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Preface xxi
Overview xxv
Programs used in the preparation of this book xxix
1. Second Quantization 1
1.1 The Fock space 1
1.2 Creation and annihilation operators 2
1.3 Number-conserving operators 6
1.4 The representation of one- and two-electron operators 9
1.5 Products of operators in second quantization 14
1.6 First- and second-quantization operators compared 18
1.7 Density matrices 19
1.8 Commutators and anticommutators 25
1.9 Nonorthogonal spin orbitals 27
2. Spin in Second Quantization 34
2.1 Spin functions 34
2.2 Operators in the orbital basis 35
2.3 Spin tensor operators 41
2.4 Spin properties of determinants 46
2.5 Configuration state functions 51
2.6 The genealogical coupling scheme 53
2.7 Density matrices 61
3. Orbital Rotations 80
3.1 Unitary transformations and matrix exponentials 80
3.2 Unitary spin-orbital transformations 86
3.3 Symmetry-restricted unitary transformations 89
3.4 The logarithmic matrix function 93
4. Exact and Approximate Wave Functions 107
4.1 Characteristics of the exact wave function 107
4.2 The variation principle 111
4.3 Size-extensivity 126
4.4 Symmetry constraints 135
5. The Standard Models 142
5.1 One- and N-electron expansions 143
5.2 A model system: the hydrogen molecule in a minimal basis 146
5.3 Exact wave functions in Fock space 162
5.4 The Hartree-Fock approximation 167
5.5 Multiconfigurational self-consistent field theory 176
5.6 Configuration-interaction theory 181
5.7 Coupled-cluster theory 186
5.8 Perturbation theory 192
6. Atomic Basis Functions 201
6.1 Requirements on one-electron basis functions 201
6.2 One- and many-centre expansions 203
6.3 The one-electron central-field system 204
6.4 The angular basis 207
6.5 Exponential radial functions 218
6.6 Gaussian radial functions 229
7. Short-Range Interactions and Orbital Expansions 256
7.1 The Coulomb hole 256
7.2 The Coulomb cusp 259
7.3 Approximate treatments of the ground-state helium atom 262
7.4 The partial-wave expansion of the ground-state helium atom 267
7.5 The principal expansion of the ground-state helium atom 273
7.6 Electron-correlation effects summarized 278
8. Gaussian Basis Sets 287
8.1 Gaussian basis functions 287
8.2 Gaussian basis sets for Hartree-Fock calculations 288
8.3 Gaussian basis sets for correlated calculations 300
8.4 Basis-set convergence 315
8.5 Basis-set superposition error 327
9. Molecular Integral Evaluation 336
9.1 Contracted spherical-harmonic Gaussians 336
9.2 Cartesian Gaussians 338
9.3 The Obara-Saika scheme for simple integrals 344
9.4 Hermite Gaussians 349
9.5 The McMurchie-Davidson scheme for simple integrals 352
9.6 Gaussian quadrature for simple integrals 357
9.7 Coulomb integra;s over spherical Gaussians 361
9.8 The Boys function 365
9.9 The McMurchie-Davidson scheme for Coulomb integrals 372
9.10 The Obara-Saika scheme for Coulomb integrals 381
9.11 Rys quadrature for Coulomb integrals 387
9.12 Scaling properties of the molecular integrals 398
9.13 The multipole method for Coulomb integrals 405
9.14 The multipole method for large systems 417
10. Hartree-Fock Theory 433
10.1 Parametrization of the wave function and the energy 433
10.2 The Hartree-Fock wave function 438
10.3 Canonical Hartree-Fock theory 443
10.4 The RHF total energy and orbital energies 450
10.5 Koopmans’ theorem 454
10.6 The Roothaan-Hall self-consistent field equations 458
10.7 Density-based Hartree-Fock theory 465
10.8 Second-order optimization 478
10.9 The SCF method as an approximate second-order method 490
10.10 Singlet and triplet instabilities in RHF theory 496
10.11 Multiple solutions in Hartree-Fock theory 504
11. Configuration-Interaction Theory 523
11.1 The CI model 523
11.2 Size-extensivity and the CI model 527
11.3 A CI model system for noninteracting hydrogen molecules 535
11.4 Parametrization of the CI model 540
11.5 Optimization of the CI wave function 543
11.6 Slater determinants as products of alpha and beta strings 550
11.7 The determinantal representation of the Hamiltonian operator 552
11.8 Direct CI methods 554
11.9 CI orbital transformations 569
11.10 Symmetry-broken CI solutions 573
12. Multiconfigurational Self-Consistent Field Theory 498
12.1 The MCSCF model 498
12.2 The MCSCF energy and wave function 600
12.3 The MCSCF Newton trust-region method 610
12.4 The Newton cigenvector method 616
12.5 Computational considerations 621
12.6 Exponential parametrization of the configuration space 630
12.7 MCSCF theory for several electronic states 637
12.8 Removal of RHF instabilities in MCSCF theory 640
13. Coupled-Cluster Theory 648
13.1 The coupled-cluster model 648
13.2 The coupled-cluster exponential ansatz 654
13.3 Size-extensivity in coupled-cluster theory 665
13.4 Coupled-cluster optimization techniques 670
13.5 The coupled-cluster variational Lagrangian 674
13.6 The equation-of-motion coupled-cluster method 677
13.7 The closed-shell CCSD model 685
13.8 Special treatments of coupled-cluster theory 698
13.9 High-spin open-shell coupled-cluster theory 704
14. Perturbation Theory 724
14.1 Rayleigh-Schrödinger perturbation theory 725
14.2 Møller-Plesset perturbation theory 739
14.3 Coupled-cluster perturbation theory 749
14.4 Møller-Plesset theory for closed-shell systems 759
14.5 Convergence in perturbation theory 769
14.6 Perturbative treatments of coupled-cluster wave functions 783
14.7 Multiconfigurational perturbation theory 796
15. Calibration of the Electronic-Structure Models 817
15.1 The sample molecules 817
15.2 Errors in quantum-chemical calculations 819
15.3 Molecular equilibrium structures: bond distances 821
15.4 Molecular equilibrium structures; bond angles 832
15.5 Molecular dipole moments 836
15.6 Molecular and atomic energies 840
15.7 Atomization energies 854
15.8 Reaction enthalpies 865
15.9 Conformational barriers 874
15.10 Conclusions 879
List of Acronyms 885
Index 887