Methods of Molecular Quantum Mechanics An Introduction to Electronic Molecular Structure

by Magnasco, Valerio

• ISBN13: 9780470684412
• ISBN10: 0470684410
• Edition: 1st
• Format: Paperback
• Copyright: 2009-12-21
• Publisher: Wiley

Summary

This provides an advanced text introducing graduate students to the mathematical foundations of methods needed to do working applications in molecular quantum mechanics. It contains a consistent use of atomic units from the very beginning for simplifying mathematical formulae, and presents a unified presentation of basic elements of atomic and molecular interactions, with particular emphasis on practical use of second-order calculation techniques.

Author Biography

Dr Valerio Magnasco, MRSC, is full Professor of Theoretical Chemistry at the University of Genoa, Italy, presently at the Department of Chemistry and Industrial Chemistry (DCCI) of the Faculty of Mathematical, Physical and Natural Sciences of the University. He is Member of the Royal Society of Chemistry (UK, RSC), the American Institute of Physics (US, AIP), the Physical Chemistry Division of the Italian Chemical Society (Italy, SCI), the Class of Sciences of Accademia Ligure di Scienze e Lettere (Italy, Genova). He is supervising a research group working on the theoretical study of atomic and molecular interactions, and is author or co-author of over 170 scientific papers mostly published on international journals, and of one book on Molecular Quantum Mechanics.

Table of Contents

Preface	p. xiii
Principles	p. 1
The Orbital Model	p. 1
Mathematical Methods	p. 2
Dirac Notation	p. 2
Normalization	p. 2
Orthogonality	p. 3
Set of Orthonormal Functions	p. 3
Linear Independence	p. 3
Basis Set	p. 4
Linear Operators	p. 4
Sum and Product of Operators	p. 4
Eigenvalue Equation	p. 5
Hermitian Operators	p. 5
Anti-Hermitian Operators	p. 6
Expansion Theorem	p. 6
From Operators to Matrices	p. 6
Properties of the Operator ∇	p. 7
Transformations in Coordinate Space	p. 9
Basic Postulates	p. 12
Correspondence between Physical Observables and Hermitian Operators	p. 12
State Function and Average Value of Observables	p. 15
Time Evolution of the State function	p. 16
Physical Interpretation of the Basic Principles	p. 17
Matrices	p. 21
Definitions and Elementary Properties	p. 21
Properties of Determinants	p. 23
Special Matrices	p. 24
The Matrix Eigenvalue Problem	p. 25
Atomic Orbitals	p. 31
Atomic Orbitals as a Basis for Molecular Calculations	p. 31
Hydrogen-like Atomic Orbitals	p. 32
Choice of an Appropriate Coordinate System	p. 32
Solution of the Radial Equation	p. 33
Solution of the Angular Equation	p. 37
Some Properties of the Hydrogen-like Atomic Orbitals	p. 41
Real Form of the Atomic Orbitals	p. 43
Slater-type Orbitals	p. 46
Gaussian-type Orbitals	p. 49
Spherical Gaussians	p. 49
Cartesian Gaussians	p. 50
The Variation Method	p. 53
Variational Principles	p. 53
Nonlinear Parameters	p. 57
Ground State of the Hydrogenic System	p. 57
The First Excited State of Spherical Symmetry of the Hydrogenic System	p. 59
The First Excited 2p State of the Hydrogenic System	p. 61
The Ground State of the He-like System	p. 61
Linear Parameters and the Ritz Method	p. 64
Applications of the Ritz Method	p. 67
The First 1s2s Excited State of the He-like Atom	p. 67
The First 1s2p State of the He-like Atom	p. 69
Appendix: The Integrals J, K, &Jprime; and &Kprime;	p. 71
Spin	p. 75
The Zeeman Effect	p. 75
The Pauli Equations for One-electron Spin	p. 78
The Dirac Formula for N-electron Spin	p. 79
Antisymmetry of Many-electron Wavefunctions	p. 85
Antisymmetry Requirement and the Pauli Principle	p. 85
Slater Determinants	p. 87
Distribution Functions	p. 89
One- and Two-electron Distribution Functions	p. 89
Electron and Spin Densities	p. 91
Average Values of Operators	p. 95
Self-consistent-field Calculations and Model Hamiltonians	p. 99
Elements of Hartree-Fock Theory for Closed Shells	p. 100
The Fock-Dirac Density Matrix	p. 100
Electronic Energy Expression	p. 102
Roothaan Formulation of the LCAO-MO-SCF Equations	p. 104
Molecular Self-consistent-field Calculations	p. 108
Hückel Theory	p. 112
Ethylene (N = 2)	p. 114
The Allyl Radical (N = 3)	p. 115
Butadiene (N = 4)	p. 119
Cyclobutadiene (N = 4)	p. 120
Hexatriene (N = 6)	p. 124
Benzene (N = 6)	p. 126
A Model for the One-dimensional Crystal	p. 129
Post-Hartree-Fock Methods	p. 133
Configuration Interaction	p. 133
Multiconfiguration Self-consistent-field	p. 135
Møller-Plesset Theory	p. 135
The MP2-R12 Method	p. 136
The CC-R12 Method	p. 137
Density Functiona1 Theory	p. 138
Valence Bond Theory and the Chemical Bond	p. 141
The Born-Oppenheimer Approximation	p. 142
The Hydrogen: Molecule H2	p. 144
Molecular Orbital Theory	p. 145
Heitler-London Theory	p. 148
The Origin of the Chemical Bond	p. 150
Valence Bond Theory and the Chemical Bond	p. 153
Schematization of Valence Bond Theory	p. 153
Schematization of Molecular Orbital Theory	p. 154
Advantages of the Valence Bond Method	p. 154
Disadvantages of the Valence Bond Method	p. 154
Construction of Valence Bond Structures	p. 156
Hybridization and Molecular Structure	p. 162
The H2O Molecule	p. 162
Properties of Hybridization	p. 164
Pauling's Formula for Conjugated and Aromatic Hydrocarbons	p. 166
Ethylene (One ¿-Bond, n = 1)	p. 169
Cyclobutadiene (n = 2)	p. 169
Butadiene (Open Chain, n = 2)	p. 171
The Allyl Radical (N = 3)	p. 173
Benzene (n = 3)	p. 176
Elements of Rayleigh-Schroedinger Perturbation Theory	p. 183
Rayleigh-Schroedinger Perturbation Equations up to Third Order	p. 183
First-order Theory	p. 186
Second-order Theory	p. 187
Approximate E2 Calculations: The Hylleraas Functional	p. 190
Linear Pseudostates and Molecular Properties	p. 191
Single Pseudostate	p. 193
N-term Approximation	p. 195
Quantum Theory of Magnetic Susceptibilities	p. 196
Diamagnetic Susceptibilities	p. 199
Paramagnetic Susceptibilities	p. 203
Appendix: Evaluation of ¿ and ¿	p. 212
Atomic and Molecular Interactions	p. 215
The H-H Nonexpanded Interactions up to Second Order	p. 216
The H-H Expanded Interactions up to Second Order	p. 220
Molecular Interactions	p. 225
Nonexpanded Energy Corrections up to Second Order	p. 226
Expanded Energy Corrections up to Second Order	p. 227
Other Expanded Interactions	p. 235
Van der Waals and Hydrogen Bonds	p. 237
The Keesom Interaction	p. 239
Symmetry	p. 247
Molecular Symmetry	p. 247
Group Theoretical Methods	p. 252
Isomorphism	p. 254
Conjugation and Classes	p. 254
Representations and Characters	p. 255
Three Theorems on Irreducible Representations	p. 255
Number of Irreps in a Reducible Representation	p. 256
Construction of Symmetry-adapted Functions	p. 256
Illustrative Examples	p. 257
Use of Symmetry in Ground-state H2O (1A1)	p. 257
Use of Symmetry in Ground-state NH3 (1A1)	p. 260
References	p. 267
Author Index	p. 275
Subject Index	p. 279
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