Trygve Helgaker, Department of Chemistry, University of Oslo, Norway. Poul Jorgensen and Jeppe Olsen Department of Chemistry, University of Aarhus, Denmark.

• Preface xxiOverview xxvPrograms used in the preparation of this book xxix1. Second Quantization 11.1 The Fock space 11.2 Creation and annihilation operators 21.3 Number-conserving operators 61.4 The representation of one- and two-electron operators 91.5 Products of operators in second quantization 141.6 First- and second-quantization operators compared 181.7 Density matrices 191.8 Commutators and anticommutators 251.9 Nonorthogonal spin orbitals 272. Spin in Second Quantization 342.1 Spin functions 342.2 Operators in the orbital basis 352.3 Spin tensor operators 412.4 Spin properties of determinants 462.5 Configuration state functions 512.6 The genealogical coupling scheme 532.7 Density matrices 613. Orbital Rotations 803.1 Unitary transformations and matrix exponentials 803.2 Unitary spin-orbital transformations 863.3 Symmetry-restricted unitary transformations 893.4 The logarithmic matrix function 934. Exact and Approximate Wave Functions 1074.1 Characteristics of the exact wave function 1074.2 The variation principle 1114.3 Size-extensivity 1264.4 Symmetry constraints 1355. The Standard Models 1425.1 One- and N-electron expansions 1435.2 A model system: the hydrogen molecule in a minimal basis 1465.3 Exact wave functions in Fock space 1625.4 The Hartree-Fock approximation 1675.5 Multiconfigurational self-consistent field theory 1765.6 Configuration-interaction theory 1815.7 Coupled-cluster theory 1865.8 Perturbation theory 1926. Atomic Basis Functions 2016.1 Requirements on one-electron basis functions 2016.2 One- and many-centre expansions 2036.3 The one-electron central-field system 2046.4 The angular basis 2076.5 Exponential radial functions 2186.6 Gaussian radial functions 2297. Short-Range Interactions and Orbital Expansions 2567.1 The Coulomb hole 2567.2 The Coulomb cusp 2597.3 Approximate treatments of the ground-state helium atom 2627.4 The partial-wave expansion of the ground-state helium atom 2677.5 The principal expansion of the ground-state helium atom 2737.6 Electron-correlation effects summarized 2788. Gaussian Basis Sets 2878.1 Gaussian basis functions 2878.2 Gaussian basis sets for Hartree-Fock calculations 2888.3 Gaussian basis sets for correlated calculations 3008.4 Basis-set convergence 3158.5 Basis-set superposition error 3279. Molecular Integral Evaluation 3369.1 Contracted spherical-harmonic Gaussians 3369.2 Cartesian Gaussians 3389.3 The Obara-Saika scheme for simple integrals 3449.4 Hermite Gaussians 3499.5 The McMurchie-Davidson scheme for simple integrals 3529.6 Gaussian quadrature for simple integrals 3579.7 Coulomb integra;s over spherical Gaussians 3619.8 The Boys function 3659.9 The McMurchie-Davidson scheme for Coulomb integrals 3729.10 The Obara-Saika scheme for Coulomb integrals 3819.11 Rys quadrature for Coulomb integrals 3879.12 Scaling properties of the molecular integrals 3989.13 The multipole method for Coulomb integrals 4059.14 The multipole method for large systems 41710. Hartree-Fock Theory 43310.1 Parametrization of the wave function and the energy 43310.2 The Hartree-Fock wave function 43810.3 Canonical Hartree-Fock theory 44310.4 The RHF total energy and orbital energies 45010.5 Koopmans’ theorem 45410.6 The Roothaan-Hall self-consistent field equations 45810.7 Density-based Hartree-Fock theory 46510.8 Second-order optimization 47810.9 The SCF method as an approximate second-order method 49010.10 Singlet and triplet instabilities in RHF theory 49610.11 Multiple solutions in Hartree-Fock theory 50411. Configuration-Interaction Theory 52311.1 The CI model 52311.2 Size-extensivity and the CI model 52711.3 A CI model system for noninteracting hydrogen molecules 53511.4 Parametrization of the CI model 54011.5 Optimization of the CI wave function 54311.6 Slater determinants as products of alpha and beta strings 55011.7 The determinantal representation of the Hamiltonian operator 55211.8 Direct CI methods 55411.9 CI orbital transformations 56911.10 Symmetry-broken CI solutions 57312. Multiconfigurational Self-Consistent Field Theory 49812.1 The MCSCF model 49812.2 The MCSCF energy and wave function  60012.3 The MCSCF Newton trust-region method 61012.4 The Newton cigenvector method 61612.5 Computational considerations 62112.6 Exponential parametrization of the configuration space 63012.7 MCSCF theory for several electronic states 63712.8 Removal of RHF instabilities in MCSCF theory 64013. Coupled-Cluster Theory 64813.1 The coupled-cluster model 64813.2 The coupled-cluster exponential ansatz 65413.3 Size-extensivity in coupled-cluster theory 66513.4 Coupled-cluster optimization techniques 67013.5 The coupled-cluster variational Lagrangian 67413.6 The equation-of-motion coupled-cluster method 67713.7 The closed-shell CCSD model 68513.8 Special treatments of coupled-cluster theory 69813.9 High-spin open-shell coupled-cluster theory 70414. Perturbation Theory 72414.1 Rayleigh-Schrödinger perturbation theory 72514.2 Møller-Plesset perturbation theory 73914.3 Coupled-cluster perturbation theory 74914.4 Møller-Plesset theory for closed-shell systems 75914.5 Convergence in perturbation theory 76914.6 Perturbative treatments of coupled-cluster wave functions 78314.7 Multiconfigurational perturbation theory 79615. Calibration of the Electronic-Structure Models 81715.1 The sample molecules 81715.2 Errors in quantum-chemical calculations 81915.3 Molecular equilibrium structures: bond distances 82115.4 Molecular equilibrium structures; bond angles 83215.5 Molecular dipole moments 83615.6 Molecular and atomic energies 84015.7 Atomization energies 85415.8 Reaction enthalpies 86515.9 Conformational barriers 87415.10 Conclusions 879List of Acronyms 885Index 887

Trygve Helgaker, Department of Chemistry, University of Oslo, Norway. Poul Jorgensen and Jeppe Olsen Department of Chemistry, University of Aarhus, Denmark.

• Preface xxiOverview xxvPrograms used in the preparation of this book xxix1. Second Quantization 11.1 The Fock space 11.2 Creation and annihilation operators 21.3 Number-conserving operators 61.4 The representation of one- and two-electron operators 91.5 Products of operators in second quantization 141.6 First- and second-quantization operators compared 181.7 Density matrices 191.8 Commutators and anticommutators 251.9 Nonorthogonal spin orbitals 272. Spin in Second Quantization 342.1 Spin functions 342.2 Operators in the orbital basis 352.3 Spin tensor operators 412.4 Spin properties of determinants 462.5 Configuration state functions 512.6 The genealogical coupling scheme 532.7 Density matrices 613. Orbital Rotations 803.1 Unitary transformations and matrix exponentials 803.2 Unitary spin-orbital transformations 863.3 Symmetry-restricted unitary transformations 893.4 The logarithmic matrix function 934. Exact and Approximate Wave Functions 1074.1 Characteristics of the exact wave function 1074.2 The variation principle 1114.3 Size-extensivity 1264.4 Symmetry constraints 1355. The Standard Models 1425.1 One- and N-electron expansions 1435.2 A model system: the hydrogen molecule in a minimal basis 1465.3 Exact wave functions in Fock space 1625.4 The Hartree-Fock approximation 1675.5 Multiconfigurational self-consistent field theory 1765.6 Configuration-interaction theory 1815.7 Coupled-cluster theory 1865.8 Perturbation theory 1926. Atomic Basis Functions 2016.1 Requirements on one-electron basis functions 2016.2 One- and many-centre expansions 2036.3 The one-electron central-field system 2046.4 The angular basis 2076.5 Exponential radial functions 2186.6 Gaussian radial functions 2297. Short-Range Interactions and Orbital Expansions 2567.1 The Coulomb hole 2567.2 The Coulomb cusp 2597.3 Approximate treatments of the ground-state helium atom 2627.4 The partial-wave expansion of the ground-state helium atom 2677.5 The principal expansion of the ground-state helium atom 2737.6 Electron-correlation effects summarized 2788. Gaussian Basis Sets 2878.1 Gaussian basis functions 2878.2 Gaussian basis sets for Hartree-Fock calculations 2888.3 Gaussian basis sets for correlated calculations 3008.4 Basis-set convergence 3158.5 Basis-set superposition error 3279. Molecular Integral Evaluation 3369.1 Contracted spherical-harmonic Gaussians 3369.2 Cartesian Gaussians 3389.3 The Obara-Saika scheme for simple integrals 3449.4 Hermite Gaussians 3499.5 The McMurchie-Davidson scheme for simple integrals 3529.6 Gaussian quadrature for simple integrals 3579.7 Coulomb integra;s over spherical Gaussians 3619.8 The Boys function 3659.9 The McMurchie-Davidson scheme for Coulomb integrals 3729.10 The Obara-Saika scheme for Coulomb integrals 3819.11 Rys quadrature for Coulomb integrals 3879.12 Scaling properties of the molecular integrals 3989.13 The multipole method for Coulomb integrals 4059.14 The multipole method for large systems 41710. Hartree-Fock Theory 43310.1 Parametrization of the wave function and the energy 43310.2 The Hartree-Fock wave function 43810.3 Canonical Hartree-Fock theory 44310.4 The RHF total energy and orbital energies 45010.5 Koopmans’ theorem 45410.6 The Roothaan-Hall self-consistent field equations 45810.7 Density-based Hartree-Fock theory 46510.8 Second-order optimization 47810.9 The SCF method as an approximate second-order method 49010.10 Singlet and triplet instabilities in RHF theory 49610.11 Multiple solutions in Hartree-Fock theory 50411. Configuration-Interaction Theory 52311.1 The CI model 52311.2 Size-extensivity and the CI model 52711.3 A CI model system for noninteracting hydrogen molecules 53511.4 Parametrization of the CI model 54011.5 Optimization of the CI wave function 54311.6 Slater determinants as products of alpha and beta strings 55011.7 The determinantal representation of the Hamiltonian operator 55211.8 Direct CI methods 55411.9 CI orbital transformations 56911.10 Symmetry-broken CI solutions 57312. Multiconfigurational Self-Consistent Field Theory 49812.1 The MCSCF model 49812.2 The MCSCF energy and wave function  60012.3 The MCSCF Newton trust-region method 61012.4 The Newton cigenvector method 61612.5 Computational considerations 62112.6 Exponential parametrization of the configuration space 63012.7 MCSCF theory for several electronic states 63712.8 Removal of RHF instabilities in MCSCF theory 64013. Coupled-Cluster Theory 64813.1 The coupled-cluster model 64813.2 The coupled-cluster exponential ansatz 65413.3 Size-extensivity in coupled-cluster theory 66513.4 Coupled-cluster optimization techniques 67013.5 The coupled-cluster variational Lagrangian 67413.6 The equation-of-motion coupled-cluster method 67713.7 The closed-shell CCSD model 68513.8 Special treatments of coupled-cluster theory 69813.9 High-spin open-shell coupled-cluster theory 70414. Perturbation Theory 72414.1 Rayleigh-Schrödinger perturbation theory 72514.2 Møller-Plesset perturbation theory 73914.3 Coupled-cluster perturbation theory 74914.4 Møller-Plesset theory for closed-shell systems 75914.5 Convergence in perturbation theory 76914.6 Perturbative treatments of coupled-cluster wave functions 78314.7 Multiconfigurational perturbation theory 79615. Calibration of the Electronic-Structure Models 81715.1 The sample molecules 81715.2 Errors in quantum-chemical calculations 81915.3 Molecular equilibrium structures: bond distances 82115.4 Molecular equilibrium structures; bond angles 83215.5 Molecular dipole moments 83615.6 Molecular and atomic energies 84015.7 Atomization energies 85415.8 Reaction enthalpies 86515.9 Conformational barriers 87415.10 Conclusions 879List of Acronyms 885Index 887

Trygve Helgaker, Department of Chemistry, University of Oslo, Norway. Poul Jorgensen and Jeppe Olsen Department of Chemistry, University of Aarhus, Denmark.

• Preface xxiOverview xxvPrograms used in the preparation of this book xxix1. Second Quantization 11.1 The Fock space 11.2 Creation and annihilation operators 21.3 Number-conserving operators 61.4 The representation of one- and two-electron operators 91.5 Products of operators in second quantization 141.6 First- and second-quantization operators compared 181.7 Density matrices 191.8 Commutators and anticommutators 251.9 Nonorthogonal spin orbitals 272. Spin in Second Quantization 342.1 Spin functions 342.2 Operators in the orbital basis 352.3 Spin tensor operators 412.4 Spin properties of determinants 462.5 Configuration state functions 512.6 The genealogical coupling scheme 532.7 Density matrices 613. Orbital Rotations 803.1 Unitary transformations and matrix exponentials 803.2 Unitary spin-orbital transformations 863.3 Symmetry-restricted unitary transformations 893.4 The logarithmic matrix function 934. Exact and Approximate Wave Functions 1074.1 Characteristics of the exact wave function 1074.2 The variation principle 1114.3 Size-extensivity 1264.4 Symmetry constraints 1355. The Standard Models 1425.1 One- and N-electron expansions 1435.2 A model system: the hydrogen molecule in a minimal basis 1465.3 Exact wave functions in Fock space 1625.4 The Hartree-Fock approximation 1675.5 Multiconfigurational self-consistent field theory 1765.6 Configuration-interaction theory 1815.7 Coupled-cluster theory 1865.8 Perturbation theory 1926. Atomic Basis Functions 2016.1 Requirements on one-electron basis functions 2016.2 One- and many-centre expansions 2036.3 The one-electron central-field system 2046.4 The angular basis 2076.5 Exponential radial functions 2186.6 Gaussian radial functions 2297. Short-Range Interactions and Orbital Expansions 2567.1 The Coulomb hole 2567.2 The Coulomb cusp 2597.3 Approximate treatments of the ground-state helium atom 2627.4 The partial-wave expansion of the ground-state helium atom 2677.5 The principal expansion of the ground-state helium atom 2737.6 Electron-correlation effects summarized 2788. Gaussian Basis Sets 2878.1 Gaussian basis functions 2878.2 Gaussian basis sets for Hartree-Fock calculations 2888.3 Gaussian basis sets for correlated calculations 3008.4 Basis-set convergence 3158.5 Basis-set superposition error 3279. Molecular Integral Evaluation 3369.1 Contracted spherical-harmonic Gaussians 3369.2 Cartesian Gaussians 3389.3 The Obara-Saika scheme for simple integrals 3449.4 Hermite Gaussians 3499.5 The McMurchie-Davidson scheme for simple integrals 3529.6 Gaussian quadrature for simple integrals 3579.7 Coulomb integra;s over spherical Gaussians 3619.8 The Boys function 3659.9 The McMurchie-Davidson scheme for Coulomb integrals 3729.10 The Obara-Saika scheme for Coulomb integrals 3819.11 Rys quadrature for Coulomb integrals 3879.12 Scaling properties of the molecular integrals 3989.13 The multipole method for Coulomb integrals 4059.14 The multipole method for large systems 41710. Hartree-Fock Theory 43310.1 Parametrization of the wave function and the energy 43310.2 The Hartree-Fock wave function 43810.3 Canonical Hartree-Fock theory 44310.4 The RHF total energy and orbital energies 45010.5 Koopmans’ theorem 45410.6 The Roothaan-Hall self-consistent field equations 45810.7 Density-based Hartree-Fock theory 46510.8 Second-order optimization 47810.9 The SCF method as an approximate second-order method 49010.10 Singlet and triplet instabilities in RHF theory 49610.11 Multiple solutions in Hartree-Fock theory 50411. Configuration-Interaction Theory 52311.1 The CI model 52311.2 Size-extensivity and the CI model 52711.3 A CI model system for noninteracting hydrogen molecules 53511.4 Parametrization of the CI model 54011.5 Optimization of the CI wave function 54311.6 Slater determinants as products of alpha and beta strings 55011.7 The determinantal representation of the Hamiltonian operator 55211.8 Direct CI methods 55411.9 CI orbital transformations 56911.10 Symmetry-broken CI solutions 57312. Multiconfigurational Self-Consistent Field Theory 49812.1 The MCSCF model 49812.2 The MCSCF energy and wave function  60012.3 The MCSCF Newton trust-region method 61012.4 The Newton cigenvector method 61612.5 Computational considerations 62112.6 Exponential parametrization of the configuration space 63012.7 MCSCF theory for several electronic states 63712.8 Removal of RHF instabilities in MCSCF theory 64013. Coupled-Cluster Theory 64813.1 The coupled-cluster model 64813.2 The coupled-cluster exponential ansatz 65413.3 Size-extensivity in coupled-cluster theory 66513.4 Coupled-cluster optimization techniques 67013.5 The coupled-cluster variational Lagrangian 67413.6 The equation-of-motion coupled-cluster method 67713.7 The closed-shell CCSD model 68513.8 Special treatments of coupled-cluster theory 69813.9 High-spin open-shell coupled-cluster theory 70414. Perturbation Theory 72414.1 Rayleigh-Schrödinger perturbation theory 72514.2 Møller-Plesset perturbation theory 73914.3 Coupled-cluster perturbation theory 74914.4 Møller-Plesset theory for closed-shell systems 75914.5 Convergence in perturbation theory 76914.6 Perturbative treatments of coupled-cluster wave functions 78314.7 Multiconfigurational perturbation theory 79615. Calibration of the Electronic-Structure Models 81715.1 The sample molecules 81715.2 Errors in quantum-chemical calculations 81915.3 Molecular equilibrium structures: bond distances 82115.4 Molecular equilibrium structures; bond angles 83215.5 Molecular dipole moments 83615.6 Molecular and atomic energies 84015.7 Atomization energies 85415.8 Reaction enthalpies 86515.9 Conformational barriers 87415.10 Conclusions 879List of Acronyms 885Index 887