# Elementary Molecular Quantum Mechanics

## 2nd Edition

### Mathematical Methods and Applications

**Authors:**Valerio Magnasco

**eBook ISBN:**9780444626660

**Hardcover ISBN:**9780444626479

**Imprint:**Elsevier Science

**Published Date:**1st August 2013

**Page Count:**1012

## Description

The second edition of * Elementary Molecular Quantum Mechanics* shows the methods of molecular quantum mechanics for graduate University students of Chemistry and Physics. This readable book teaches in detail the mathematical methods needed to do working applications in molecular quantum mechanics, as a preliminary step before using commercial programmes doing quantum chemistry calculations.

This book aims to bridge the gap between the classic Coulson’s Valence, where application of wave mechanical principles to valence theory is presented in a fully non-mathematical way, and McWeeny’s Methods of Molecular Quantum Mechanics, where recent advances in the application of quantum mechanical methods to molecular problems are presented at a research level in a full mathematical way. Many examples and mathematical points are given as problems at the end of each chapter, with a hint for their solution. Solutions are then worked out in detail in the last section of each Chapter.

## Key Features

- Uses clear and simplified examples to demonstrate the methods of molecular quantum mechanics
- Simplifies all mathematical formulae for the reader
- Provides educational training in basic methodology

## Readership

Students in chemical physics, theoretical and quantum chemistry also graduate students of chemistry/physics and undergraduate students of physical sciences

## Table of Contents

Dedication

Preface

Part 1: Mathematical Methods

Chapter 1. Mathematical foundations and approximation methods

1.1 Mathematical Foundations

1.2 The Variational Method

1.3 Perturbative Methods for Stationary States

1.4 The Wentzel–Kramers–Brillouin Method

1.5 Problems 1

1.6 Solved Problems

Chapter 2. Coordinate systems

2.1 Introduction

2.2 Systems of Orthogonal Coordinates

2.3 Generalized Coordinates

2.4 Cartesian Coordinates (*x*,*y*,*z*)

2.5 Spherical Coordinates (*r*,*θ*,*φ*)

2.6 Spheroidal Coordinates (*μ*,*ν*,*φ*)

2.7 Parabolic Coordinates (*ξ*,*η*,*φ*)

2.8 Problems 2

2.9 Solved Problems

Chapter 3. Differential equations in quantum mechanics

3.1 Introduction

3.2 Partial Differential Equations

3.3 Separation of Variables

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

3.8 The Hydrogen Atom in an Electric Field

3.9 The Hydrogen Molecular Ion H2+

3.10 The Stark Effect in Atomic Hydrogen

3.11 Appendix: Checking the Solutions

3.12 Problems 3

3.13 Solved Problems

Chapter 4. Special functions

4.1 Introduction

4.2 Legendre Functions

4.3 Laguerre Functions

4.4 Hermite Functions

4.5 Hypergeometric Functions

4.6 Bessel Functions

4.7 Functions Defined by Integrals

4.8 The Dirac *δ*-Function

4.9 The Fourier Transform

4.10 The Laplace Transform

4.11 Spherical Tensors

4.12 Orthogonal Polynomials

4.13 Padé Approximants

4.14 Green’s Functions

4.15 Problems 4

4.16 Solved Problems

Chapter 5. Functions of a complex variable

5.1 Functions of a Complex Variable

5.2 Complex Integral Calculus

5.3 Calculus of Residues

5.4 Problems 5

5.5 Solved Problems

Chapter 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

6.7 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

6.11 Problems 6

6.12 Solved Problems

Chapter 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

7.5 Applications

7.6 Problems 7

7.7 Solved Problems

Chapter 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

8.16 Problems 8

8.17 Solved Problems

Chapter 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

Chapter 10. Angular momentum methods for atoms

10.1 Introduction

10.2 The Vector Model

10.3 Construction of States of Definite Angular Momentum

10.4 An Outline of Advanced Methods for Coupling Angular Momenta

10.5 Problems 10

10.6 Solved Problems

Part 2: Applications

Chapter 11. The physical principles of quantum mechanics

11.1 The Orbital Model

11.2 The Fundamental Postulates of Quantum Mechanics

11.3 The Physical Principles of Quantum Mechanics

11.4 Problems 11

11.5 Solved Problems

Chapter 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

Chapter 13. Variational calculations

13.1 Introduction

13.2 The Variational Method

13.3 Non-linear Parameters

13.4 linear Parameters and the Ritz Method

13.5 Atomic Applications of the Ritz Method

13.6 Molecular Applications of the Ritz Method

13.7 Variational Principles in Second Order

13.8 Problems 13

13.9 Solved Problems

Chapter 14. Many-electron wavefunctions and model Hamiltonians

14.1 Introduction

14.2 Antisymmetry of the Electronic Wavefunction and the Pauli’s Principle

14.3 Electron Distribution Functions

14.4 Average Values of One- and Two-Electron Operators

14.5 The Slater’s Rules

14.6 Pople’s Two-Dimensional Chart of Quantum Chemistry

14.7 Hartree–Fock Theory for Closed Shells

14.8 Hückel’s Theory

14.9 Semiempirical MO Methods

14.10 Problems 14

14.11 Solved Problems

Chapter 15. Valence bond theory and the chemical bond

15.1 Introduction

15.2 The Chemical Bond in H2

15.3 Elementary VB Methods

15.4 Pauling’s VB Theory for Conjugated and Aromatic Hydrocarbons

15.5 Hybridization and Directed Valency in Polyatomic Molecules

15.6 Problems 15

15.7 Solved Problems

Chapter 16. Post-Hartree–Fock methods

16.1 Introduction

16.2 Matrix Elements between Slater Determinants

16.3 Spinless Pair Functions and the Correlation Problem

16.4 Configurational Interaction Methods

16.5 Multiconfigurational-SCF Method

16.6 Møller-Plesset Perturbation Theory

16.7 Second Quantization

16.8 Diagrammatic Theory

16.9 The Density Functional Theory

16.10 Problems 16

16.11 Solved Problems

Chapter 17. Atomic and molecular interactions

17.1 Introduction

17.2 Electric Properties of Molecules

17.3 Interatomic Potentials

17.4 Molecular Interactions

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

17.9 Problems 17

17.10 Solved problems

Chapter 18. Evaluation of molecular integrals

18.1 Introduction

18.2 The Basic Integrals

18.3 One-centre Integrals

18.4 Evaluation of the Electrostatic Potential *J*1*S*

18.5 The (1*S*2|1*S*2) Electron Repulsion Integral

18.6 General Formula for One-centre Two-electron Integrals

18.7 Two-centre Integrals Over 1*S* STOS

18.8 On the General Formulae for Two-centre Integrals

18.9 A Short Note on Multicentre Integrals

18.10 Molecular Integrals Over GTOS

18.11 Problems 18

18.12 Solved Problems

Chapter 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

19.7 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

Chapter 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

References

Author Index

Subject Index

## Details

- No. of pages:
- 1012

- Language:
- English

- Copyright:
- © Elsevier Science 2013

- Published:
- 1st August 2013

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780444626660

- Hardcover ISBN:
- 9780444626479

## About the Author

### Valerio Magnasco

Professor of Theoretical Chemistry at the Department of Chemistry and Industrial Chemistry, (DCCI) University of Genoa, Italy.

### Affiliations and Expertise

Professor of Theoretical Chemistry at the Department of Chemistry and Industrial Chemistry, (DCCI) University of Genoa, Italy.

## Reviews

"This interesting book is devoted to both the mathematical methods and applications of Quantum Mechanics in Chemistry...very useful not only for students but for scientists and researchers." **--Zentralblatt MATH**

"Magnasco presents this detailed quantum chemistry text, building up from basic mathematical foundations to multi-atom systems and molecular vibrations. The first several chapters cover linear algebra, approximation methods, coordinate systems, differential equations, special functions, functions of a complex variable, and matrix mechanics." **--ProtoView.com, April 2014**