e 169.86 125.22 128.34 151.76 126.85  7.29 7.50 12.84 8.64  See Fig. 5.5 for the numbering of the atoms. GIAO/CSGT results. Average absolute error with respect to the 13 C NMR experimental data. Only GIAO results can be obtained from MP2 calculations within the Gaussian 03 package. From the supporting information of Ref. [362a].  151  Appendix D. Supporting Information of Chapter 5 Table D.3: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-2 in vacuum, calculated with the GIAO and CSGT method with the 6-311++G(2df,p) basis set at the HF, MP2, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C1a C2 C3 C5 C8 δc H10 H11 H12 H16 a b c d  e  HF 126.247 /126.054b 133.269 /133.203 148.972 /148.913 158.504 /158.560 159.511 /159.242 22.3/22.3 7.537 /7.055 8.430 /7.938 6.574 /6.004 3.121 /3.067  MP2d 118.412 131.922 141.883 158.894 149.318 20.3 7.434 8.124 6.408 2.883  B3LYP 124.515 /124.219b 134.254 /134.17 146.712 /146.473 160.458 /160.365 156.333 /155.217 22.2/21.9 7.394 /6.932 8.304 /7.803 6.602 /5.984 2.894 /2.794  PBEPBE 120.717 /120.577b 131.894 /131.993 143.497 /143.400 157.222 /157.303 152.196 /151.060 20.4/20.2 7.362 /6.913 8.302 /7.809 6.692 /6.066 2.921 /2.801  Expt.e 169.86 125.22 128.34 151.76 126.85  7.29 7.50 8.64 12.84  See Fig. 5.5 for the numbering of the atoms. GIAO/CSGT results. Average absolute error with respect to the 13 C NMR experimental data. Only GIAO results can be obtained from MP2 calculations within the Gaussian 03 package. From the supporting information of Ref. [362a].  152  Appendix D. Supporting Information of Chapter 5 Table D.4: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-d-1 in vacuum, calculated with the GIAO and CSGT method with the 6-311++G(2df,p) basis set at the HF, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C2a C3 C6 C12 C14 δc H5 H13 H15 a b c d  HF 134.183/133.957b 157.818/158.057 160.046/159.869 148.321/148.224 132.913/132.800 9.7/9.6 6.445/6.033 8.421/7.957 7.036/6.641  B3LYP 135.426/134.820b 158.448/158.849 157.103/156.582 146.519/146.422 133.216/132.639 9.2/8.9 6.526/6.084 8.317/7.858 6.935/6.511  PBEPBE 131.763/131.295b 154.779/155.462 153.097/152.750 143.487/143.582 130.561/130.097 6.4/6.4 6.660/6.216 8.327/7.875 6.922/6.493  Expt.d 133.16 148.99 145.69 139.25 117.58 9.8 7.87 7.10  See Fig. 5.6 for the numbering of the atoms. GIAO/CSGT results. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  Table D.5: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-d-2 in vacuum, calculated with the GIAO and CSGT method with the 6-311++G(2df,p) basis set at the HF, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C2a C3 C5 C11 C13 δc H12 H14 H29 a b c d  HF 153.015/152.030b 177.881/177.963 154.187/154.067 112.238/112.566 135.734/135.858 20.5/20.2 6.630/6.143 7.707/7.273 9.942/9.441  B3LYP 158.818/157.491b 172.921/172.612 146.277/145.638 109.975/110.814 131.933/131.849 18.8/18.1 6.365/5.934 7.502/7.040 9.146/8.679  PBEPBE 155.360/154.227b 168.044/167.967 140.429/139.564 107.136/108.035 128.450/128.521 17.9/17.7 6.301/5.887 7.492/7.032 9.008/8.549  Expt.d 133.16 148.99 145.69 139.25 117.58 7.87 7.10 9.8  See Fig. 5.6 for the numbering of the atoms. GIAO/CSGT results. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  153  Appendix D. Supporting Information of Chapter 5 Table D.6: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-1 in DMSO, calculated with the CSGT method with the 6-311++G(2df,p) basis set at the HF, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C1a C2 C3 C5 C8 δb H10 H11 H12 H13 a b  c  HF 193.616 129.805 145.240 155.339 145.163 13.4 7.536 7.915 10.143 7.266  B3LYP 180.779 134.230 134.263 161.551 134.771 8.7 7.471 7.253 9.436 7.188  PBEPBE 171.604 132.209 128.824 159.015 128.923 3.7 7.429 7.083 9.194 7.185  Expt.c 169.86 125.22 128.34 151.76 126.85 7.29 7.50 12.84 8.64  See Fig. 5.5 for the numbering of the atoms. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  154  Appendix D. Supporting Information of Chapter 5 Table D.7: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-2 in DMSO, calculated with the CSGT method with the 6-311++G(2df,p) basis set at the HF, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C1a C2 C3 C5 C8 δb H10 H11 H12 H16 a b  c  HF 132.864 134.908 150.593 158.870 159.589 21.8 7.575 8.143 6.432 5.235  B3LYP 130.600 136.311 147.337 161.494 155.088 21.5 7.431 7.921 6.337 4.796  PBEPBE 126.670 134.092 143.987 158.587 150.836 19.7 7.401 7.902 6.383 4.780  Expt.c 169.86 125.22 128.34 151.76 126.85 7.29 7.50 8.64 12.84  See Fig. 5.5 for the numbering of the atoms. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  Table D.8: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-d-1 in DMSO, calculated with the CSGT method with the 6-311++G(2df,p) basis set at the HF, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  155  Appendix D. Supporting Information of Chapter 5  C2a C3 C6 C12 C14 δb H5 H13 H15 a b  c  HF 137.264 159.041 160.801 149.657 135.281 11.5 6.916 8.117 7.001  B3LYP 137.380 160.464 157.288 147.352 135.633 10.7 6.875 7.945 6.853  PBEPBE 133.44 157.241 153.485 144.310 133.092 7.4 6.961 7.941 6.820  Expt.c 133.16 148.99 145.69 139.25 117.58 9.8 7.87 7.10  See Fig. 5.6 for the numbering of the atoms. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  Table D.9: Theoretical 13 C and 1 H NMR chemical shifts (in ppm) of Hmppt-d-2 in DMSO, calculated with the CSGT method with the 6-311++G(2df,p) basis set at the HF, B3LYP, and PBEPBE levels of theory, compared with experimental results obtained in DMSO.  C2a C3 C5 C11 C13 δb H12 H14 H29 a b  c  HF 144.065 179.028 159.829 123.118 138.387 18.4 7.150 7.563 11.704  B3LYP 148.731 175.047 152.403 120.432 136.426 17.2 6.830 7.353 10.765  PBEPBE 145.537 170.295 146.946 117.330 133.412 14.5 6.743 7.333 10.521  Expt.c 133.16 148.99 145.69 139.25 117.58 7.87 7.10 9.8  See Fig. 5.6 for the numbering of the atoms. Average absolute error with respect to the 13 C NMR experimental data. From the supporting information of Ref. [362a].  156  Appendix D. Supporting Information of Chapter 5 Table D.10: Theoretical UV-Vis absorption positions (in nm) and peak oscillator strengths (in parentheses) for single excitations of thiol and thione tautomers of Hmppt and Hmppt-dimer in vacuum and H2 O, calculated at B3LYP/6-31G(d,p) and PBEPBE/6-31G(d,p) levels of theory, compared with experimental data obtained in H2 O. Transitions with large CI coefficients are shown below each peak position.  Tautomers  Hmppt-1 (B3LYP/ 6-31G(d,p))  Hmppt-1 (PBEPBE/ 6-31G(d,p))  Hmppt-2 (B3LYP/ 6-31G(d,p))  Vacuum 200 (0.1990) Hb − 3 →Lb −0.38 H−2 →L+1 0.53 245 (0.1082) H−3 →L+1 0.12 H−2 →L 0.62 H−2 →L+1 −0.12 H→L+1 0.11 282 (0.0002) H−1 →L+1 0.70 314 (0.2959) H−2 →L+1 0.12 H→L 0.59 213 (0.2095) H−3 →L −0.35 H−2 →L+1 0.51 H→L+4 0.10 244 (0.00004) H−1 →L+2 0.70 246 (0.00006) H→L+2 0.70 264 (0.0773) H−3 →L+1 0.11 H−2 →L 0.61 H−2 →L+1 −0.16  H2 O 200 (0.2707) H−3 →L −0.29 H−2 →L+1 0.59 241 (0.1703) H−3 →L+1 0.14 H−2 →L 0.59 H→L+1 0.22 272 (0.0003) H−2 →L −0.25 H→L+1 0.64 307 (0.3413) H−2 →L+1 0.11 H→L 0.63 212 (0.2650) H−3 →L −0.28 H−2 →L+1 0.56 H→L+5 −0.13  Assignmenta 210 (4.13)  245 (3.92)  276 (3.43)  326 (4.21)  -  259 (0.1157) H−3 →L+1 0.15 H−2 →L 0.61 H−2 →L+1 −0.12 H→L+1 0.11 328 (0.2159) 323 (0.2997) H→L 0.54 H→L+1 H−2 →L+1 0.12 0.22 H→L 0.60 189 (0.1346) 212 (0.0812) Continued on next page 157  Appendix D. Supporting Information of Chapter 5 Table D.10 – Continued from previous page Tautomers Vacuum H2 O Assignmenta →L 0.22 H−3 →L 0.33 H−3 H−3 →L+1 0.21 H−2 →L+1 −0.32 →L 0.17 H−3 →L+2 0.10 H−1 0.43 H−2 →L 0.10 H→L+1 H−2 →L+1 −0.33 H→L+2 0.28 H−1 →L+2 0.30 H→L+1 0.21 207 (0.0142) H−3 →L 0.28 H→L+1 −0.35 H→L+2 0.50 234 (0.0276) H−3 →L 0.14 H−2 →L −0.32 H−1 →L 0.52 H−1 →L+1 −0.13 H→L+1 −0.25 246 (0.0145) H−2 →L 0.57 H−1 →L 0.32 H→L −0.14 258 (0.0912) 257 (0.1109) H−3 →L+1 0.21 H−3 →L+1 −0.14 H−2 →L+1 0.10 H−2 →L 0.15 H→L 0.63 H−1 →L+1 0.18 H→L 0.61 263 (0.0040) H−1 →L 0.67 204 (0.0427) 216 (0.1081) H−3 →L 0.23 H−3 →L 0.21 H−3 →L+2 0.11 H−3 →L+1 −0.11 Hmppt-2 H→L+1 0.12 H−2 →L+1 −0.33 (PBEPBE/ H→L+3 0.61 H−1 →L+1 −0.19 6-31G(d,p)) H−1 →L+2 −0.23 H→L+2 0.37 H→L+2 −0.18 235 (0.0067) 251 (0.0144) Continued on next page 158  Appendix D. Supporting Information of Chapter 5 Table D.10 – Continued from previous page Tautomers Vacuum H2 O Assignmenta H−3 →L+1 −0.12 H−2 →L 0.22 H→L+1 0.18 H−2 →L+1 −0.12 H→L+2 0.23 H−1 →L+1 0.60 H→L+1 0.20 279 (0.0691) 269 (0.0314) H−3 →L+1 0.21 H−2 →L −0.15 H−1 →L −0.10 H−2 →L+1 −0.14 H→L 0.61 H−1 →L 0.59 H→L+2 0.15 H→L −0.24 297 (0.0020) 280 (0.0722) H−1 →L 0.67 H−3 →L+1 0.13 H→L 0.11 H−1 →L 0.31 H→L 0.56 200 (0.0866) 201 (0.1452) →L 0.15 H−15 →L 0.14 H−13 201 (4.27) →L 0.38 H−11 →L 0.20 H−11 H−7 →L+4 0.34 H−7 →L+4 0.30 H−6 →L+2 −0.27 H−6 →L+3 0.25 H−5 →L+4 −0.29 H−5 →L+4 −0.11 H−1 →L+3 0.11 H−4 →L+5 0.11 Hmppt-d-1 H−1 →L+4 −0.13 H−1 →L+2 0.13 (PBEPBE/ H−1 →L+4 −0.15 H→L+2 −0.16 6-31G(d,p)) H→L+3 −0.15 199 (0.0735) 199 (0.0293) H−14 →L 0.53 H−14 →L −0.19 H−13 →L 0.35 H−12 →L 0.62 H−5 →L+5 0.12 H−4 →L+6 0.15 H−4 →L+6 0.15 247 (0.0956) 249 (0.1119) 238 (4.43) Continued on next page  159  Appendix D. Supporting Information of Chapter 5 Table D.10 – Continued from previous page Tautomers Vacuum H2 O Assignmenta H−7 →L+2 0.11 H−6 →L+2 0.12 H−13 →L 0.35 H−4 →L+2 −0.20 H−6 →L+3 −0.13 H−3 →L+2 −0.13 H−6 →L+4 0.12 H−1 →L+1 −0.18 H−2 →L+1 −0.14 H−1 →L+3 −0.23 0.35 H−1 →L+1 −0.16 H→L+2 H−1 →L+2 −0.13 H→L+4 0.39 H→L+3 0.41 H→L+4 0.40 279 (0.0118) 268 (0.0106) 270 (3.71) H−7 →L 0.57 H−1 →L+1 −0.15 H−5 →L+1 −0.19 H−1 →L+3 0.45 H−1 →L+3 −0.14 H→L+2 0.43 H→L+1 −0.26 H→L+3 −0.26 322 (0.0051) 322 (0.0021) H−5 →L 0.68 H−5 →L 0.68 336 (4.32) 306 (0.2314) 300 (0.2669) H−6 →L 0.65 H−6 →L 0.65 H−1 →L+1 0.15 201 (0.1180) 204 (0.1251) H−13 →L −0.11 H−11 →L 0.48 H−10 →L −0.53 H−8 →L 0.12 H−3 →L+8 −0.38 H−7 →L+1 −0.17 H−6 →L+2 0.39 219 (0.1204) 220 (0.1775) →L 0.31 H−11 →L −0.19 H−8 Hmppt-d-2 H−13 →L 0.35 H−7 →L+1 0.15 (PBEPBE/ H−9 →L 0.37 H−2 →L+5 0.58 6-31G(d,p)) H−8 →L+1 0.15 H−1 →L+6 −0.10 H−7 →L 0.20 H−7 →L+2 0.13 H−7 →L+1 0.28 H−5 →L+3 0.16 H−5 →L+5 0.13 H−3 →L+8 −0.12 H−1 →L+4 0.13 237 (0.0438) 221 (0.1987) Continued on next page 160  Appendix D. Supporting Information of Chapter 5 Table D.10 – Continued from previous page Tautomers Vacuum H2 O Assignmenta H−8 →L 0.23 H−10 →L −0.18 →L 0.22 H−7 →L+1 0.34 H−9 →L 0.17 H−5 →L+3 0.44 H−7 H−4 →L+4 −0.22 H−7 →L+2 0.21 H−6 →L+1 0.35 H→L+3 −0.13 H−1 →L+4 0.16 H−1 →L+5 −0.18 H→L+3 −0.14 H→L+6 0.25 243 (0.0200) H−7 →L+1 0.16 H−5 →L+4 0.29 H−4 →L+3 0.48 H−3 →L+3 −0.12 H−1 →L+3 0.21 H→L+4 −0.18 248 (0.0772) H−7 →L+1 −0.16 H−6 →L 0.13 H−5 →L+2 0.62 H−5 →L+4 −0.11 268 (0.0324) 278 (0.0324) H−7 →L+1 −0.14 H−7 →L+1 −0.10 →L 0.42 H−6 →L 0.45 H−6 H−5 →L+2 0.29 H−5 →L+2 −0.13 H−4 →L+4 −0.42 H−3 →L+3 −0.41 H−2 →L+4 −0.28 H→L+3 0.11 336 (0.0244) 318 (0.0476) H−4 →L+1 −0.13 H−6 →L+1 −0.15 H−2 →L+2 0.59 H−4 →L+1 −0.37 H→L+2 0.32 H−3 →L 0.10 H−2 →L 0.28 H−1 →L+2 −0.16 H→L+3 0.40 H→L+4 −0.12 327 (0.0245) Continued on next page  161  Appendix D. Supporting Information of Chapter 5 Table D.10 – Continued from previous page Tautomers Vacuum H2 O Assignmenta H−4 →L+1 0.55 H−1 →L+12 −0.11 H→L+2 0.39 a  See the supporting information of Ref. [362a]. “H” and “L” denote the HOMO and the LUMO, respectively. “H−m” and “L+n” denote the mth orbital below the HOMO and the nth orbital above the LUMO, respectively.  b  162  Appendix E  General Mathematical Review In this appendix we provide a brief review on the related mathematical concepts in the previous chapters.  E.1 E.1.1  Basic Concepts Related to Functional Map  Map is the basic concept all through mathematics. It roots in the set theory and usually has the following definition [390]: Definition E.1.1. Let X and Y be two sets. A map f from X to Y (usually written as f : X → Y ) is a subset of X × Y so that for each x ∈ X there is a unique y ∈ Y for which (x, y) ∈ f . A map is also called a function (specially, a function can also be defined as a map from a number set to another number set) or a mapping. In elementary mathematics we often write y = f (x) instead of (x, y) ∈ f . The set X on which f is defined is called the domain of the map, and the subset f (X) = {y ∈ Y | y = f (x), x ∈ X}  (E.1)  is called its range.  E.1.2  Field and Scalar  Field is an abstract algebraic concept origins from the elementary arithmetic. It is an algebraic structure in which the abstract operations of addition and multiplication (not necessarily the ordinary arithmetic operations) can be performed. If we use the symbols + and × to denote the abstract addition and multiplication operations respectively, a field F is exactly defined by the following properties [391]: 163  Appendix E. General Mathematical Review Definition E.1.2. (1) + and × are closed on F . That is, for any two elements a and b ∈ F , a + b and a × b both ∈ F . (2) + and × are associative on F . That is, for any three elements a, b and c ∈ F , a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c. (3) + and × are commutative on F . That is, for any three elements a, b and c ∈ F , a + b = b + a and a × b = b × a. (4) × is distributive over +. That is, for any three elements a, b and c ∈ F , a × (b + c) = (a × b) + (a × c). (5) There is an additive identity in F . That is, there exist an special element 0 in F so that for any a ∈ F , a + 0 = a. (6) There is a multiplicative identity in F . That is, there exist an special element 1 in F so that for any a ∈ F , a × 1 = a. (7) There is an additive inverse for each element in F . That is, for any a ∈ F , the element −a, which makes the equation a + (−a) = 0 hold, is also in F . (8) There is a multiplicative inverse for each element in F . That is, for any a = 0 in F , the element a−1 , which makes the equation a × a−1 = 1 hold, is also in F . For the elements in F , we call them scalars. The common examples for fields are the real number set R, the complex number set C and the rational number set Q.  E.1.3  Vector Space  Intuitively, a vector space (or linear space) is a collection of abstract objects (called vectors) which can be scaled and added. By abstract we mean the vectors can be anything which fulfills the mathematical requirements. The formal definition of a vector space is [392]: Definition E.1.3. Suppose F is a field and X is a set, two types of operations are defined as • Vector addition, which is a map X × X → X. For any two elements a and b, their vector addition is a + b. 164  Appendix E. General Mathematical Review • Scalar multiplication, which is a map F × X → X. For any a ∈ F and b ∈ X, their scalar multiplication is ab. To be a vector space, X must satisfy the following axioms, which are similar to those in the definition of a field. (1) Vector addition is associative. (2) Vector addition is commutative. (3) There is a vector additive identity element 0 in X. (4) Vector addition has inverse elements in X. (5) Scalar multiplication is distributive over vector addition. Namely, for any a ∈ F and b, c ∈ X, a(b + c) = ab + ac. (6) Vector addition is distributive over scalar multiplication. (7) There is a scalar multiplicative identity element 1 in F . (8) For every a, b ∈ F and c ∈ X, we have a(bc) = (ab)c. The most common example of vector space is the 3-dimensional Euclidean space in which we live. Here the vectors are the positional vectors.  E.1.4  Definitions of Functional  The concept of functional is widely used in the field of mathematics, physics and theoretical chemistry. It can be defined at different levels. One of the intuitive definition is: Definition E.1.4. A functional is a map from a set of functions to a set of real or complex numbers. Or alternatively, we can say a functional is a “function of function”. More rigorously and generally, we can define a functional as [390] Definition E.1.5. A functional is a map from a vector space to the field corresponding to the vector space. We can call the first definition the “physicist’s definition” and the second “mathematician’s definition”. Both definitions work well in their fields.  165  Appendix E. General Mathematical Review Usually we deal with the integral-type functional in physics. For example, suppose a planar curve is represented by y(x), so the arc length in the interval x ∈ [a, b] is given by S=  1 + [y ′ (x)]2 dx.  (E.2)  But a functional is not necessarily to be in an integral-type. For another example, M = max y(x) (E.3) x∈[a, b]  is not a integral-type functional.  E.1.5  Linear and Antilinear Map  A linear map (also called a linear transformation or linear operator) is a operation which commutes with the vector addition and scalar scalar multiplication. The formal definition is [393] Definition E.1.6. Let X and Y be two vector spaces over the field F . A map f : X → Y is said to be a linear map if for any x ∈ X, y ∈ Y and a, b ∈ F , the following condition is satisfied: f (ax + by) = af (x) + bf (y). Similarly, an antilinear map can be defined as Definition E.1.7. Let X and Y be two vector spaces over the field F . A map f ∗ : X → Y is said to be a antilinear map if for any x ∈ X, y ∈ Y and a, b ∈ F , the following condition is satisfied: f ∗ (ax + by) = a∗ f (x) + b∗ f (y), where a∗ , b∗ mean their complex conjugate. The antilnear map f ∗ is also a linear map from X to Y ∗ , which is the complex conjugate vector space of Y.  E.1.6  Linear Functional  Similar to a linear map, for a linear functional we have the following definition [390]: Definition E.1.8. A linear functional is a linear map from a vector space to its field. 166  Appendix E. General Mathematical Review  E.2 E.2.1  Functional Calculus An Nonrigorous Definition of Functional Derivative  We focus on the integral-type functional. For simplicity we work on the onedimensional case. Extensions to the higher dimensional cases are straightforward. So the general form of an integral-type functional is (integral limits omitted) F =  f (x) dx.  (E.4)  Like we did in the calculus, let the function f (x) vary infinitesimally, then F must also change infinitesimally if F is well-behaved. The linear (major) part of this change can be written as δF =  δF δf (x) dx, δf (x)  (E.5)  where the quantity δfδF(x) is the functional derivative of F with respect to the function f . It has a parameter dependency on x. Or we can define it explicitly as [394] Definition E.2.1. Given an arbitrary function g(x) in the interval x ∈ [a, b], the limit lim  λ→0  F [f + λg(x)] − F [f ] λ  =  δF g(x) dx δf (x)  (E.6)  defines the functional derivative of F with respect to f .  E.2.2  Properties of Functional Derivative  δ shares the same properties with the It is easy to show that the operator δf ordinary derivative operator. Say it is linear and for the product of two functionals F and G we have  δF δG δ (F G) = G+ F. δf δf δf  (E.7)  For composite functionals, for instance, the functional F [f [x, g(x)]], where F is a functional of f , and f is a functional of g(x) yet, the functional derivative of F with respect to g can be evaluated through a chain rule [395] like δF δf (x′ ) ′ δF = dx . (E.8) δg(x) δf (x′ ) δg(x) 167  Appendix E. General Mathematical Review Remember here x′ is only a dummy variable for integration. A function can be thought as a special case of a functional. So when F is only a function but not a functional of f , its functional derivative can still be evaluated as dF δF (f (x)) = δ(x − x′ ), (E.9) δf (x′ ) df where δ(x − x′ ) is the Dirac delta function. Using this rule, when F is a function of f , and f is a function of g(x), we can find the functional derivative of F with respect to g is δF (f (x′′ )) δg(x)  = =  dF df δ(x′′ − x′ ) δ(x − x′ ) dx′ df dg dF dF df δ(x′′ − x) = δ(x′′ − x), df dg dg  (E.10)  which is a recovery of the ordinary chain rule in calculus. As an analogy to the results in calculus, we can also define higher order functional derivatives and the Taylor expansion of a functional should be F [f + ∆f ] = F [f ] +  δF ∆f (x) dx δf (x)  1 δ2F ∆f (x1 )∆f (x2 ) dx1 dx2 + · · · 2 δf (x1 )δf (x2 ) δnF 1 ··· + n! δf (x1 )δf (x2 ) · · · f (xn ) × ∆f (x1 )∆f (x2 ) · · · ∆f (xn ) dx1 dx2 · · · dxn , +  (E.11)  where ∆f is the change of f . This Taylor expansion expression is very important in applying perturbation techniques to DFT.  E.2.3  Euler-Lagrange Equation  Suppose we have an integral-type functional which depends on a function ρ(x)(not necessarily to be a density function) and its derivative ρ′ (x), namely F [ρ] =  f (x, ρ(x), ρ′ (x)) dx.  (E.12)  Let ρ(x) vary infinitesimally, we have F [ρ + δρ] = =  f (x, ρ + δρ, (ρ + δρ)′ ) dx f (x, ρ, ρ′ ) dx +  (  ∂f ∂f δρ + ′ (δρ)′ ) dx. ∂ρ ∂ρ  (E.13) 168  Appendix E. General Mathematical Review Using integration by parts, we can show ∂f ∂f (δρ)′ dx = ′ δρ − ′ ∂ρ ∂ρ  d ∂f δρ dx. dx ∂ρ′  (E.14)  The first term on the right-hand side of the equation above vanishes on the integration boundary if ρ is a well-behaved function. It is usually so in physics. So according to Eq. (E.13) and Eq. (E.14), we have F [ρ + δρ] − F [ρ] =  (  d ∂f ∂f − )δρ dx. ∂ρ dx ∂ρ′  (E.15)  Now it is clear that the functional F in Eq. (E.12) has a functional derivative δF ∂f d ∂f = − . δρ ∂ρ dx ∂ρ′  (E.16)  For the more general case, in which the functional can depend on a vector function as well as its higher order partial derivatives, namely F [ρ(r)] =  f (r, ρ(r), ∇ρ(r), ∇2 ρ(r), · · · , ∇n ρ(r)) dr,  (E.17)  where ∇i is the vector operator whose components are ∂ i /(∂r1i1 ∂r2i2 · · · ∂rnin ). Here i1 + i2 + · · · + in = i and n, i, i1 , i2 , · · · , in are all integers. The general form of the functional derivative is [395] ∂f ∂f ∂f ∂f δF n n = −∇· + ∇2 · 2 − · · · + (−1) ∇ · ∂∇n ρ . δρ ∂ρ ∂∇ρ ∂∇ ρ  (E.18)  This is the famous Euler-Lagrange equation in calculus of variation. This equation has deep physical backgrounds. For example, in classic mechanics one has the principle of least action, so the action S, which is defined by the Lagrangian of the system L(t, q, q) ˙ as S=  L(t, q, q) ˙ dt,  (E.19)  must be stationary in order to give real dynamical trajectories. Here t is the time variable and q, q˙ are the sets of general coordinates and general momentums, respectively. Recalling Eq. (E.12), we find that S is a functional of q of the same type as that in Eq. (E.12). The stationary condition requires δS = 0, (E.20) δρ 169  Appendix E. General Mathematical Review so applying the rule in Eq. (E.16), we can show the the stationary condition is equivalent to d ∂L ∂L − = 0. (E.21) ∂q dt ∂ q˙ This is just the famous Lagrange equation, which is the master equation in classic dynamics.  E.2.4  Evaluation of Functional Derivative  For specific examples of functional derivatives in DFT, one can observe Eq. (1.7) and Eq. (1.11), apply the rules introduced above. It is easy to show that δTTF 5 = CF ρ2/3 , δρ 3 1 |∇ρ|2 1 ∇2 ρ δTvW = − . (E.22) δρ 8 ρ2 4 ρ One thing we must keep in mind for evaluating functional derivatives is that the functional derivative is with respect to some function, no matter what variables the function depends on. Taking the functional derivative of the Coulomb energy functional J[ρ] (see Eq. (1.8) as an example, we have δJ 1 = δρ 2  ∂ ρ(r1 )ρ(r2 ) dr2 = ∂ρ |r1 − r2 |  ρ(r2 ) dr2 . |r1 − r2 |  (E.23)  The derivative is taken on the function ρ twice so that the prefactor half disappears.  E.3 E.3.1  Basic Concepts Related to Function Spaces Sesquilinear Form  A sesquilinear form has two arguments. It is linear in one but antilinear in the other. The formal definition is Definition E.3.1. Let X be a complex vector space and C be the field of complex number. A map f : X → C is said to be a sesquilinear form if for any x1 , x2 , y1 , y2 ∈ X and a, b ∈ C, the following conditions are satisfied: f (x1 + x2 , y1 + y2 ) = f (x1 , y1 ) + f (x1 , y2 ) + f (x2 , y1 ) + f (x2 , y2 ) f (ax1 , by1 ) = a∗ bf (x1 , y1 ). The choice of first argument to be antilinear is only conventional. 170  Appendix E. General Mathematical Review  E.3.2  Inner Product Space  Definition E.3.2. A inner product space is a vector space X over a field F with a special map ·, · : X × X → F , which is called the inner product, defined to satisfy the following conditions [396]: (1) It is conjugate symmetric. That is, for any x, y ∈ X, x, y = y, x ∗ . (2) It is positive definite. So x, x > 0 for any x = 0 and x ∈ X; the only element which makes x, x = 0 is x = 0. (3) It is a sesquilinear form. Inner product space is a generalization of the two or three-dimensional Euclidean space. Many concepts in Euclidean geometry, say the length of a vector, the angle between two vectors and the orthogonality between vectors, all can be defined. One thing need to mention is in the definition of the inner product, we say nothing about how to calculate the inner product. It might have many ways to specify the inner product, and it is not always the common dot product of spacial vectors in the Euclidean space.  E.3.3  Metric Space  Definition E.3.3. A metric space [397] is an ordered pair (X, d) where X is a set and d is a distance function (metric) on X, that is, d : X × X → R with the conditions (1) d(x, y)  0;  (2) d(x, y) = 0 if and only if x = y; (3) d(x, y) = d(y, x); (4) d(x, z)  d(x, y) + d(y, z),  where x, y, z ∈ X. The last condition is also called the triangle inequality. It resembles the geometric inequality relation between the sides of a triangle in Euclidean geometry.  171  Appendix E. General Mathematical Review  E.3.4  Complete Metric Space  In order to understand the completeness of a metric space, one must understand the concept of Cauchy sequence first. Its definition is: Definition E.3.4. Given a metric space (X, d), a sequence x1 , x2 , x3 · · · in X is called a Cauchy sequence [398] if for any positive real number ε > 0 there exists a positive integer N such that for any two integers m, n > N , the distance d(xm , xn ) < ε. Or we can understand a Cauchy sequence in the way that its terms are getting closer and closer as the number of the terms goes to infinity. A Cauchy sequence should have a limit. But this limit is not necessarily in the metric space X. If every Cauchy sequence in X has a limit in X, we call X a complete metric space. The set of real number is complete while the set of rational number is not. The common example is that the Cauchy sequence 1 + 1!1 + 2!1 + · · · has a limit of an irrational number e.  E.3.5  Normed Space  The concept “norm” can be viewed as an extension of the concept “length” in two- or three-dimensional Euclidean space. The rigorous definition is: Definition E.3.5. Given a vector space X over a field F of the complex numbers, a norm [399] on X is a function · : X → R with the following properties: For any a in F and x, y in X, (1)  x  0;  (2)  x = 0 if and only if x = 0;  (3)  ax = |a| x , where | · | means the absolute value;  (4)  x+y  x + y , the triangle inequality holds.  If the second condition is released so that some non-zero vectors also can have zero norm, this defines a seminorm. The most common norm is the Euclidean norm which has the expression x = x21 + x22 + · · · + x2n , (E.24) where x1 , x2 , · · · , xn are coordinates of the vector x. But we still have the freedom to choose other norms, say the very useful p-norm x  p  = (xp1 + xp2 + · · · + xpn )1/p ,  (E.25) 172  Appendix E. General Mathematical Review where p is a real number and p 1. So until now we can introduce the concept of a normed space as: Definition E.3.6. A normed space is a pair (X, · ), where X is a vector space and · is a norm defined on X. Not all vector spaces are normable.  E.4 E.4.1  More Concepts Related to Functional Convex Set and Convex Functional  The concept of a convex set is very important if we are trying to optimize some object functional on some domain. The definition is: Definition E.4.1. A set X is said to be a convex set if for any two elements x, y in X, the element z = λx + (1 − λ)y is also in X, where 0 λ 1. The concept of a convex functional (function) is similar. Definition E.4.2. A functional F on a convex set X is said to be a convex functional if for any x, y ∈ X and 0 λ 1 F [λx + (1 − λ)y]  λF [x] + (1 − λ)F [y].  (E.26)  The above inequality is a special case of the Jensen’s inequality [396]. We have already mentioned that FLieb is a convex functional in ρ in Sec. 1.4.2. If the inequal sign in Eq. (E.26) is reversed, we call it a concave functional. For example, the ground-state energy functional of the external potential is a concave functional [400].  E.4.2  Convergence and Weak Convergence  To understand functionals more deeply, it is useful to renew the concept of convergence in function spaces (see Sec. E.5), that is: Definition E.4.3. fi , i = 1, 2, · · · is a sequence of functions and f is some function in Lp (X). If lim fi − f p → 0, (E.27) i→∞  we call fi converges to f (denoted by fi → f ); and in addition, if for every functional F in the dual space of Lp (X) there is lim F [fi − f ] → 0,  i→∞  (E.28) 173  Appendix E. General Mathematical Review we call fi weakly converges to f (denoted by fi ⇀ f ). Obviously convergence implies weak convergence.  E.4.3  Continuity and Lower Semicontinuity  Like we do on functions, we want to study the continuity of a functional. The definition is: Definition E.4.4. fi , i = 1, 2, · · · is a sequence of functions and f is some function in Lp (X). A functional F on Lp (X) is continuous at f if and only if when fi → f , F [fi ] → F [f ]. Sometimes the concept of continuity is not enough in the study of functionals, so we extend this concept to (weak) lower semicontinuity, which is Definition E.4.5. fi , i = 1, 2, · · · is a sequence of functions and f is some function in Lp (X). If fi → f (or fi ⇀ f ), then for a functional F F [f ]  lim inf fi .  i→∞  (E.29)  We call F is (weakly)24 lower semicontinuous at f . As Lieb revealed in Theorem 3.6 in Ref. [400], FLieb is (weakly) lower semicontinuous.  E.4.4  Convex Envelope  A convex envelope of a functional is an extension of the geometric concept of the convex envelope or convex hull for a set of point. The definition is as the following: Definition E.4.6. X is a subset of the Banach space Y . f is a function in X and F is a functional defined on X. Specially, F can go to +∞ at some f , but not all of them. {G[f ]} is a set of functionals on X whose element obeys (1) G is weakly lower semicontinuous. (2) G is convex. 24  To define weak continuity one needs the concept of nets [401]. Here we just show its property which fi ⇀ f implies F [fi ] → F [f ].  174  Appendix E. General Mathematical Review (3) G[f ]  F [f ] for all f .  Then we say the convex envelope of F is sup{G[f ]}. From the definition above one can deduce that FLieb is the convex envelope of FLevy [400] for all ρ ∈ D = L3 ∩ L1 (see Sec. 1.4.2 of Chap. 1).  E.4.5  Tangent Functional  The tangent functional is an extension of the tangent line to a curve at a given point for a functional. The definition is: Definition E.4.7. X is a subset of the Banach space Y . F is a functional on X and f0 is a function in X. If there is a linear functional Tl [f ] on Y so that for any f ∈ X F [f ]  F [f0 ] + Tl [f − f0 ],  (E.30)  then Tl is a tangent functional of F at f0 . Tl might be continuous or not, and generally it is not unique. But it is guaranteed by the Hahn-Banach theorem [402] that a convex functional F has at least one tangent functional at the point where F is finite. The importance of tangent functional in establishing the Gˆ ateaux differentiability (see Sec. E.6.1) of a functional can be seen through the following theorem proved by van Leeuwen [403]: Theorem E.4.8. X is a convex subset of the Banach space Y , and F is a lower semicontinuous convex functional on Y . In addition F is finite on X. If F has a unique continuous tangent functional Tl at some f0 ∈ X, then F is Gˆ ateaux differentiable at f0 and the Gˆ ateaux derivative is Tl .  E.5 E.5.1  Important Function Spaces Hilbert Space  The Hilbert space [404] is an algebraic structure suitable for quantum mechanics and it is also widely used in the field of pure mathematics and engineering. It typically appears as an infinite-dimensional function space. With the concepts introduced in the previous sections, we can easily define it as: Definition E.5.1. A Hilbert space is a complete inner product space. 175  Appendix E. General Mathematical Review In such a space, one can introduce an orthonormal basis and set up a coordinate system, as we do in the common Euclidean space. In quantum mechanics, quantum states can be viewed as vectors in a Hilbert space, and observables are expressed by linear operators on this Hilbert space.  E.5.2  Banach Space  Definition E.5.2. A Banach space [405] is a complete normed space. Since we can define the norm according to the inner product, namely x = x, x ,  (E.31)  every Hilbert space is a Banach space. But the converse is not always true.  E.5.3  Lp space  The most important examples of Banach spaces are the Lp spaces [396]. They consist of p-power integrable functions. They have broad applications in pure mathematics, physics, engineering, and especially, help to set up the mathematical foundations of DFT. Its definition is Definition E.5.3. Suppose a real number p 1 and (X, µ) is a measure 25 space with the measure µ, if the p-norm defined on the the set of all measurable functions f from X to C (or R) satisfies that 1/p  f  p  =  |f |p dµ  < ∞,  (E.32)  we call this set of measurable functions an Lp space. as  Here the integral is in the Lebesgue sense. For the ∞-norm, it is defined f  ∞  = lim f p→∞  p  = sup f.  (E.33)  An Lp space is a Banach space and only when p = 2 is it a Hilbert space. Usually the wave-functions we assigned to electronic states of a system in quantum mechanics are square-integrable functions, which conform to the 25  For briefness we will not talk about measure and measure space and other related concepts (e. g. the Lebesgue integral) here. Interested readers can refer to the standard textbook [396] on these topics.  176  Appendix E. General Mathematical Review definition of an L2 space. But the plane-wave basis are exceptions since they can not be normalized in the common sense. Another important concept about Lp space is its dual space [406]. It can be explained as the following: Definition E.5.4. Given a vector space X over some field F , all linear functionals on X also form a vector space X ∗ . We call X ∗ the dual space of X. The most familiar example of this dual relationship to physicists is the bra-ket space widely used in quantum mechanics. For example, we can write a sate function f in the ket notation |f , with its dual, in the bra notation f |. Now the functionals are defines as f |· =  f ∗ {·}dτ.  (E.34)  More generally, for a p-integrable function g in Lp , its dual functional L in the dual space of Lp can be defined as L(g) =  f gµ(dτ ),  (E.35)  where f is some unique function in the dual of Lp , dτ is the volume element and µ is the measure. With this functional definition, we have a well-known result: Theorem E.5.5. The dual space of Lp is Lq , where 1/p + 1/q = 1. For the proof of this theorem please see Ref. [407]. Now the norm of the functional L in Eq. (E.35) is a q-norm as L = f  q  .  (E.36)  For L1 , its dual is L∞ , but not vice versa. The dual of L∞ is complicated and less useful [407] so that we will omit it. Not so rigorously, we can understand this theorem in the way that a p-normed function g and a q-normed function f will make the integral in Eq. (E.35) finite. We can also find that L2 is the dual of itself. In this case we call L2 is a self-dual space. In Lp spaces, there are many inequality relations. One of the most important is the H¨ older’s inequality [408] which states  177  Appendix E. General Mathematical Review Theorem E.5.6. If (X, µ) is a measure space and f, g are measurable realor complex-valued functions on X; 1 p, q ∞ and 1/p + 1/q = 1, then the following inequality fg 1 f p g q (E.37) holds. When p = q = 2, the H¨ older’s inequality turns into a special case called the Cauchy-Schwarz inequality, which can be used to derive lower bounds of the kinetic energy functional in DFT [400].  E.5.4  Sobolev Space  The concept of a Sobolev space was introduced into mathematics since it is the natural space formed by solutions of some partial differential equations. Suppose a function f has its derivatives26 up to some order k, the k, p-norm can be defined as f k,p = f p + f (k) p , (E.38) where k [409] as  0, and p  1. So we can give a definition for a Sobolev space  Definition E.5.7. A Sobolev space W k,p is a subset of the Lp space with a finite k,p-norm. Sobolev space has many applications in quantum mechanics. For example, to keep a finite kinetic energy, the wave-functions as well as the square root of the electron density must be in a Sobolev space W 1,2 (R3 ) = {f |f ∈ L2 , ∇f ∈ L2 }. To obtain some estimations of the derivative norms, the Sobolev inequality is helpful. It states that Sobolev Inequality. Assume f is a continuously differentiable function with compact support27 from Rn to R. Then for 1 p < n there is a constant C such that (E.39) f p∗ C Df p , where 1/p∗ = 1/p − 1/n and p∗ > p; Df is the derivative operator in Rn and C depends on n and p. 26  Generally these derivatives should be weak derivatives, which are defined through the integration by parts formula in a Lebesgue sense. But for simplicity we just treat it as the normal derivative here. 27 A function f on X with a compact support means the set {x ∈ X : f (x) = 0} is compact.  178  Appendix E. General Mathematical Review With the aid of this inequality, one can show that [410] 1/3  ρ(r)3 dr  C  |∇ ρ(r)|2 dr.  (E.40)  That is, if ρ1/2 is in W 1,2 (R3 ), then ρ must be in L3 (R3 ), which explain why BN ⊂ D in Sec. 1.4.2.  E.6  Functional Differentiation  The definition of a functional derivative in Sec. E.2 may be made much more mathematically precise and formal [411]. Generally speaking, there are two types of functional derivatives in our study. One is the Gˆateaux derivative (weak derivative) and the other is the Fr´echet derivative (strong derivative).  E.6.1  Gˆ ateaux Derivative  The Gˆ ateaux derivative resembles the directional derivative in common differential calculus, but it is more general. Its definition is Definition E.6.1. Let X and Y be Banach spaces28 and Z is an open subset of X; F is a functional from X to Y ; f is some point in Z and g is the test function (test direction) in X. So the Gˆ ateaux derivative dF [f, g] of F at f in the direction g is defined as dF [f, g] = lim  λ→0  d F [f + λg] − F [f ] = F [f + λg] λ dλ  .  (E.41)  λ=0  If the limit exists for all directions g, then we say F is Gˆ ateaux differentiable at f . This definition is similar to Definition E.2.1. Unlike the Fr´echet derivative, the Gˆ ateaux derivative is not necessarily linear. We can think the Gˆ ateaux derivative as an operator of the test function g. It is homogeneous with respect to g, that is dF [f, ag] = a dF [f, g] ,  (E.42)  28  Strictly speaking the Gˆ ateaux derivative is defined on locally convex topological vector spaces of which Banach space is only a special case, so it is more general than the Fr´echet derivative which is defined on Banach space. But Banach space is sufficient for our use now.  179  Appendix E. General Mathematical Review where a is a constant. But this operator is not additive, namely, dF [f, g1 + g2 ] = dF [f, g1 ] + dF [f, g2 ],  (E.43)  where g1 and g2 are some test functions in general cases. So the Gˆateaux derivative might be nonlinear.  E.6.2  Fr´ echet Derivative  The Fr´echet derivative is defined on Banach spaces. Fr´echet differentiability is stronger than Gˆ ateaux differentiability since it requires linear approximations of the functions on Banach spaces. Definition E.6.2. Suppose X and Y are Banach spaces, and Z is an open subset of X. f is some point in X and g is the test function in X. A functional F : X → Y is called Fr´ echet differentiable at f if there exists a bounded linear operator A : X → Y such that lim  g→0  F [f + g] − F [f ] − A[g] g X  Y  = 0,  (E.44)  where · X and · Y represent the norms in X and Y , respectively. If the above limit exists for all possible g, we write the Fr´ echet derivative of functional F at f as dF [f ] = A. The major difference between the Gˆ ateaux derivative and the Fr´echet derivative is that the Fr´echet derivative is a linear operator. Obviously a functional is Gˆ ateaux differentiable if it is Fr´echet differentiable but not vice versa. To be Fr´echet differentiable, the Gˆ ateaux derivative must be linear and bounded at each point of the open subset Z and also continuous.  E.7 E.7.1  Legendre Transform Legendre Transform on Functions  In physics, it is often convenient to transform a function (or a functional) into another function with different variables. A common tool for this purpose is the Legendre transform [412]. One of the explicit definition is Definition E.7.1. For a function f (x), let p = f ′ (x), the Legendre transform of f (x) is f˜(p) = p x(p) − f (x(p)). (E.45) 180  Appendix E. General Mathematical Review This definition also gives a practical procedure to calculate the Legendre transform of f (x): find the inverse function of f ′ (x) and then substitute it into Eq. (E.45). From the definition we can know that Legendre transform is the inverse of its own and has a natural relation f (x) + f˜(p) = x p.  (E.46)  Usually we call x and p conjugate variables. In some fields (e. g. thermodynamics) we adopt another convention which is f (x) − f˜(p) = x p.  (E.47)  This convention will not change the nature of the Legendre transform. One common example of Legendre transform in physics is the introduction of enthalpy function in thermodynamics. The internal energy U = U (S, V, {Ni })  (E.48)  is a function of entropy, volume and the numbers of different particles. Sometimes we want to study a system under a constant pressure. Remember the pressure ∂U , (E.49) P =− ∂V S so we can do a Legendre transform on U with respect to −P V , namely H = U + P V = H(S, P, {Ni }).  (E.50)  Now we obtain a new function H which depends on entropy, pressure and the numbers of different particles. This function is suitable for studying the constant pressure system and we give it a name enthalpy.  E.7.2  Legendre Transform on Functionals  The definition above can be made more general for functionals. Definition E.7.2. f and g are functions in the Banach space X and F is a concave29 functional on X. The Legendre transform of F is F˜ [g] = inf f  f ∗ g dµ − F [f ] .  (E.51)  The searching for the infimum is on variations of f while g is fixed. R If F is convex, the following definition should be changed to F˜ [g] = supf { f ∗ g dµ − F [f ]}. 29  181  Appendix E. General Mathematical Review With this definition two conjugate functions do not necessarily obey a differential relation like that in definition E.7.1. According to this understanding, we can carry a Legendre transform on the total energy functional of the external potential E[υext ] with the thermodynamic convention, then we have a new functional F [ρ] = sup E[υext ] −  ρυext dr υext ∈ L3/2 + L∞ ,  (E.52)  which is just the Lieb functional defined in Eq. (1.29). Actually the first HK theorem is just an existence theorem of a Legendre-transformed functional F [ρ] of the ground-state energy functional E[υext ]. E[υext ] is a known functional of some unknown quantity (e. g. the two-particle cumulant [413]), while after a Legendre transform it turns into a unknown functional F [ρ] of the known quantity ρ. Finding good approximations for F [ρ] is the central problem of DFT.  E.8  Asymptotic Analysis  Asymptotic analysis is a method of describing limiting behavior of functions. It is widely used in algorithms analysis, real and complex analysis, differential equations, and engineering [414, 415].  E.8.1  Notation of Asymptotic Analysis  The most important concept in asymptotic analysis is the asymptotic equivalence relation. We say “the function f (x) is much smaller than the function g(x)”, which is noted by f (x) ≪ g(x), x → x0 , when lim  x→x0  f (x) = 0. g(x)  (E.53)  (E.54)  So the notation of asymptotic equivalence, that is f (x) ∼ g(x), x → x0 ,  (E.55)  f (x) − g(x) ∼ g(x), x → x0 .  (E.56)  means  182  Appendix E. General Mathematical Review Or equivalently, we have the condition lim  x→x0  f (x) = 1. g(x)  (E.57)  Remember the definitions above rule out the possibility of that any f (x) ∼ 0 since g(x) can not be zero. Obviously if f (x) ∼ g(x) then g(x) ∼ f (x). But the asymptotic equivalence relation does not mean approximately equal. For examples, x2 +x ∼ x2 when x → ∞, but their difference still goes to infinity; x2 and x are approximately equal when x → 0 since their difference goes to zero, but x2 ≁ x. There are also another notation system called the Big-O nation (or BigOh notation, Landau notation and Bachmann-Landau notation) in asymptotic analysis. In this system, symbols O, o, Ω, ω, and Θ are used for various upper, lower, and tight bounds. We say f (x) = O(g(x)) as x → a or f (x) is O(g(x)) as x → a if and only if ∃ δ > 0, ∃ M > 0 such that |f (x)|  M |g(x)| for |x − a| < δ.  (E.58)  The little-o notation f (x) = o(g(x)) has the same meaning of f (x) ≪ g(x). For the meaning of other symbols, interested readers are directed to the famous book written by Knuth [416]. Big-O notation is widely used in algorithm complexity analysis by computer scientists.  E.8.2  Asymptotic Expansion  The asymptotic expansion of a function is a formal series of functions which has the property that truncating the series after a finite number of terms provides an good approximation to the given function. The major difference between the asymptotic expansion with the common Taylor expansion is that an asymptotic expansion need not to be convergent. Divergent series are ubiquitous in asymptotic expansions, e. g. the famous Stirling series √  1 1 139 571 + − − + ··· , 2 3 12n 288n 51840n 2488320n4 (E.59) which provides a good approximation for large factorials. Theory about asymptotic expansions can be found in Ref. [417]. n! =  E.8.3  2πn  n e  n  1+  Method of Dominant Balance  A common problem in asymptotic analysis is finding the leading asymptotic behavior of the solution of some differential equation at some singular points. 183  Appendix E. General Mathematical Review The most general and powerful procedure available for this kind of problem is the method of dominant balance. We will illustrate this method by working on an example. Say we have a differential equation y (E.60) y ′′ = 5 x in hand, and we want to find the leading asymptotic behavior of the solution when x → 0+. First, following what Liouville and others did two centuries ago, we write the solution in the form y(x) = eS(x) .  (E.61)  Substituting this back into Eq. (E.60), we find a equation for S(x), that is S ′′ + (S ′ )2 =  1 . x5  (E.62)  Then we want to know which term is dominant as x → 0+. There are two possibilities: (a) (S ′ )2 ≪ S ′′ . So (S ′ )2 can be dropped from the left-hand side of Eq. (E.62), which gives the asymptotic relation S ′′ ∼  1 . x5  (E.63)  1 This means S ∼ 12x 3 as x → 0+, which is inconsistent with the assump′ 2 ′′ tion (S ) ≪ S we made before. So this entry is not possible.  (b) S ′′ ≪ (S ′ )2 . Then the asymptotic relation becomes (S ′ )2 ∼  1 . x5  (E.64)  This is equivalent to S ∼ ± 23 x−2/3 . Checking this approximate solution with the assumption S ′′ ≪ (S ′ )2 , we find it consistent. So it is the right answer. Furthermore, we can write 2 S(x) = ± x−2/3 + C(x). 3  (E.65)  Substituting this back into Eq. (E.60) again, we get a equation for C(x), that is 5 C ′′ − 2x−5/2 C ′ + (C ′ )2 + x−7/2 = 0. (E.66) 2 184  Appendix E. General Mathematical Review This equation looks complicated but after several trial choices for the dominant term, we find the only consistent approximate solution is C ∼ 45 ln(x). So the leading behavior of the solution we found is 2 −2/3  y ∼ c x5/4 e± 3 x  , x → 0+,  (E.67)  where c is some constant. Of course we can make further approximations such that C(x) =  5 ln(x) + D(x), 4  (E.68)  and discuss the asymptotic behavior of D(x). The method of dominant balance is circular until you are satisfied with the answer you get. For further discussions and more examples on this method, please see Ref. [418].  185  Bibliography  Bibliography [390] A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover Publications, Inc., Mineola, New York, 1999). [391] W. Rudin, Principles of Mathematical Analysis (McGraw-Hill Publishing Co., New York, 1976), 3rd ed. [392] G. Strang, Introduction to Linear Algebra (Wellesley-Cambridge Press, Wellesley MA, 2003), 3rd ed. [393] P. R. Halmos, Finite-Dimensional Vector Spaces (Springer-Verlag, New York, 1993), 2nd ed. [394] R. G. Parr and W. Yang, Density-functional Theory of Atoms and Molecules (Oxford University Press, Inc., 200 Madison Avenue, New York, NY 10016, USA, 1989), 1st ed. [395] I. M. Gelfand and S. V. Fomin, Calculus of Variations (Dover Publications, Mineola, New York, 2000). [396] W. Rudin, Real and Complex Analysis (McGraw-Hill Publishing Co., New York, 1986), 3rd ed. [397] M. ´ o Searc´ oid, Metric Spaces (Springer, New York, 2006), 1st ed. [398] M. Spivak, Calculus (Publish or Perish, Berkeley, CA, 1994), 3rd ed. [399] H. Schaefer and M. Wolff, Topological Vector Spaces (Springer-Verlag, New York, 1999). [400] E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983). [401] S. Willard, General Topology (Addison-Wesley, Reading Massachusetts, 1970). [402] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis (Academic Press, San Diego, 1980). [403] R. van Leeuwen, Adv. Chem. Phys. 43, 25 (2003). [404] L. Debnath and P. Mikusi´ nski, Introduction to Hilbert Spaces with Applications (Academic Press, New York, 1999), 2nd ed.  186  Bibliography [405] R. E. Megginson, An Introduction to Banach Space Theory (SpringerVerlag, New York, 1998), 1st ed. [406] W. Rudin, Functional Analysis (McGraw-Hill Publishing Co., New York, 1991), 2nd ed. [407] E. H. Lieb and M. Loss, Analysis (American Mathematical Society, Providence, Rhode Island, 2001), 2nd ed. [408] G. H. Hardy, J. E. Littlewood, and G. P´olya, Inequalities (Cambridge University Press, Cambridge, UK, 1952), 2nd ed. [409] R. A. Adams and J. J. F. Fournier, Sobolev Spaces (Academic Press, Boston, 2003), 2nd ed. [410] E. H. Lieb, Rev. Mod. Phys. 48, 553 (1976). [411] L. A. Lusternik and V. J. Sobolev, Elements of functional analysis (John Wiley & Sons, Inc., New York, 1974), 2nd ed. [412] V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer Inc., 233 Spring Street, New York, NY 10013, USA, 1989), 2nd ed. [413] W. Kutzelnigg, J. Chem. Phys. 125, 171101 (2006). [414] N. G. de Bruijn, Asymptotic Methods in Analysis (Dover Publications, New York, 1981). [415] P. D. Miller, Applied Asymptotic Analysis (American Mathematical Society, Providence, Rhode Island, 2006). [416] D. Knuth, The Art of Computer Programming, Volume 1: Fundamental Algorithms (Addison-Wesley, Reading, Massachusetts, 1997), 3rd ed. [417] A. Erd´elyi, Asymptotic Expansions (Dover Publications, New York, 1956). [418] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory (McGraw Hill Inc., New York, 1978).  187  

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