Elementary Molecular Quantum Mechanics: Mathematical Methods and Applications

Elementary Molecular Quantum Mechanics: Mathematical Methods and Applications
Copyright
Dedication
Preface
1. Mathematical foundations and approximation methods
1.1 Mathematical Foundations
1.1.1 Regular functions
1.1.2 Schmidt orthogonalization
1.1.3 Löwdin orthogonalization
1.1.4 Set of orthonormal functions and basis set
1.1.5 Linear operators
1.1.6 Hermitian operators
1.1.7 Expansion theorem
1.1.8 Basic principles of quantum mechanics
1.2 The Variational Method
1.2.1 Non-linear parameters
1.2.2 Linear parameters: the Ritz method
1.3 Perturbative Methods for Stationary States
1.3.1 RS perturbation theory
1.3.2 Second-order approximation methods in RS perturbation theory
1.3.2.1 The Unsöld approximation
1.3.2.2 The Hylleraas method
1.3.2.3 The Kirkwood method
1.3.2.4 The Ritz method for Etilde2: linear pseudostates
1.3.3 BW perturbation theory
1.3.4 Perturbation methods without partitioning of the Hamiltonian
1.3.4.1 LS perturbation theory
1.3.4.2 EN perturbation theory
1.3.5 Perturbation theories including exchange
1.3.6 The moment method
1.4 The Wentzel–Kramers–Brillouin Method
1.5 Problems 1
1.6 Solved Problems
2. Coordinate systems
2.1 Introduction
2.2 Systems of Orthogonal Coordinates
2.3 Generalized Coordinates
2.4 Cartesian Coordinates
2.5 Spherical Coordinates
2.6 Spheroidal Coordinates 밀Ⰳ봀Ⰳ픀
2.7 Parabolic Coordinates 븀Ⰳ뜀Ⰳ픀
2.8 Problems 2
2.9 Solved Problems
3. Differential equations in quantum mechanics
3.1 Introduction
3.2 Partial Differential Equations
3.3 Separation of Variables
3.3.1 The particle in a three-dimensional box
3.3.2 The three-dimensional harmonic oscillator
3.3.3 The atomic one-electron system
3.3.4 The molecular one-electron system
3.3.5 The hydrogen atom in a uniform electric field
3.4 Solution by Series Expansion
3.5 Solution Near Singular Points
3.6 The One-dimensional Harmonic Oscillator
3.7 The Atomic One-electron System
3.7.1 Solution of the radial equation
3.7.2 Solution of the Φ-equation
3.7.3 Solution of the ̿-equation
3.7.4 The hydrogen-like atomic orbitals
3.8 The Hydrogen Atom in an Electric Field
3.9 The Hydrogen Molecular Ion H2+
3.10 The Stark Effect in Atomic Hydrogen
3.10.1 Solution of the ξ-equation in the zero-field case
3.10.2 The first-order Stark effect
3.11 Appendix: Checking the Solutions
3.11.1 The radial equation of the H-atom in spherical coordinates
3.11.2 The ̿-equation of the H-atom in spherical coordinates
3.11.3 The ξ-equation of the H-atom in parabolic coordinates
3.12 Problems 3
3.13 Solved Problems
4. Special functions
4.1 Introduction
4.2 Legendre Functions
4.2.1 Legendre polynomials and associated Legendre polynomials
4.2.2 Recurrence relations for Legendre polynomials
4.2.3 Series of Legendre polynomials
4.2.4 Legendre functions of first and second kind
4.2.5 Neumann’s formula for the Legendre functions
4.3 Laguerre Functions
4.3.1 Laguerre polynomials and Laguerre functions
4.3.2 Associated Laguerre polynomials
4.3.3 Basic integrals over associated Laguerre functions
4.4 Hermite Functions
4.4.1 Hermite polynomials
4.4.2 Hermite functions
4.4.3 Integrals over Hermite functions
4.5 Hypergeometric Functions
4.5.1 Hypergeometric series and differential equation
4.5.2 Confluent hypergeometric functions
4.6 Bessel Functions
4.6.1 Bessel functions of integral order
4.6.2 Bessel functions of half-integral order
4.6.3 Spherical Bessel functions
4.6.4 Modified Bessel functions
4.7 Functions Defined by Integrals
4.7.1 The gamma function
4.7.2 The incomplete gamma function
4.7.3 From the gamma function to the exponential integral function
4.7.4 The exponential integral function
4.7.5 The generalized exponential integral function
4.7.6 Further functions
4.8 The Dirac δ-Function
4.9 The Fourier Transform
4.10 The Laplace Transform
4.11 Spherical Tensors
4.11.1 Spherical tensors in complex form
4.11.2 Spherical tensors in real form
4.11.3 Generalized spherical tensors
4.12 Orthogonal Polynomials
4.13 Padé Approximants
4.14 Green’s Functions
4.15 Problems 4
4.16 Solved Problems
5. Functions of a complex variable
5.1 Functions of a Complex Variable
5.1.1 Complex numbers
5.1.2 Functions of a complex variable
5.1.3 Regular functions
5.1.4 Elementary operations
5.1.5 Power series of elementary functions
5.1.6 Many-valued functions
5.2 Complex Integral Calculus
5.2.1 Line integrals
5.2.2 Integrals in the complex plane and the Cauchy theorem
5.2.3 Integration over a not simply connected domain
5.2.4 Cauchy’s integral representation
5.2.5 Taylor’s expansion around a singularity
5.2.6 Laurent’s expansion
5.2.7 Zeros of a regular function
5.2.8 Analytic continuation
5.3 Calculus of Residues
5.3.1 The residue theorem
5.3.2 The Jordan lemma
5.3.3 Sum of non-convergent series
5.3.4 Evaluation of integrals of functions of real variable
5.4 Problems 5
5.5 Solved Problems
6. Matrices
6.1 Definitions and Elementary Properties
6.2 The Partitioning of Matrices
6.3 Properties of Determinants
6.4 Special Matrices
6.5 The Matrix Eigenvalue Problem
6.6 Functions of Hermitian Matrices
6.6.1 Analytic functions
6.6.2 Projectors and canonical form
6.6.3 Examples
6.7 The Matrix Pseudoeigenvalue Problem
6.8 The Lagrange Interpolation Formula
6.9 The Cayley–Hamilton Theorem
6.10 The Eigenvalue Problem in Hückel’s Theory of the π Electrons of Benzene
6.10.1 General considerations
6.10.2 Unitary transformation diagonalizing the Hückel’s matrix
6.11 Problems 6
6.12 Solved Problems
7. Molecular symmetry
7.1 Introduction
7.2 Symmetry and Quantum Mechanics
7.3 Molecular Symmetry
7.4 Symmetry Operations as Transformation of the Coordinate Axes
7.4.1 Passive and active representations of symmetry operations
7.4.2 Symmetry transformations in coordinate space
7.4.3 Symmetry operators and transformations in function space
7.4.3.1 Rotation of a function by Cα+
7.4.3.2 Reflection of a function across the plane σα
7.4.4 Matrix representatives of symmetry operators
7.4.5 Similarity transformations
7.5 Applications
7.5.1 The fundamental theorem of symmetry
7.5.2 Selection rules
7.5.3 Ground state electron configuration of polyatomic molecules
7.6 Problems 7
7.7 Solved Problems
8. Abstract group theory
8.1 Introduction
8.2 Axioms of Group Theory
8.3 Examples of Groups
8.4 Multiplication Table
8.5 Subgroups
8.6 Isomorphism
8.7 Conjugation and Classes
8.8 Direct-Product Groups
8.9 Representations and Characters
8.10 Irreducible Representations
8.11 Projectors and Symmetry-Adapted Functions
8.12 The Symmetric Group
8.13 Molecular Point Groups
8.14 Continuous Groups
8.15 Rotation Groups
8.15.1 Axial groups
8.15.2 The spherical group
8.15.3 Transformation properties of spherical harmonics
8.16 Problems 8
8.17 Solved Problems
9. The electron spin
9.1 Introduction
9.2 Electron Spin according to Pauli and the Zeeman Effect
9.3 Theory of One-Electron Spin
9.4 Matrix Representation of Spin Operators
9.5 Theory of Two-Electron Spin
9.6 Theory of Many-Electron Spin
9.7 The Kotani’ Synthetic Method
9.8 Löwdin’ Spin Projection Operators
9.9 Problems 9
9.10 Solved Problems
10. Angular momentum methods for atoms
10.1 Introduction
10.2 The Vector Model
10.2.1 Coupling of angular momenta
10.2.2 LS coupling and multiplet structure
10.3 Construction of States of Definite Angular Momentum
10.3.1 The matrix method
10.3.2 The projection operator method
10.4 An Outline of Advanced Methods for Coupling Angular Momenta
10.4.1 Clebsch–Gordan coefficients and Wigner 3-j and 9-j symbols
10.4.2 Gaunt coefficients and coupling rules
10.5 Problems 10
10.6 Solved Problems
11. The physical principles of quantum mechanics
11.1 The Orbital Model
11.2 The Fundamental Postulates of Quantum Mechanics
11.2.1 Correspondence between observables and operators
11.2.2 State function and average values of observables
11.2.3 Time evolution of state function
11.3 The Physical Principles of Quantum Mechanics
11.3.1 Wave-particle dualism
11.3.2 Atomicity of matter
11.3.3 Schroedinger’s wave equation
11.3.4 Born interpretation
11.3.5 Measure of observables
11.4 Problems 11
11.5 Solved Problems
12. Atomic orbitals
12.1 Introduction
12.2 Hydrogen-like Atomic Orbitals
12.3 Slater-type Orbitals
12.4 Gaussian-type Orbitals
12.5 Problems 12
12.6 Solved Problems
13. Variational calculations
13.1 Introduction
13.2 The Variational Method
13.2.1 Variational principles in first order
13.2.2 Variational approximations
13.2.3 Basis functions and variational parameters
13.3 Non-linear Parameters
13.3.1 The 1s ground state of the atomic one-electron system
13.3.2 The first 2s, 2p excited states of the atomic one-electron system
13.3.3 The 1s2 ground state of the atomic two-electron system
13.4 linear Parameters and the Ritz Method
13.5 Atomic Applications of the Ritz Method
13.5.1 The first 1s2s excited state of the atomic two-electron system
13.5.2 The first 1s2p excited state of the atomic two-electron system
13.5.3 Results for hydrogen-like AOs
13.6 Molecular Applications of the Ritz Method
13.6.1 The ground and first excited state of the H2+ molecular ion
13.6.2 The interaction energy and its components
13.7 Variational Principles in Second Order
13.7.1 The dipole polarizability of the H atom
13.7.2 The London attraction between two ground-state H atoms
13.8 Problems 13
13.9 Solved Problems
14. Many-electron wavefunctions and model Hamiltonians
14.1 Introduction
14.2 Antisymmetry of the Electronic Wavefunction and the Pauli’s Principle
14.2.1 Two-electron wavefunctions
14.2.2 Many-electron wavefunctions and the Slater method
14.3 Electron Distribution Functions
14.3.1 One-electron distribution functions: general definitions
14.3.2 Electron density and spin density
14.3.3 Two-electron distribution functions: general definitions
14.4 Average Values of One- and Two-Electron Operators
14.4.1 Symmetrical sums of one-electron operators
14.4.2 Symmetrical sums of two-electron operators
14.4.3 Average value of the electronic energy
14.5 The Slater’s Rules
14.6 Pople’s Two-Dimensional Chart of Quantum Chemistry
14.7 Hartree–Fock Theory for Closed Shells
14.7.1 Basic theory and properties of the fundamental invariant ρ
14.7.2 Electronic energy for the HF wavefunction
14.7.3 Roothaan’s variational derivation of the HF equations
14.7.4 Hall–Roothaan’s formulation of the LCAO-MO-SCF equations
14.7.5 Mulliken population analysis
14.7.6 Atomic bases in quantum chemical calculations
14.7.7 Localization of molecular orbitals
14.8 Hückel’s Theory
14.8.1 Recurrence relation for the linear chain
14.8.2 General solution for the linear chain
14.8.3 General solution for the closed chain
14.8.4 Alternant hydrocarbons
14.8.5 An introduction to band theory of solids
14.9 Semiempirical MO Methods
14.9.1 Extended Hückel’s theory
14.9.2 The CNDO method
14.9.3 The INDO method
14.9.4 The ZINDO method
14.10 Problems 14
14.11 Solved Problems
15. Valence bond theory and the chemical bond
15.1 Introduction
15.2 The Chemical Bond in H2
15.2.1 Failure of the MO theory for ground-state H2
15.2.2 The Heitler–London theory for H2
15.2.3 Equivalence between MO-CI and full VB for ground-state H2 and improvements in the wavefunction
15.2.4 The orthogonality catastrophe in the covalent VB theory for ground-state H2
15.3 Elementary VB Methods
15.3.1 General formulation of VB theory
15.3.2 Construction of VB structures for multiple bonds
15.3.3 The allyl radical
15.3.4 Cyclobutadiene
15.3.5 VB description of simple molecules
15.4 Pauling’s VB Theory for Conjugated and Aromatic Hydrocarbons
15.4.1 Pauling’s formula for the matrix elements of singlet covalent VB structures
15.4.2 Cyclobutadiene
15.4.3 Butadiene
15.4.4 Allyl radical
15.4.5 Benzene
15.4.6 Naphthalene
15.4.7 Derivation of The Pauling’s formula for H2 and cyclobutadiene
15.5 Hybridization and Directed Valency in Polyatomic Molecules
15.5.1 sp2 Hybridization in H2O
15.5.2 VB description of H2O
15.5.3 Properties of hybridization
15.5.4 The principle of maximum overlap in VB theory
15.6 Problems 15
15.7 Solved Problems
16. Post-Hartree–Fock methods
16.1 Introduction
16.2 Matrix Elements between Slater Determinants
16.2.1 Slater’s rules for orthonormal determinants
16.2.2 Löwdin’s density matrices for non-orthogonal determinants
16.3 Spinless Pair Functions and the Correlation Problem
16.4 Configurational Interaction Methods
16.4.1 Configuration interaction
16.4.2 Large-scale CI methods
16.4.3 Generalized valence bond methods
16.4.4 Cusp-corrected configurational interaction
16.4.5 Kołos–Wolniewicz wavefunctions
16.5 Multiconfigurational-SCF Method
16.6 Møller-Plesset Perturbation Theory
16.7 Second Quantization
16.7.1 Creation and annihilation operators
16.7.2 One-electron operators
16.7.3 Two-electron operators
16.7.4 Energy expressions
16.7.5 The Fock space
16.8 Diagrammatic Theory
16.8.1 Second- and third-order diagrammatic theory
16.8.2 Fourth-order diagrammatic theory
16.8.3 Padé approximants and perturbation expansions
16.8.4 Coupled-cluster many-body perturbation theory
16.8.5 CC-R12-MBPT
16.9 The Density Functional Theory
16.10 Problems 16
16.11 Solved Problems
17. Atomic and molecular interactions
17.1 Introduction
17.2 Electric Properties of Molecules
17.2.1 Molecular moments and polarizabilities
17.2.2 Molecular moments
17.2.3 Polarizabilities
17.3 Interatomic Potentials
17.3.1 The H–H+ non-expanded interaction up to second order
17.3.2 The H–H non-expanded interaction up to second order
17.3.3 The multipole analysis of the H–H non-expanded second-order induction energy
17.3.4 The multipole analysis of the H–H non-expanded second-order dispersion energy
17.3.5 The H–H expanded interaction up to second order
17.3.6 Higher-order terms in the H–H long-range dispersion interaction
17.3.7 The expanded dispersion interaction for many-electron atoms
17.4 Molecular Interactions
17.4.1 Non-expanded molecular energy corrections up to second order
17.4.2 Expanded molecular energy corrections up to second order
17.4.3 Multipole expansion of the first-order electrostatic energy in
17.5 The Pauli Repulsion Between Closed Shells
17.6 The Van der Waals Bond
17.7 Accurate Theoretical Results for Simple Diatomic Systems
17.8 A Generalized Multipole Expansion for Molecular Interactions
17.8.1 Generalized expansion of the intermolecular potential
17.8.2 Generalized molecular moments and polarizabilities
17.8.3 Molecular interaction energies
17.8.4 The damping of dispersion in the
17.9 Problems 17
17.10 Solved problems
18. Evaluation of molecular integrals
18.1 Introduction
18.2 The Basic Integrals
18.2.1 The indefinite integral
18.2.2 Definite integrals and auxiliary functions
18.3 One-centre Integrals
18.3.1 One-electron integrals
18.3.2 Two-electron integrals
18.4 Evaluation of the Electrostatic Potential J1s
18.4.1 Spherical coordinates
18.4.2 Spheroidal coordinates
18.5 The
18.5.1 Same orbital exponent
18.5.2 Different orbital exponents
18.6 General Formula for One-centre Two-electron Integrals
18.7 Two-centre Integrals Over 1s STOs
18.7.1 One-electron integrals
18.7.2 Two-electron integrals
18.7.3 Limiting values of two-centre integrals
18.8 On the General Formulae for Two-centre Integrals
18.8.1 Spheroidal coordinates
18.8.2 Spherical coordinates
18.9 A Short Note on Multicentre Integrals
18.9.1 Three-centre one-electron integral over 1s STOs
18.9.2 Four-centre two-electron integral over 1s STOs
18.10 Molecular Integrals Over GTOs
18.10.1 Some properties of Gaussian functions
18.10.2 Integrals of Gaussian functions
18.10.3 Integral transforms
18.10.4 Molecular integrals
18.11 Problems 18
18.12 Solved Problems
19. Relativistic molecular quantum mechanics
19.1 Introduction
19.2 The Schroedinger’s Relativistic Equation
19.3 The Klein–Gordon Relativistic Equation
19.4 Dirac’s Relativistic Equation for the Electron
19.5 Spinors: Small and Large Components
19.6 Dirac’s Equation for a Central Field
19.6.1 Separation of the radial equation
19.6.2 The hydrogen-like atom
19.7 One-Electron Molecular Systems: H2+ and HHe+2
19.8 Two-Electron Atomic System: The He Atom
19.9 Two-Electron Molecular Systems: H2 and HHe+
19.10 Many-Electron Atoms and Molecules
19.11 Problems 19
19.12 Solved Problems
20. Molecular vibrations
20.1 Introduction
20.2 Separation of Translational and Rotational Motions
20.3 Normal Coordinates in Classical and Quantum Mechanics
20.4 The Born–Oppenheimer Approximation
20.5 Electronically Degenerate States and the Renner's Effect in NH2
20.6 The Jahn–Teller Effect in CH4+
20.7 The Von Neumann–Wigner Non-crossing Rule in Diatomics
20.8 Conical Intersections in Polyatomic Molecules
20.9 Problems 20
20.10 Solved Problems
Author Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
R
S
T
U
V
W
X
Y
Z
Subject Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Y
Z