I THE FRAMEWORK
1 Essential Quantum Mechanics ............................... 3
1.1 The Time-independent Schrödinger Equation .................. 3
1.1.1 The Born-Oppenheimer Approximation .................. 5
1.1.2 Atomic Units ........................................ 5
1.1.3 Parts of the Total Energy ........................... 6
1.2 Wave-mechanics of Non-interacting Fermions ................. 7
1.2.1 Mean Field Theory ................................... 7
1.2.2 Spacial and Spin Parts in Single-particle
Wavefunctions ....................................... 8
1.2.3 Determinant Wavefunctions ........................... 9
1.3 Basis Vectors and Representations ......................... 11
1.3.1 Bras and Kets ...................................... 11
1.3.2 Expansion Coefficients ............................. 15
1.3.3 Eigenstates of Momentum and Matrix Elements
of Operators ....................................... 16
1.3.4 Matrix Elements of the Coulomb Potential ........... 17
1.3.5 Momentum Space and k-space ......................... 18
1.4 Periodic Boundary Conditions .............................. 20
1.4.1 Fourier Transformation of a Wavefunctioh ........... 20
1.4.2 Cells and Supercells ............................... 21
1.4.3 The Reciprocal Lattice ............................. 23
1.4.4 The First Brillouin Zone ........................... 24
1.4.5 Bloch's Theorem .................................... 24
1.4.6 Expansion in Plane Waves ........................... 25
1.5 Local Orbitals and Spherical Harmonics .................... 26
1.5.1 Spherical Harmonics ................................ 27
1.5.2 Local Atomic-like Orbitals ......................... 30
1.5.3 Expansion of a Wavefunction in a Non-orthogonal
Basis .............................................. 32
1.5.4 Expansion of an Operator ........................... 32
1.5.5 Spherical Harmonics and Multipoles ................. 33
1.6 The Variational Principle and the Schrödinger Equation .... 37
1.6.1 The Single-particle Schrödinger Equation ........... 37
1.6.2 The Hartree-Fock Approximation ..................... 39
1.7 The Density Matrix and the Charge Density ................. 42
1.7.1 The Fermi Distribution ............................. 42
1.7.2 Matrix Representations of the Density Operator ..... 43
1.7.3 Mulliken Charges, Bond Charges and Bond Orders ..... 45
1.8 The Density of States ..................................... 48
1.8.1 Definition ......................................... 48
1.8.2 The Green Function ................................. 50
1.9 Jellium ................................................... 52
1.9.1 Cancellation of Electrostatic Energies ............. 53
1.9.2 The Kinetic Energy ................................. 53
1.9.3 Exchange and Correlation Energy .................... 55
1.10 The Matrix Eigenvalue Problem ............................. 55
1.10.1 Local Orbitals or Plane-waves? ..................... 56
1.10.2 Approaches to Solving the Matrix Eigenvalue
Problem ............................................ 57
1.11 Pseudopotentials .......................................... 61
1.11.1 Basic Ideas ........................................ 61
1.11.2 The Ashcroft Empty-core Pseudopotential ............ 62
2 Essential Density Functional Theory ....................... 64
2.1 What is a Functional? ..................................... 64
2.2 Functional Derivatives .................................... 65
2.3 The Thomas-Fermi Model .................................... 68
2.3.1 Description of the Thomas-Fermi Functional ......... 68
2.3.2 The Euler-Lagrange Equation ........................ 68
2.3.3 The Local Density Approximation .................... 69
2.4 The Kohn-Sham Equations ................................... 70
2.4.1 The Existence of a Density Functional .............. 70
2.4.2 The Hohenberg-Kohn-Sham Functional ................. 71
2.4.3 The Kohn and Sham Trick ............................ 72
2.4.4 Self-consistent Solution of the Kohn-Sham
Equations .......................................... 75
2.4.5 Approximating the Exchange and Correlation:
The LDA ............................................ 77
3 Exploiting the Variational Principle ...................... 79
3.1 The Hellmann-Feynman Theorem .............................. 79
3.1.1 Statement and Proof ................................ 79
3.1.2 The van der Waals Interaction ...................... 81
3.1.3 Another Example: The Image Potential .............. 82
3.2 Perturbation Theory with the Density ...................... 84
3.2.1 Switching on an External Potential ................. 84
3.2.2 First-order Perturbation Theory .................... 85
3.2.3 Second-order Perturbation Theory ................... 85
3.3 The Second-order HKS Functional ........................... 87
3.3.1 Derivation of E(2) ................................. 87
3.3.2 The Error in E(2) .................................. 90
3.4 The Harris-Foulkes Functional and its Generalizations ..... 92
3.4.1 The First-order Functional and the Harris-Foulkes
Functional ......................................... 92
3.4.2 Generalization of the Harris-Foulkes Functional .... 93
4 Linear response theory .................................... 96
4.1 Definition of the Response Function χе(r, r') ............. 96
4.2 Relationship to HKS Density Functional .................... 97
4.2.1 Matrix and Vector Algebra Notation for Integrals ... 99
4.3 The Non-interacting Response Function .................... 100
4.4 The Dielectric Function .................................. 101
4.5 The Error in the Harris-Foulkes Functional ............... 102
4.6 Linear Response and the Green Function ................... 105
4.7 Linear Response in Jellium ............................... 107
4.8 Electron-Electron Interactions in the Jellium Response ... 111
4.9 The Long Wavelength Limit of Response Functions in
Jellium .................................................. 114
4.9.1 The Thomas-Fermi Response Function ................ 114
4.9.2 The Compressibility Sum Rule ...................... 115
4.10 Linear Response in a Perfect Crystal ..................... 118
4.11 Non-local Potentials ..................................... 120
4.11.1 General Ideas .................................... 120
4.11.2 Non-local Perturbations in Jellium ............... 122
4.11.3 Second-order Energy in Jellium ................... 125
II MODELLING ATOMS WITHIN SOLIDS
5 Testing an interatomic force model ....................... 129
5.1 The Cohesive Energy and Crystal Structures ............... 130
5.2 The Structural Energy Difference Theorem ................. 131
5.2.1 Examples of the SEDT .............................. 133
5.3 Elastic Constants ........................................ 133
5.3.1 Cubic Crystals .................................... 136
5.3.2 Some Subtleties: Pair Potentials and Cauchy
Pressure .......................................... 141
5.4 Phonons .................................................. 147
5.4.1 Lattice Dynamics in the Harmonie Approximation .... 147
5.4.2 Calculating Force Constants ....................... 150
5.5 Point Defects ............................................ 152
5.5.1 Definition of Vacancy Formation Energies .......... 154
6 Pairwise Potentials in Simple Metals ..................... 158
6.1 Introduction ............................................. 158
6.2 The Energy in Terms of Pseudopotentials .................. 161
6.2.1 Structure Factors ................................. 161
6.2.2 The q = 0 Problem ................................. 162
6.2.3 The Madelung Energy ΔEZZ .......................... 166
6.2.4 The Total Energy and the Energy-wavenumber
Characteristic .................................... 169
6.3 Periodic Boundary Conditions ............................. 171
6.4 The Effective Pairwise Interaction ....................... 172
6.5 Example: The Ashcroft Empty-core Potential ............... 175
6.6 Asymptotic Forms of the Pair Potential ................... 178
6.7 The Pseudoatom Picture ................................... 180
6.7.1 The Energy of a Pseudoatom and the Local
Density ........................................... 184
7 Tight Binding ............................................ 187
7.1 Introduction ............................................. 187
7.1.1 Predicting the Past ............................... 188
7.1.2 Ab Initio Tight Binding ........................... 189
7.1.3 One-, Two- and Three-centre integrals ............. 190
7.2 Non-self-consistent Tight Binding ........................ 191
7.3 Slater-Koster Parameters ................................. 194
7.3.1 The Symmetries of Two-centre Integrals ............ 194
7.3.2 Distance Dependence of the Bond Integrals ......... 197
7.4 The Repulsive Energy ..................................... 198
7.5 The Tight-Binding Bond Model ............................. 200
7.5.1 Basic Ideas ....................................... 200
7.5.2 Development of the Model .......................... 202
7.5.3 A Closer Look at the Energy in a Non-Orthogonal
Basis ............................................. 204
7.6 Hellmann-Feynman Forces .................................. 207
7.6.1 The Pitfall of Incomplete Bases ................... 207
7.6.2 The Force on an Ion ............................... 208
7.7 Self-consistent Tight-Binding ............................ 211
7.7.1 The Self-consistent Charge Transfer Model ........ 211
7.7.2 Local Charge Neutrality ........................... 214
7.7.3 Including Atomic Polarization ..................... 215
7.8 Moments of the Density of States ......................... 218
7.9 The Recursion Method ..................................... 220
7.9.1 Block Recursion ................................... 229
7.10 Second-moment Models ..................................... 230
7.10.1 General Ideas ..................................... 230
7.10.2 The TBBM with a Gaussian Density of States ....... 231
7.10.3 An Effective Pairwise Potential ................... 236
7.11 Fourth-moment Models ..................................... 237
7.12 Bond-order Potentials .................................... 242
7.12.1 A Multi-atom Expansion of the Bond Energy ......... 242
7.12.2 Example: Analytic Bond Orders in a p-Bonded
Trimer ............................................ 247
8 Hybrid Schemes ........................................... 253
8.1 Generalized Pseudopotential Theory ....................... 253
8.2 Effective Medium Theory .................................. 257
9 Ionic models ............................................. 263
9.1 Introduction ............................................. 263
9.2 The Rigid Ion Model Derived .............................. 264
9.3 Beyond the Rigid Ion Model ............................... 270
9.3.1 The Basic Second-order Model ...................... 270
9.3.2 Deformable Ions ................................... 270
9.3.3 Variable Charge Transfer Models ................... 273
Bibliography .................................................. 275
Index ......................................................... 283
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