A primer in density functional theory : [lectures presented at the second Coimbra School on Computational Physics, which took place in late August of 2001

density Functional Theory (dft) Is By Now A Well-established Method For Tackling The Quantum Mechanics Of Many-body Systems. Originally Applied To Compute Properties Of Atoms And Simple Molecules, Dft Has Quickly Become A Work Horse For More Complex Applications In The Chemical And Materials Sciences. The Present Set Of Lectures, Spanning The Whole Range From Basic Principles To Relativistic And Time-dependent Extensions Of The Theory, Is The Ideal Introduction For Graduate Students Or Nonspecialist Researchers Wishing To Familiarize Themselves With Both The Basic And Most Advanced Techniques In This Field. 3540030832......Page 1 Lecture Notes in Physics 620......Page 2 Editors......Page 5 Preface......Page 6 List of Contributors......Page 9 Table of Contents......Page 10 1.1.1 Quantum Mechanical Many-Electron Problem......Page 14 1.1.2 Summary of Kohn--Sham Spin-Density Functional Theory......Page 15 1.2.1 Wavefunctions and Their Interpretation......Page 18 1.2.2 Wavefunctions for Non-interacting Electrons......Page 20 1.2.3 Wavefunction Variational Principle......Page 21 1.2.4 Hellmann--Feynman Theorem......Page 22 1.2.5 Virial Theorem......Page 23 1.3.1 Introduction to Density Functionals......Page 24 1.3.2 Density Variational Principle......Page 25 1.3.3 Kohn--Sham Non-interacting System......Page 26 1.3.4 Exchange Energy and Correlation Energy......Page 27 1.3.5 Coupling-Constant Integration......Page 29 1.4.1 Uniform Coordinate Scaling......Page 33 1.4.2 Local Lower Bounds......Page 35 1.4.4 Size Consistency......Page 36 1.4.5 Derivative Discontinuity......Page 37 1.5.1 Kinetic Energy......Page 38 1.5.2 Exchange Energy......Page 39 1.5.3 Correlation Energy......Page 40 1.5.4 Linear Response......Page 42 1.6.1 Local Spin Density Approximation......Page 45 1.6.2 Gradient Expansion......Page 49 1.6.3 History of Several Generalized Gradient Approximations......Page 53 1.6.4 Construction of a ``GGA Made Simple''......Page 55 1.6.5 GGA Nonlocality: Its Character, Origins, and Effects......Page 57 1.6.6 Hybrid Functionals......Page 62 1.6.7 Meta-generalized Gradient Approximations......Page 63 1.6.8 Jacob's Ladder of Density Functional Approximations......Page 64 References......Page 65 2.1 Introduction......Page 69 2.1.1 Preliminaries and Notation......Page 70 2.1.2 Motivation for Orbital-Dependent Functionals......Page 72 2.1.3 Basic Concept of Orbital-Dependent Functionals......Page 76 2.2 Optimized Potential Method (OPM)......Page 77 2.2.1 Direct Functional Derivative......Page 78 2.2.2 Total Energy Minimization......Page 79 2.2.3 Invariance of the Density......Page 80 2.2.4 Exact Relations Related to OPM......Page 83 2.2.5 Krieger–Li–Iafrate Approximation......Page 84 2.3.1 Accuracy of the KLI Approximation......Page 86 2.3.2 Properties of the Exact Exchange: Comparison with Explicit Density Functionals......Page 94 2.4.1 Many-Body Theory on the Basis of the Kohn–Sham System: Exact Expression for E_{xc}......Page 103 2.4.2 Perturbative Approach to the Sham–Schluter Equation: Second Order Correlation Functional......Page 107 2.4.3 Extensions of the Second Order Functional......Page 110 2.5.1 Self-interaction Corrected LDA......Page 112 2.5.2 Colle–Salvetti Functional......Page 113 2.6.1 Description of Dispersion Forces by Second Order Correlation Functional......Page 114 2.6.2 Comparison of Available Orbital-Dependent Approximations for E_c......Page 119 2.6.3 Analysis of the Second Order Correlation Potential......Page 124 2.7 Final Remarks......Page 130 References......Page 133 3.1 Summary......Page 136 3.2 Foundations......Page 139 3.3 Functionals......Page 142 3.4 Results......Page 149 3.5 Further Results......Page 153 References......Page 155 4.1 Introduction......Page 157 4.2.1 Preliminaries......Page 158 4.2.2 The Runge–Gross Theorem......Page 161 4.2.3 Time-Dependent Kohn–Sham Equations......Page 164 4.2.4 XC Functionals......Page 166 4.2.5 Numerical Considerations......Page 169 4.3.1 Basic Theory......Page 171 4.3.2 The XC Kernel......Page 174 4.4.1 DFT Techniques to Calculate Excitations......Page 177 4.4.2 Full Solution of the Kohn–Sham Equations......Page 179 4.4.3 Excitations from Linear-Response Theory......Page 181 4.4.4 When Does It Not Work?......Page 186 4.5.1 What Is a “Strong” Laser?......Page 187 4.5.2 High-Harmonic Generation......Page 189 4.5.3 Multi-photon Ionization......Page 190 4.5.4 Ionization Yields from TDDFT......Page 192 4.6 Conclusion......Page 194 References......Page 195 5.1 Introduction......Page 198 5.2.1 Green’s Function and Self-energy Operator......Page 199 5.2.2 Many-Body Perturbation Theory and the GW Approximation......Page 202 5.3 Pathologies of the Kohn–Sham xc Functional......Page 207 5.3.1 The Band Gap Problem......Page 209 5.3.2 Widely Separated Open Shell Atoms......Page 212 5.3.3 The Exchange-Correlation Electric Field......Page 213 5.4.1 Theoretical Background......Page 216 5.4.2 Applications......Page 220 5.4.3 Generalised KS Schemes and Self-energy Models......Page 225 References......Page 227 6.1 Introduction......Page 231 6.2.1 Generalities......Page 232 6.2.2 Atoms......Page 237 6.2.3 Plane-Waves......Page 238 6.2.4 Real-Space......Page 242 6.3.1 The Pseudo-potential Concept......Page 243 6.3.2 Empirical Pseudo-potentials......Page 245 6.3.3 Ab-initio Pseudo-potentials......Page 246 6.3.4 Hamann Potential......Page 247 6.3.5 Troullier–Martins Potential......Page 248 6.3.6 Non-local Core Corrections......Page 249 6.3.7 Pseudo-potential Transferability......Page 250 6.3.8 Kleinman and Bylander Form of the Pseudo-potential......Page 251 6.4 Atomic Calculations......Page 252 6.5 Plane-Wave Calculations......Page 257 6.6 Real-Space Calculations......Page 264 References......Page 267 The material world of everyday experience, as studied by chemistry and condensed-matter physics, is built up from electrons and a few (or at most a few hundred) kinds of nuclei .