Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory

052181832X......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1.1 Scope......Page 17
1.2 Conventions and notation......Page 18
1.3 The independent-particle approximation......Page 19
1.4 Electron correlation......Page 23
1.5 Configuration interaction......Page 25
1.6 Motivation......Page 26
1.7 Extensivity......Page 27
1.8 Disconnected clusters and extensivity......Page 31
2.2.1 The perturbation Ansatz......Page 34
2.2.2 Indeterminacy of the solution......Page 36
2.2.4 Order-by-order expansion......Page 37
2.2.5 Expansion in zero-order functions......Page 39
2.2.6 Wigner’s 2n + 1 rule......Page 40
2.2.7 The Hylleraas variation principle for the first-order wave function......Page 41
2.3 Projection operators......Page 43
2.4.1 General formalism......Page 45
2.4.2 Brillouin–Wigner perturbation theory......Page 49
2.4.3 Demonstration of non-extensivity of finite-order BWPT......Page 50
2.4.4 Formal Rayleigh–Schrodinger perturbation theory......Page 53
2.4.5 The general (non-diagonal) case......Page 56
2.4.6 Bracketing procedure for RSPT......Page 57
2.4.7 Summary of formal RSPT results......Page 59
2.4.8 Extensivity of Rayleigh–Schrodinger perturbation theory......Page 61
2.5 Similarity transformation derivation of the formal perturbation equations and quasidegenerate PT......Page 62
2.6 Other approaches......Page 69
3.1 Background......Page 70
3.2.1 Definitions......Page 71
3.2.2 Anticommutation relations......Page 73
3.2.3 Representation of operators......Page 75
3.2.4 Invariance under unitary transformations......Page 81
3.3.1 Normal products of operators......Page 83
3.3.2 Contractions......Page 84
3.3.3 Time-independent Wick's theorem......Page 85
3.3.4 Outline of proof of Wick's theorem......Page 86
3.4.1 The reference state......Page 87
3.4.2 Normal products and Wick’s theorem relative to the Fermi vacuum......Page 90
3.5 Partitioning of the Hamiltonian......Page 91
3.6.1 One-electron operators......Page 96
3.6.2 Two-electron operators......Page 97
3.6.3 The normal-product Hamiltonian......Page 99
3.7 Generalized time-independent Wick's theorem......Page 101
3.8 Evaluation of matrix elements......Page 102
4.1 Time ordering......Page 106
4.2 Slater determinants......Page 107
4.3.1 Representation of one-particle operators and contractions......Page 108
4.3.2 Rules of interpretation......Page 111
4.3.3 The complete one-particle operator......Page 113
4.3.4 Products of one-particle operators......Page 116
4.3.5 Phase factors......Page 122
4.4.1 Goldstone diagrams for a two-particle operator......Page 127
4.4.2 Vacuum expectation values of products of two-particle operators......Page 129
4.4.3 Hugenholtz diagrams......Page 134
4.4.4 Antisymmetrized Goldstone diagrams......Page 137
4.4.5 Representation of operators not in normal-product form......Page 139
4.4.6 The RSPT perturbation operator......Page 144
5.1 Resolvent operator and denominators......Page 146
5.3 Second-order energy......Page 147
5.4 Third-order energy......Page 148
5.5 Conjugate diagrams......Page 150
5.6.1 First-order wave function......Page 151
5.6.2 Second-order wave function......Page 152
5.7.1 Energy formula......Page 154
5.7.2 Diagrams for E(4) in the canonical HF case......Page 155
5.7.3 Non-HF diagrams for E(4)......Page 160
5.7.4 Cancellation of the unlinked diagrams in E(4)......Page 162
5.7.5 Role of the EPV terms......Page 166
5.8 Linked-diagram theorem......Page 168
5.9 Numerical example......Page 169
5.10.1 Extensivity implications......Page 172
5.10.2 The dependence of diagrams......Page 173
5.10.3 Relationship to configuration interaction (CI)......Page 175
6.1 The factorization theorem......Page 181
6.2 The linked-diagram theorem......Page 188
7.1 Techniques of diagram summation......Page 193
7.2 Factorization of fourth-order quadruple-excitation diagrams......Page 196
7.3 Spin summations......Page 198
8.1 Formal quasidegenerate perturbation theory (QDPT)......Page 201
8.2 The Fermi vacuum and the model states......Page 208
8.3 Normal-product form of the generalized Bloch equations......Page 210
8.4 Diagrammatic notation for QDPT......Page 211
8.5 Schematic representation of the generalized Bloch equation......Page 214
8.6.1 Zero and first order......Page 219
8.6.2 Second-order level-shift operator......Page 221
8.6.3 Second-order wave operator......Page 223
8.6.4 Weight factors and phases......Page 227
8.6.5 Folded diagrams......Page 231
8.6.6 Third-order level-shift operator......Page 236
8.6.7 Third-order wave operator......Page 241
8.7.1 The Hose–Kaldor approach......Page 243
8.7.2 The one-electron interaction......Page 244
8.7.3 First-order diagrams......Page 246
8.7.4 Second-order diagrams......Page 248
8.7.5 Third-order level-shift diagrams......Page 252
9.1 Coupled-cluster theory for noninteracting He atoms......Page 267
9.2.1 The exponential Ansatz and extensivity......Page 270
9.2.2 The cluster operators......Page 271
9.3.1 Coupled-cluster doubles equations: configuration-space derivation......Page 274
9.3.2 Coupled-cluster doubles equations: algebraic derivation......Page 279
9.4 Exponential Ansatz and the linked-diagram theorem of MBPT......Page 288
9.5 Diagrammatic derivation of the CCD equations......Page 295
10.1 The connected form of the CC equations......Page 308
10.2 The general form of CC diagrams......Page 311
10.3 Systematic generation of CC diagrams......Page 313
10.4 The coupled-cluster singles and doubles (CCSD) equations......Page 315
10.5 Coupled-cluster singles, doubles and triples (CCSDT) equations......Page 324
10.6 Coupled-cluster singles, doubles, triples and quadruples (CCSDTQ) equations......Page 337
10.7 Coupled-cluster effective-Hamiltonian diagrams......Page 344
10.8 Results of various CC methods compared with full CI......Page 356
11.1 Expectation value for a CC wave function......Page 363
11.2 Reduced density matrices......Page 368
11.3 The response treatment of properties......Page 377
11.4 The CC energy functional......Page 382
11.5 The Λ equations......Page 383
11.6 Effective-Hamiltonian form of the Λ equations
......Page 392
11.7 Response treatment of the density matrices......Page 397
11.8 The perturbed reference function......Page 401
11.9 The CC correlation-energy derivative......Page 412
12.1 Spin summations and computational considerations......Page 422
12.2 Coupled-cluster theory with an arbitrary single-determinant reference function......Page 427
12.3 Generalized many-body perturbation theory......Page 431
12.4 Brueckner orbitals and alternative treatments of T1......Page 434
12.5 Monitoring multiplicities in open-shell coupled-cluster calculations......Page 438
12.6 The A and B response matrices from the viewpoint of CCS......Page 441
12.7 Noniterative approximations based on the CC energy functional......Page 443
12.8 The nature of the solutions of CC equations......Page 445
13.1 Introduction......Page 447
13.2 The EOM-CC Ansatz......Page 448
13.3 Diagrammatic treatment of the EE-EOM-CC equations......Page 453
13.4 EOM-CC treatment of ionization and electron attachment......Page 461
13.5 EOM-CC treatment of higher-order properties......Page 465
13.6 EOM-CC treatment of frequency-dependent properties......Page 470
14.1 Introduction......Page 478
14.2 Hilbert-space state-universal MRCC......Page 481
14.3 Hilbert-space state-specific MRCC......Page 487
14.4.1 The Fock-space approach......Page 491
14.4.2 The valence-universal wave operator......Page 493
14.4.3 The Fock-space Bloch equations......Page 499
14.4.4 Relationship to EOM-CC......Page 504
14.5 Intermediate-Hamiltonian Fock-space MRCC......Page 506
References......Page 512
Author index......Page 537