Modern Quantum Chemistry | Neil S Ostlund

Preface to Revised edition Preface1. Mathematical Review 1.1 Linear Algebra1.1. 1 Three-dimensional vector algebra1.1. 2 Matrices1.1. 3 Determinants1.1. 4 N-Dimensional Complex Vector spaces1.1. 5 Change of Basis1.1. 6 The Eigenvalue Problem1.1. 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 Reading2. Many Electron Wave functions and operators 2.1 The Electronic Problem2.1. 1 Atomic Units2.1. 2 The Born-Oppenheimer Approximation2.1. 3 The Antisymmetry or Pauli Exclusion Principle 2.2 Orbitals, Slater Determinants, and Basis functions2.2. 1 Spin Orbitals and Spatial Orbitals2.2. 2 Hartree Products2.2. 3 Slater Determinants2.2. 4 The Hartree-Fock Approximation2.2. 5 The Minimal Basis H subscript 2 Model2.2. 6 Excited Determinants2.2. 7 Form of the Exact Wave function and Configuration Interaction 2.3 Operators and Matrix Elements2.3. 1 Minimal Basis H subscript 2 matrix Elements2.3. 2 Notations for One- and Two-Electron Integrals2.3. 3 General Rules for Matrix Elements2.3. 4 Derivation of the Rules for Matrix Elements2.3. 5 Transition from Spin Orbitals to Spatial Orbitals2.3. 6 Coulomb and Exchange Integrals2.3. 7 Pseudo-Classical interpretation of Determinantal Energies 2.4 Second Quantization2.4. 1 Creation and annihilation Operators and Their Anticommutation Relations2.4. 2 Second-Quantized Operators and Their Matrix Elements 2.5 Spin-Adapted Configurations2.5. 1 Spin Operators2.5. 2 Restricted Determinants and Spin-Adapted Configurations2.5. 3 Unrestricted Determinants Notes, Further Reading3. The Hartree-Fock Approximation 3.1 The Hartree-Fock Equations3.1. 1 The Coulomb and Exchange Operators3.1. 2 The Fock Operator 3.2 Derivation of the Hartree-Fock Equations3.2. 1 Functional Variation3.2. 2 Minimization of the Energy of a Single Determinant3.2. 3 The Canonical Hartree-Fock Equations 3.3 Interpretation of Solutions to the Hartree-Fock Equations3.3. 1 Orbital energies and Koopmans' Theorem3.3. 2 Brillouin's Theorem3.3. 3 The Hartree-Fock Hamiltonian 3.4 Restricted Closed-Shell hartree-Fock: The Roothaan Equations3.4. 1 Closed-Shell Hartree-Fock: Restricted Spin Orbitals3.4. 2 Introduction of a Basis: The Roothaan Equations3.4. 3 The Charge Density3.4. 4 Expression for the Fock Matrix3.4. 5 Orthogonalization of the Basis3.4. 6 The SCF Procedure3.4. 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 Set3.5. 2 STO-3G H subscript 23.5. 3 An SCF Calculation on STO-3G HeH superscript + 3.6 Polyatomic Basis Sets3.6. 1 Contracted Gaussian functions3.6. 2 Minimal Basis Sets: STO-3G3.6. 3 Double Zeta Basis Sets: 4-31G3.6. 4 Polarized Basis Sets: 6-31G and 6-31G 3.7 Some Illustrative Closed-Shell Calculations3.7. 1 Total Energies3.7. 2 Ionization Potentials3.7. 3 Equilibrium Geometries3.7. 4 Population Analysis and Dipole Moments 3.8 Unrestricted Open-Shell Hartree-Fock: The Pople-Nesbet Equations3.8. 1 Open-Shell Hartree-Fock: Unrestricted Spin Orbitals3.8. 2 Introduction of a Basis: The Pople-Nesbet Equations3.8. 3 Unrestricted Density Matrices3.8. 4 Expression for the Fock Matrices3.8. 5 Solution of the Unrestricted SCF Equations3.8. 6 Illustrative Unrestricted Calculations3.8. 7 The Dissociation Problem and its Unrestricted Solution Notes, Further Reading4. Configuration Interaction 4.1 Multiconfigurational Wave Functions and the Structure of the Full CI Matrix4.1. 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 Reading5. Pair and Coupled-Pair Theories 5.1 The Independent Electron Pair Approximation (IEPA)5.1. 1 Invariance under Unitary Transformations: an example5.1. 2 Some Illustrative Calculations 5.2 Coupled-Pair Theories5.2. 1 The Coupled Cluster Approximation (CCA)5.2. 2 The Cluster Expansion of the Wave Function5.2. 3 Linear CCA and the Coupled Electron Pair Approximation (CEPA)5.2. 4 Some Illustrative Calculations 5.3 Many-Electron Theories with Single Particle Hamiltonians5.3. 1 The Relaxation Energy via CI, IEPA, CCA, and CEPA5.3. 2 The Resonance Energy of Polyenes in Hückel Theory Notes, Further Reading6. Many-Body Perturbation Theory 6.1 Rayleigh-Schrödinger (RS) Perturbation Theory 6.2 Diagrammatic Representation of RS Perturbation Theory6.2. 1 Diagrammatic Perturbation Theory for 2 States6.2. 2 Diagrammatic Perturbation Theory for N States6.2. 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 Energy6.7. 1 Hugenholtz Diagrams6.7. 2 Goldstone Diagrams6.7. 3 Summation of Diagrams6.7. 4 What Is the Linked Cluster Theorem? 6.8 Some Illustrative Calculations Notes, Further Reading7. 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 Function7.2. 1 The Self-Energy7.2. 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 ReadingAppendix A. Integral Evaluation with 1s Primitive GaussiansAppendix B. Two-Electron Self-Consistent-Field ProgramAppendix C. Analytic Derivative methods and Geometry OptimizationAppendix D. Molecular Integrals for H subscript 2 as a Function of Bond Length Index