Methods of Molecular Quantum Mechanics

Methods of Molecular Quantum Mechanics: An Introduction to Electronic Molecular Structure......Page 2
Contents......Page 10
Preface......Page 16
1.1 THE ORBITAL MODEL......Page 18
1.2.2 Normalization......Page 19
1.2.5 Linear Independence......Page 20
1.2.8 Sum and Product of Operators......Page 21
1.2.10 Hermitian Operators......Page 22
1.2.13 From Operators to Matrices......Page 23
1.2.14 Properties of the Operator......Page 24
1.2.15 Transformations in Coordinate Space......Page 26
1.3.1 Correspondence between Physical Obervables and Hermitian Operators......Page 29
1.3.2 State Function and Average Value of Observables......Page 32
1.3.3 Time Evolution of the State Function......Page 33
1.4 PHYSICAL INTERPRETATION OF THE BASIC PRINCIPLES......Page 34
2.1 DEFINITIONS AND ELEMENTARY PROPERTIES......Page 38
2.2 PROPERTIES OF DETERMINANTS......Page 40
2.3 SPECIAL MATRICES......Page 41
2.4 THE MATRIX EIGENVALUE PROBLEM......Page 42
3.1 ATOMICORBITALS ASABASISFORMOLECULAR CALCULATIONS......Page 48
3.2.1 Choice of an Appropriate Coordinate System......Page 49
3.2.2 Solution of the Radial Equation......Page 50
3.2.3 Solution of the Angular Equation......Page 54
3.2.4 Some Properties of the Hydrogen-like Atomic Orbitals......Page 58
3.2.5 Real Form of the Atomic Orbitals......Page 60
3.3 SLATER-TYPE ORBITALS......Page 63
3.4.1 Spherical Gaussians......Page 66
3.4.2 Cartesian Gaussians......Page 67
4.1 VARIATIONAL PRINCIPLES......Page 70
4.2.1 Ground State of the Hydrogenic System......Page 74
4.2.2 The First Excited State of Spherical Symmetry of the Hydrogenic System......Page 76
4.2.4 The Ground State of the He-like System......Page 78
4.3 LINEAR PARAMETERS AND THE RITZ METHOD......Page 81
4.4.1 The First 1s2s Excited State of the He-like Atom......Page 84
4.4.2 The First 1s2p State of the He-like Atom......Page 86
APPENDIX: THE INTEGRALS J, K, J\' AND K\'......Page 88
5.1 THE ZEEMAN EFFECT......Page 92
5.2 THE PAULI EQUATIONS FOR ONE-ELECTRON SPIN......Page 95
5.3 THE DIRAC FORMULA FOR N-ELECTRON SPIN......Page 96
6.1 ANTISYMMETRY REQUIREMENT AND THE PAULI PRINCIPLE......Page 102
6.2 SLATER DETERMINANTS......Page 104
6.3.1 One- and Two-electron Distribution Functions......Page 106
6.3.2 Electron and Spin Densities......Page 108
6.4 AVERAGE VALUES OF OPERATORS......Page 112
7 Self-consistent-field Calculations and Model Hamiltonians......Page 116
7.1.1 The Fock–Dirac Density Matrix......Page 117
7.1.2 Electronic Energy Expression......Page 119
7.2 ROOTHAAN FORMULATION OF THE LCAO– MO–SCF EQUATIONS......Page 121
7.3 MOLECULAR SELF-CONSISTENT-FIELD CALCULATIONS......Page 125
7.4 HÜCKEL THEORY......Page 129
7.4.1 Ethylene (N = 2)......Page 131
7.4.2 The Allyl Radical (N = 3)......Page 132
7.4.3 Butadiene (N = 4)......Page 136
7.4.4 Cyclobutadiene (N = 4)......Page 137
7.4.5 Hexatriene (N = 6)......Page 141
7.4.6 Benzene (N = 6)......Page 143
7.5 A MODEL FOR THE ONE-DIMENSIONAL CRYSTAL......Page 146
8.1 CONFIGURATION INTERACTION......Page 150
8.3 MØLLER–PLESSET THEORY......Page 152
8.4 THE MP2-R12 METHOD......Page 153
8.5 THE CC-R12 METHOD......Page 154
8.6 DENSITY FUNCTIONAL THEORY......Page 155
9 Valence Bond Theory and the Chemical Bond......Page 158
9.1 THE BORN–OPPENHEIMER APPROXIMATION......Page 159
9.2 THE HYDROGEN MOLECULE H2......Page 161
9.2.1 Molecular Orbital Theory......Page 162
9.2.2 Heitler–London Theory......Page 165
9.3 THE ORIGIN OF THE CHEMICAL BOND......Page 167
9.4.1 Schematization of Valence Bond Theory......Page 170
9.4.4 Disadvantages of the Valence Bond Method......Page 171
9.4.5 Construction of Valence Bond Structures......Page 173
9.5.1 The H2O Molecule......Page 179
9.5.2 Properties of Hybridization......Page 181
9.6 PAULING’S FORMULA FOR CONJUGATED AND AROMATIC HYDROCARBONS......Page 183
9.6.2 Cyclobutadiene (n = 2)......Page 186
9.6.3 Butadiene (Open Chain, n = 2)......Page 188
9.6.4 The Allyl Radical (N = 3)......Page 190
9.6.5 Benzene (n = 3)......Page 193
10.1 RAYLEIGH–SCHROEDINGER PERTURBATION EQUATIONS UP TO THIRD ORDER......Page 200
10.2 FIRST-ORDER THEORY......Page 203
10.3 SECOND-ORDER THEORY......Page 204
10.4 APPROXIMATE E2 CALCULATIONS: THE HYLLERAAS FUNCTIONAL......Page 207
10.5 LINEAR PSEUDOSTATES AND MOLECULAR PROPERTIES......Page 208
10.5.1 Single Pseudostate......Page 210
10.5.2 N-term Approximation......Page 212
10.6 QUANTUM THEORY OF MAGNETIC SUSCEPTIBILITIES......Page 213
10.6.1 Diamagnetic Susceptibilities......Page 216
10.6.2 Paramagnetic Susceptibilities......Page 220
APPENDIX: EVALUATION OF μ AND ε......Page 229
11 Atomic and Molecular Interactions......Page 232
11.1 THE H–H NONEXPANDED INTERACTIONS UP TO SECOND ORDER......Page 233
11.2 THE H–H EXPANDED INTERACTIONS UP TO SECOND ORDER......Page 237
11.3 MOLECULAR INTERACTIONS......Page 242
11.3.1 Nonexpanded Energy Corrections up to Second Order......Page 243
11.3.2 Expanded Energy Corrections up to Second Order......Page 244
11.3.3 Other Expanded Interactions......Page 252
11.4 VAN DER WAALS AND HYDROGEN BONDS......Page 254
11.5 THE KEESOM INTERACTION......Page 256
12.1 MOLECULAR SYMMETRY......Page 264
12.2 GROUP THEORETICAL METHODS......Page 269
12.2.2 Conjugation and Classes......Page 271
12.2.4 Three Theorems on Irreducible Representations......Page 272
12.2.6 Construction of Symmetry-adapted Functions......Page 273
12.3.1 Use of Symmetry in Ground-state H2O (1A1)......Page 274
12.3.2 Use of Symmetry in Ground-state NH3 (1A1)......Page 277
References......Page 284
Author Index......Page 292
Subject Index......Page 296