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[资源] Methods of Molecular Quantum Mechanics

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–Fock Theory for Closed Shells 100
7.1.1 The Fock–Dirac Density Matrix 100
7.1.2 Electronic Energy Expression 102
7.2 Roothaan Formulation of the LCAO–MO–SCF
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–Fock Methods 133
8.1 Configuration Interaction 133
8.2 Multiconfiguration Self-consistent-field 135
8.3 Møller–Plesset 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–Oppenheimer Approximation 142
9.2 The Hydrogen Molecule H2 144
9.2.1 Molecular Orbital Theory 145
9.2.2 Heitler–London 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–Schroedinger Perturbation Theory 183
10.1 Rayleigh–Schroedinger 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–H Nonexpanded Interactions
up to Second Order 216
11.2 The H–H 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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附件 1 : Methods_of_Molecular_Quantum_Mechanics._An_Introduction_to_Electronic_Molecular_Structure_-_Valerio_Magnasco_(Wiley,_2010)(298p)(Isbn0470684429).pdf

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