Preface ...................................................... xvii
Acknowledgments ................................................ xx
Notation ...................................................... xxi
Part I Overview and background topics
1 Introduction ................................................. 1
Summary ...................................................... 1
1.1 Quantum theory and the origins of electronic structure .. 1
1.2 Emergence of quantitative calculations .................. 5
1.3 The greatest challenge: electron correlation ............ 8
1.4 Recent developments ..................................... 9
Select further reading ...................................... 10
2 Overview .................................................... 11
Summary ..................................................... 11
2.1 Electronic ground state: bonding and characteristic
structures ............................................. 12
2.2 Volume or pressure as the most fundamental variable .... 16
2.3 Elasticity: stress-strain relations .................... 21
2.4 Magnetism and electron-electron interactions ........... 22
2.5 Phonons and displacive phase transitions ............... 24
2.6 Thermal properties: solids, liquids, and phase
diagrams ............................................... 28
2.7 Atomic motion: diffusion, reactions, and catalysis ..... 31
2.8 Surfaces, interfaces, and defects ...................... 32
2.9 Nanomaterials: between molecules and condensed matter .. 36
2.10 Electronic excitations: bands and band gaps ............ 40
2.11 Electronic excitations: heat capacity, conductivity,
and optical spectra .................................... 44
2.12 Example of MgE: bands, phonons, and
superconductivity ...................................... 47
2.13 The continuing challenge: electron correlation ......... 50
Select further reading ...................................... 51
3 Theoretical background ...................................... 52
Summary ..................................................... 52
3.1 Basic equations for interacting electrons and nuclei ... 52
3.2 Coulomb interaction in condensed matter ................ 56
3.3 Force and stress theorems .............................. 56
3.4 Statistical mechanics and the density matrix ........... 60
3.5 Independent-electron approximations .................... 61
3.6 Exchange and correlation ............................... 65
3.7 Perturbation theory and the "2n + 1 theorem" ........... 68
Select further reading ...................................... 70
Exercises ................................................... 71
4 Periodic solids and electron bands .......................... 73
Summary ..................................................... 73
4.1 Structures of crystals: lattice + basis ................ 73
4.2 The reciprocal lattice and Brillouin zone .............. 81
4.3 Excitations and the Bloch theorem ...................... 85
4.4 Time reversal and inversion symmetries ................. 89
4.5 Point symmetries ....................................... 91
4.6 Integration over the Brillouin zone and special
points ................................................. 92
4.7 Density of states ...................................... 96
Select further reading ...................................... 96
Exercises ................................................... 97
5 Uniform electron gas and simple metals ..................... 100
Summary .................................................... 100
5.1 Non-interacting and Hartree-Fock approximations ....... 102
5.2 The correlation hole and energy ....................... 107
5.3 Binding in sp-bonded metals ........................... 112
5.4 Excitations and the Lindhard dielectric function ...... 113
Select further reading ..................................... 116
Exercises .................................................. 116
Part II Density functional theory
6 Density functional theory: foundations ..................... 119
Summary .................................................... 119
6.1 Thomas-Fermi-Dirac approximation: example of
a functional .......................................... 120
6.2 The Hohenberg-Kohn theorems ........................... 121
6.3 Constrained search formulation of density functional
theory ................................................ 125
6.4 Extensions of Hohenberg-Kohn theorems ................. 126
6.5 Intricacies of exact density functional theory ........ 129
6.6 Difficulties in proceeding from the density ........... 131
Select further reading ..................................... 132
Exercises .................................................. 133
7 The Kohn-Sham auxiliary system ............................. 135
Summary .................................................... 135
7.1 Replacing one problem with another .................... 135
7.2 The Kohn-Sham variational equations ................... 138
7.3 Јxc, VKC, and the exchange-correlation hole ........... 139
7.4 Meaning of the eigenvalues ............................ 144
7.5 Intricacies of exact Kohn-Sham theory ................. 145
7.6 Time-dependent density functional theory .............. 147
7.7 Other generalizations of the Kohn-Sham approach ....... 148
Select further reading ..................................... 149
Exercises .................................................. 149
8 Functionals for exchange and correlation ................... 152
Summary .................................................... 152
8.1 The local spin density approximation (LSDA) ........... 152
8.2 Generalized-gradient approximations (GGAs) ............ 154
8.3 LDA and GGA expressions for the potential Vxcσ(r) ..... 157
8.4 Non-collinear spin density ............................ 159
8.5 Non-local density formulations: ADA and WDA ........... 160
8.6 Orbital-dependent functionals I: SIC and LDA + U ...... 160
8.7 Orbital-dependent functionals II: OEP and EXX ......... 162
8.8 Hybrid functionals .................................... 165
8.9 Tests of functionals .................................. 166
Select further reading ..................................... 169
Exercises .................................................. 170
9 Solving Kohn-Sham equations ................................ 172
Summary .................................................... 172
9.1 Self-consistent coupled Kohn-Sham equations ........... 172
9.2 Total energy functionals .............................. 174
9.3 Achieving self-consistency ............................ 179
9.4 Force and stress ...................................... 182
Select further reading ..................................... 184
Exercises .................................................. 184
Part III Important preliminaries on atoms
10 Electronic structure of atoms .............................. 187
Summary .................................................... 187
10.1 One-electron radial Schrцdinger equation .............. 187
10.2 Independent-particle equations: spherical potentials .. 189
10.3 Open-shell atoms: non-spherical potentials ............ 190
10.4 Relativistic Dirac equation and spin-orbit
interactions .......................................... 193
10.5 Example of atomic states: transition elements ......... 195
10.6 Delta-SCF: electron addition, removal, and
interaction energies .................................. 198
10.7 Atomic sphere approximation in solids ................. 199
Select further reading ..................................... 201
Exercises .................................................. 202
11 Pseudopotentials ........................................... 204
Summary .................................................... 204
11.1 Scattering amplitudes and pseudopotentials ............ 204
11.2 Orthogonalized plane waves (OPWs) and
pseudopotentials ...................................... 207
11.3 Model ion potentials .................................. 211
11.4 Norm-conserving pseudopotentials (NCPPs) .............. 212
11.5 Generation of l-dependent norm-conserving
pseudopotentials ...................................... 215
11.6 Unscreening and core corrections ...................... 218
11.7 Transferability and hardness .......................... 219
11.8 Separable pseudopotential operators and projectors .... 220
11.9 Extended norm conservation: beyond the linear regime .. 221
11.10 Ultrasoft pseudopotentials ........................... 222
11.11 Projector augmented waves (PAWs): keeping the full
wavefunction ......................................... 225
11.12 Additional topics .................................... 227
Select further reading ..................................... 228
Exercises .................................................. 229
Part IV Determination of electronic structure: the three
basic methods
12 Plane waves and grids: basics .............................. 236
Summary .................................................... 236
12.1 The independent-particle Schrцdinger equation in
a plane wave basis .................................... 236
12.2 The Bloch theorem and electron bands .................. 238
12.3 Nearly-free-electron approximation .................... 239
12.4 Form factors and structure factors .................... 240
12.5 Approximate atomic-like potentials .................... 242
12.6 Empirical pseudopotential method (EPM) ................ 243
12.7 Calculation of density: introduction of grids ......... 246
12.8 Real-space methods .................................... 248
Select further reading ..................................... 251
Exercises .................................................. 251
13 Plane waves and grids: full calculations ................... 254
Summary .................................................... 254
13.1 initio" pseudopotential method ........................ 255
13.2 Projector augmented waves (PAWs) ...................... 258
13.3 Simple crystals: structures, bands .................... 259
13.4 Supercells: surfaces, interfaces, phonons, defects .... 265
13.5 Clusters and molecules ................................ 269
Select further reading ..................................... 270
Exercises .................................................. 271
14 Localized orbitals: tight-binding .......................... 272
Summary .................................................... 273
14.1 Localized atom-centered orbitals ...................... 273
14.2 Matrix elements with atomic orbitals .................. 274
14.3 Slater-Koster two-center approximation ................ 278
14.4 Tight-binding bands: illustrative examples ............ 279
14.5 Square lattice and QuO2 planes ........................ 282
14.6 Examples of bands: semiconductors and transition
metals ................................................ 283
14.7 Electronic states of nanotubes ........................ 285
14.8 Total energy, force, and stress in tight-binding ...... 289
14.9 Transferability: non-orthogonality and environment
dependence ............................................ 291
Select further reading ..................................... 293
Exercises .................................................. 294
15 Localized orbitals: full calculations ...................... 298
Summary .................................................... 298
15.1 Solution of Kohn-Sham equations in localized bases .... 298
15.2 Analytic basis functions: gaussians ................... 300
15.3 Gaussian methods: ground state and excitation
energies .............................................. 302
15.4 Numerical orbitals .................................... 304
15.5 Localized orbitals: total energy, force, and stress ... 307
15.6 Applications of numerical local orbitals .............. 309
15.7 Green's function and recursion methods ................ 310
15.8 Mixed basis ........................................... 310
Select further reading ..................................... 311
Exercises .................................................. 311
16 Augmented functions: APW, KKR, MTO ......................... 313
Summary .................................................... 313
16.1 Augmented plane waves (APWs) and "muffin tins" ........ 313
16.2 Solving APW equations: examples ....................... 318
16.3 The KKR or multiple-scattering theory (MST) method .... 323
16.4 Alloys and the coherent potential approximation
(CPA) ................................................. 329
16.5 Muffin-tin orbitals (MTOs) ............................ 331
16.6 Canonical bands ....................................... 333
16.7 Localized "tight-binding" MTO and KKR formulations .... 338
16.8 Total energy, force, and pressure in augmented
methods ............................................... 341
Select further reading ..................................... 342
Exercises .................................................. 342
17 Augmented functions: linear methods ........................ 345
Summary .................................................... 345
17.1 Energy derivative of the wavefunction: ψ and ψ ........ 346
17.2 General form of linearized equations .................. 348
17.3 Linearized augmented plane waves (LAPWs) .............. 350
17.4 Applications of the LAPW method ....................... 351
17.5 Linear muffin-tin orbital (LMTO) method ............... 355
17.6 "Ab initio" tight-binding ............................. 358
17.7 Applications of the LMTO method ....................... 360
17.8 Beyond linear methods: NMTO ........................... 362
17.9 Full potential in augmented methods ................... 364
Select further reading ..................................... 365
Exercises .................................................. 366
Part V Predicting properties of matter from electronic
structure - recent developments
18 Quantum molecular dynamics (QMD) ........................... 371
Summary .................................................... 371
18.1 Molecular dynamics (MD): forces from the electrons .... 371
18.2 Car-Parrinello unified algorithm for electrons and
ions .................................................. 373
18.3 Expressions for plane waves ........................... 376
18.4 Alternative approaches to density functional QMD ...... 377
18.5 Non-self-consistent QMD methods ....................... 378
18.6 Examples of simulations ............................... 379
Select further reading ..................................... 383
Exercises .................................................. 384
19 Response functions: phonons, magnons ....................... 387
Summary .................................................... 387
19.1 Lattice dynamics from electronic structure theory ..... 388
19.2 The direct approach: "frozen phonons," magnons ........ 390
19.3 Phonons and density response functions ................ 394
19.4 Green's function formulation .......................... 395
19.5 Variational expressions ............................... 396
19.6 Periodic perturbations and phonon dispersion curves ... 398
19.7 Dielectric response functions, effective charges ...... 399
19.8 Electron-phonon interactions and superconductivity .... 401
19.9 Magnons and spin response functions ................... 402
Select further reading ..................................... 403
Exercises .................................................. 404
20 Excitation spectra and optical properties .................. 406
Summary .................................................... 406
20.1 Dielectric response for non-interacting particles .... 407
20.2 Time-dependent density functional theory and linear
response .............................................. 408
20.3 Variational Green's function methods for dynamical
linear response ....................................... 411
20.4 Explicit real-time calculations ....................... 412
20.5 Beyond the adiabatic local approximation .............. 416
Select further reading ..................................... 416
Exercises .................................................. 417
21 Wannier functions .......................................... 418
Summary .................................................... 418
21.1 Definition and properties ............................. 418
21.2 "Maximally projected" Wannier functions ............... 421
21.3 Maximally localized Wannier functions ................. 422
21.4 Non-orthogonal localized functions .................... 428
21.5 Wannier functions for "entangled bands" ............... 429
Select further reading ..................................... 431
Exercises .................................................. 432
22 Polarization, localization, and Berry's phases ............. 434
Summary .................................................... 434
22.1 Polarization: the fundamental difficulty .............. 436
22.2 Geometric Berry's phase theory of polarization ........ 439
22.3 Relation to centers of Wannier functions .............. 442
22.4 Calculation of polarization in crystals ............... 442
22.5 Localization: a rigorous measure ...................... 444
22.6 Geometric Berry's phase theory of spin waves .......... 446
Select further reading ..................................... 447
Exercises .................................................. 447
23 Locality and linear scaling O(N) methods ................... 450
Summary .................................................... 450
23.1 Locality and linear scaling in many-particle quantum
systems ............................................... 451
23.2 Building the hamiltonian .............................. 454
23.3 Solution of equations: non-variational methods ........ 455
23.4 Variational density matrix methods .................... 463
23.5 Variational (generalized) Wannier function methods .... 466
23.6 Linear-scaling self-consistent density functional
calculations .......................................... 469
23.7 Factorized density matrix for large basis sets ........ 470
23.8 Combining the methods ................................. 472
Select further reading ..................................... 472
Exercises .................................................. 473
24 Where to find more ......................................... 475
Appendix A Functional equations .............................. 476
Summary .................................................... 476
A.1 Basic definitions and variational equations ........... 476
A.2 Functionals in density functional theory including
gradients ............................................. 477
Select further reading ..................................... 478
Exercises .................................................. 478
Appendix В LSDA and GGA functionals .......................... 479
Summary .................................................... 479
B.l Local spin density approximation (LSDA) ............... 479
B.2 Generalized gradient approximation (GGAs) ............. 480
B.3 GGAs: explicit РВЕ form ............................... 480
Select further reading ..................................... 481
Appendix С Adiabatic approximation ........................... 482
Summary .................................................... 482
C.1 General formulation ................................... 482
C.2 Electron-phonon interactions .......................... 484
Select further reading ..................................... 484
Exercises .................................................. 484
Appendix D Response functions and Green's functions .......... 485
Summary .................................................... 485
D.1 Static response functions ............................. 485
D.2 Response functions in self-consistent field theories .. 486
D.3 Dynamic response and Kramers-Kronig relations ......... 487
D.4 Green's functions ..................................... 489
Select further reading ..................................... 491
Exercises .................................................. 491
Appendix E Dielectric functions and optical properties ....... 492
Summary .................................................... 492
E.l Electromagnetic waves in matter ....................... 492
E.2 Conductivity and dielectric tensors ................... 494
E.3 The/sum rule .......................................... 494
E.4 Scalar longitudinal dielectric functions .............. 495
E.5 Tensor transverse dielectric functions ................ 496
E.6 Lattice contributions to dielectric response .......... 496
Select further reading ..................................... 497
Exercises .................................................. 498
Appendix F Coulomb interactions in extended systems .......... 499
Summary .................................................... 499
F.1 Basic issues .......................................... 499
F.2 Point charges in a background: Ewald sums ............. 500
F.3 Smeared nuclei or ions ................................ 505
F.4 Energy relative to neutral atoms ...................... 506
F.5 Surface and interface dipoles ......................... 507
F.6 Reducing effects of artificial image charges .......... 508
Select further reading ..................................... 510
Exercises .................................................. 510
Appendix G Stress from electronic structure .................. 512
Summary .................................................... 512
G.l Macroscopic stress and strain ......................... 512
G.2 Stress from two-body pair-wise forces ................. 514
G.3 Expressions in Fourier components ..................... 515
G.4 Internalstrain ........................................ 516
Select further reading ..................................... 517
Exercises .................................................. 518
Appendix H Energy and stress densities ....................... 519
Summary .................................................... 519
H.1 Energy density ........................................ 520
H.2 Stress density ........................................ 523
H.3 Applications .......................................... 524
Select further reading ..................................... 527
Exercises .................................................. 527
Appendix I Alternative force expressions Summary ............. 530
1.1 Variational freedom and forces ........................ 530
1.2 Energy differences .................................... 532
1.3 Pressure .............................................. 532
1.4 Force and stress ...................................... 533
1.5 Force in APW-type methods ............................. 534
Select further reading ..................................... 534
Appendix J Scattering and phase shifts ....................... 536
Summary .................................................... 536
J.l Scattering and phase shifts for spherical
potentials ............................................ 536
Select further reading ..................................... 538
Appendix К Useful relations and formulas ..................... 539
Summary .................................................... 539
K.l Bessel, Neumann, and Hankel functions ................. 539
K.2 Spherical harmonics and Legendre polynomials .......... 539
K.3 Real spherical harmonics .............................. 540
K.4 Clebsch-Gordon and Gaunt coefficients ................. 541
K.5 Chebyshev polynomials ................................. 542
Appendix L Numerical methods ................................. 543
Summary .................................................... 543
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