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Methods of Molecular Quantum Mechanics
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1 Principles 1 1.1 The Orbital Model 1 1.2 Mathematical Methods 2 1.2.1 Dirac Notation 2 1.2.2 Normalization 2 1.2.3 Orthogonality 3 1.2.4 Set of Orthonormal Functions 3 1.2.5 Linear Independence 3 1.2.6 Basis Set 4 1.2.7 Linear Operators 4 1.2.8 Sum and Product of Operators 4 1.2.9 Eigenvalue Equation 5 1.2.10 Hermitian Operators 5 1.2.11 Anti-Hermitian Operators 6 1.2.12 Expansion Theorem 6 1.2.13 From Operators to Matrices 6 1.2.14 Properties of the Operator r 7 1.2.15 Transformations in Coordinate Space 9 1.3 Basic Postulates 12 1.3.1 Correspondence between Physical Observables and Hermitian Operators 12 1.3.2 State Function and Average Value of Observables 15 1.3.3 Time Evolution of the State Function 16 1.4 Physical Interpretation of the Basic Principles 17 2 Matrices 21 2.1 Definitions and Elementary Properties 21 2.2 Properties of Determinants 23 2.3 Special Matrices 24 2.4 The Matrix Eigenvalue Problem 25 3 Atomic Orbitals 31 3.1 Atomic Orbitals as a Basis for Molecular Calculations 31 3.2 Hydrogen-like Atomic Orbitals 32 3.2.1 Choice of an Appropriate Coordinate System 32 3.2.2 Solution of the Radial Equation 33 3.2.3 Solution of the Angular Equation 37 3.2.4 Some Properties of the Hydrogen-like Atomic Orbitals 41 3.2.5 Real Form of the Atomic Orbitals 43 3.3 Slater-type Orbitals 46 3.4 Gaussian-type Orbitals 49 3.4.1 Spherical Gaussians 49 3.4.2 Cartesian Gaussians 50 4 The Variation Method 53 4.1 Variational Principles 53 4.2 Nonlinear Parameters 57 4.2.1 Ground State of the Hydrogenic System 57 4.2.2 The First Excited State of Spherical Symmetry of the Hydrogenic System 59 4.2.3 The First Excited 2p State of the Hydrogenic System 61 4.2.4 The Ground State of the He-like System 61 4.3 Linear Parameters and the Ritz Method 64 4.4 Applications of the Ritz Method 67 4.4.1 The First 1s2s Excited State of the He-like Atom 67 4.4.2 The First 1s2p State of the He-like Atom 69 Appendix: The Integrals J, K, J0 and K0 71 5 Spin 75 5.1 The Zeeman Effect 75 5.2 The Pauli Equations for One-electron Spin 78 5.3 The Dirac Formula for N-electron Spin 79 6 Antisymmetry of Many-electron Wavefunctions 85 6.1 Antisymmetry Requirement and the Pauli Principle 85 6.2 Slater Determinants 87 viii CONTENTS 6.3 Distribution Functions 89 6.3.1 One- and Two-electron Distribution Functions 89 6.3.2 Electron and Spin Densities 91 6.4 Average Values of Operators 95 7 Self-consistent-field Calculations and Model Hamiltonians 99 7.1 Elements of Hartree?CFock Theory for Closed Shells 100 7.1.1 The Fock?CDirac Density Matrix 100 7.1.2 Electronic Energy Expression 102 7.2 Roothaan Formulation of the LCAO?CMO?CSCF Equations 104 7.3 Molecular Self-consistent-field Calculations 108 7.4 H?uckel Theory 112 7.4.1 Ethylene (N¼2) 114 7.4.2 The Allyl Radical (N ¼ 3) 115 7.4.3 Butadiene (N ¼ 4) 119 7.4.4 Cyclobutadiene (N ¼ 4) 120 7.4.5 Hexatriene (N ¼ 6) 124 7.4.6 Benzene (N ¼ 6) 126 7.5 A Model for the One-dimensional Crystal 129 8 Post-Hartree?CFock Methods 133 8.1 Configuration Interaction 133 8.2 Multiconfiguration Self-consistent-field 135 8.3 Møller?CPlesset Theory 135 8.4 The MP2-R12 Method 136 8.5 The CC-R12 Method 137 8.6 Density Functional Theory 138 9 Valence Bond Theory and the Chemical Bond 141 9.1 The Born?COppenheimer Approximation 142 9.2 The Hydrogen Molecule H2 144 9.2.1 Molecular Orbital Theory 145 9.2.2 Heitler?CLondon Theory 148 9.3 The Origin of the Chemical Bond 150 9.4 Valence Bond Theory and the Chemical Bond 153 9.4.1 Schematization of Valence Bond Theory 153 9.4.2 Schematization of Molecular Orbital Theory 154 9.4.3 Advantages of the Valence Bond Method 154 9.4.4 Disadvantages of the Valence Bond Method 154 9.4.5 Construction of Valence Bond Structures 156 CONTENTS ix 9.5 Hybridization and Molecular Structure 162 9.5.1 The H2O Molecule 162 9.5.2 Properties of Hybridization 164 9.6 Pauling??s Formula for Conjugated and Aromatic Hydrocarbons 166 9.6.1 Ethylene (One p-Bond, n ¼ 1) 169 9.6.2 Cyclobutadiene (n ¼ 2) 169 9.6.3 Butadiene (Open Chain, n ¼ 2) 171 9.6.4 The Allyl Radical (N ¼ 3) 173 9.6.5 Benzene (n ¼ 3) 176 10 Elements of Rayleigh?CSchroedinger Perturbation Theory 183 10.1 Rayleigh?CSchroedinger Perturbation Equations up to Third Order 183 10.2 First-order Theory 186 10.3 Second-order Theory 187 10.4 Approximate E2 Calculations: The Hylleraas Functional 190 10.5 Linear Pseudostates and Molecular Properties 191 10.5.1 Single Pseudostate 193 10.5.2 N-term Approximation 195 10.6 Quantum Theory of Magnetic Susceptibilities 196 10.6.1 Diamagnetic Susceptibilities 199 10.6.2 Paramagnetic Susceptibilities 203 Appendix: Evaluation of m and « 212 11 Atomic and Molecular Interactions 215 11.1 The H?CH Nonexpanded Interactions up to Second Order 216 11.2 The H?CH Expanded Interactions up to Second Order 220 11.3 Molecular Interactions 225 11.3.1 Nonexpanded Energy Corrections up to Second Order 226 11.3.2 Expanded Energy Corrections up to Second Order 227 11.3.3 Other Expanded Interactions 235 11.4 Van der Waals and Hydrogen Bonds 237 11.5 The Keesom Interaction 239 12 Symmetry 247 12.1 Molecular Symmetry 247 x CONTENTS 12.2 Group Theoretical Methods 252 12.2.1 Isomorphism 254 12.2.2 Conjugation and Classes 254 12.2.3 Representations and Characters 255 12.2.4 Three Theorems on Irreducible Representations 255 12.2.5 Number of Irreps in a Reducible Representation 256 12.2.6 Construction of Symmetry-adapted Functions 256 12.3 Illustrative Examples 257 12.3.1 Use of Symmetry in Ground-state H2O (1A1) 257 12.3.2 Use of Symmetry in Ground-state NH3 (1A1) 260 References 267 Author Index 275 Subject Index 279 |
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