Modern Quantum Chemistry

Preface to Revised edition Preface 1. Mathematical Review 1.1 Linear Algebra 1.1.1 Three-dimensional vector algebra 1.1.2 Matrices 1.1.3 Determinants 1.1.4 N-Dimensional Complex Vector spaces 1.1.5 Change of Basis 1.1.6 The Eigenvalue Problem 1.1.7 Functions of Matrices 1.2 Orthogonal functions, Eigenfunctions, and Operators 1.3 The Variation Method 1.3.1 The Variation principle 1.3.2 The Linear Variational Problem Notes, Further Reading 2. Many Electron Wave functions and operators 2.1 The Electronic Problem 2.1.1 Atomic Units 2.1.2 The Born-Oppenheimer Approximation 2.1.3 The Antisymmetry or Pauli Exclusion Principle 2.2 Orbitals, Slater Determinants, and Basis functions 2.2.1 Spin Orbitals and Spatial Orbitals 2.2.2 Hartree Products 2.2.3 Slater Determinants 2.2.4 The Hartree-Fock Approximation 2.2.5 The Minimal Basis H subscript 2 Model 2.2.6 Excited Determinants 2.2.7 Form of the Exact Wave function and Configuration Interaction 2.3 Operators and Matrix Elements 2.3.1 Minimal Basis H subscript 2 matrix Elements 2.3.2 Notations for One- and Two-Electron Integrals 2.3.3 General Rules for Matrix Elements 2.3.4 Derivation of the Rules for Matrix Elements 2.3.5 Transition from Spin Orbitals to Spatial Orbitals 2.3.6 Coulomb and Exchange Integrals 2.3.7 Pseudo-Classical interpretation of Determinantal Energies 2.4 Second Quantization 2.4.1 Creation and annihilation Operators and Their Anticommutation Relations 2.4.2 Second-Quantized Operators and Their Matrix Elements 2.5 Spin-Adapted Configurations 2.5.1 Spin Operators 2.5.2 Restricted Determinants and Spin-Adapted Configurations 2.5.3 Unrestricted Determinants Notes, Further Reading 3. The Hartree-Fock Approximation 3.1 The Hartree-Fock Equations 3.1.1 The Coulomb and Exchange Operators 3.1.2 The Fock Operator 3.2 Derivation of the Hartree-Fock Equations 3.2.1 Functional Variation 3.2.2 Minimization of the Energy of a Single Determinant 3.2.3 The Canonical Hartree-Fock Equations 3.3 Interpretation of Solutions to the Hartree-Fock Equations 3.3.1 Orbital energies and Koopmans' Theorem 3.3.2 Brillouin's Theorem 3.3.3 The Hartree-Fock Hamiltonian 3.4 Restricted Closed-Shell hartree-Fock: The Roothaan Equations 3.4.1 Closed-Shell Hartree-Fock: Restricted Spin Orbitals 3.4.2 Introduction of a Basis: The Roothaan Equations 3.4.3 The Charge Density 3.4.4 Expression for the Fock Matrix 3.4.5 Orthogonalization of the Basis 3.4.6 The SCF Procedure 3.4.7 Expectation Values and Population Analysis 3.5 Model Calculations on H subscript 2 and HeH superscript + 3.5.1 The 1s Minimal STO-3G Basis Set 3.5.2 STO-3G H subscript 2 3.5.3 An SCF Calculation on STO-3G HeH superscript + 3.6 Polyatomic Basis Sets 3.6.1 Contracted Gaussian functions 3.6.2 Minimal Basis Sets: STO-3G 3.6.3 Double Zeta Basis Sets: 4-31G 3.6.4 Polarized Basis Sets: 6-31G and 6-31G 3.7 Some Illustrative Closed-Shell Calculations 3.7.1 Total Energies 3.7.2 Ionization Potentials 3.7.3 Equilibrium Geometries 3.7.4 Population Analysis and Dipole Moments 3.8 Unrestricted Open-Shell Hartree-Fock: The Pople-Nesbet Equations 3.8.1 Open-Shell Hartree-Fock: Unrestricted Spin Orbitals 3.8.2 Introduction of a Basis: The Pople-Nesbet Equations 3.8.3 Unrestricted Density Matrices 3.8.4 Expression for the Fock Matrices 3.8.5 Solution of the Unrestricted SCF Equations 3.8.6 Illustrative Unrestricted Calculations 3.8.7 The Dissociation Problem and its Unrestricted Solution Notes, Further Reading 4. Configuration Interaction 4.1 Multiconfigurational Wave Functions and the Structure of the Full CI Matrix 4.1.1 Intermediate Normalization and an Expression for the Correlation Energy 4.2 Doubly Excited CI 4.3 Some Illustrative Calculations 4.4 Natural Orbitals and the One-Particle Reduced Density Matrix 4.5 The Multiconfiguration Self-Consistent Field (MCSCF) and Generalized Valence Bond (GVB) Methods 4.6 Truncated CI and the Size-Consistency Problem Notes, Further Reading 5. Pair and Coupled-Pair Theories 5.1 The Independent Electron Pair Approximation (IEPA) 5.1.1 Invariance under Unitary Transformations: an example 5.1.2 Some Illustrative Calculations 5.2 Coupled-Pair Theories 5.2.1 The Coupled Cluster Approximation (CCA) 5.2.2 The Cluster Expansion of the Wave Function 5.2.3 Linear CCA and the Coupled Electron Pair Approximation (CEPA) 5.2.4 Some Illustrative Calculations 5.3 Many-Electron Theories with Single Particle Hamiltonians 5.3.1 The Relaxation Energy via CI, IEPA, CCA, and CEPA 5.3.2 The Resonance Energy of Polyenes in Hückel Theory Notes, Further Reading 6. Many-Body Perturbation Theory 6.1 Rayleigh-Schrödinger (RS) Perturbation Theory 6.2 Diagrammatic Representation of RS Perturbation Theory 6.2.1 Diagrammatic Perturbation Theory for 2 States 6.2.2 Diagrammatic Perturbation Theory for N States 6.2.3 Summation of Diagrams 6.3 Orbital Perturbation Theory: One-Particle Perturbations 6.4 Diagrammatic Representation of Orbital Perturbation Theory 6.5 Perturbation Expansion of the Correlation Energy 6.6 The N-Dependence of the RS Perturbation Expansion 6.7 Diagrammatic Representation of the Perturbation Expansion of the Correlation Energy 6.7.1 Hugenholtz Diagrams 6.7.2 Goldstone Diagrams 6.7.3 Summation of Diagrams 6.7.4 What Is the Linked Cluster Theorem? 6.8 Some Illustrative Calculations Notes, Further Reading 7. The One-particle Many-Body Green's Function 7.1 Green's Functions in single Particle Systems 7.2 The One-Particle Many-Body Green's Function 7.2.1 The Self-Energy 7.2.2 The solution of the Dyson Equation 7.3 Application of the formalism to H subscript 2 and HeH superscript + 7.4 Perturbation Theory and the Green's Function Method 7.5 Some Illustrative Calculations Notes, Further Reading Appendix A. Integral Evaluation with 1s Primitive Gaussians Appendix B. Two-Electron Self-Consistent-Field Program Appendix C. Analytic Derivative methods and Geometry Optimization Appendix D. Molecular Integrals for H subscript 2 as a Function of Bond Length Index