# Molecular Symmetry and Spectroscopy

## 1st Edition

**Authors:**Philip Bunker

**eBook ISBN:**9780323150255

**Imprint:**Academic Press

**Published Date:**28th March 1979

**Page Count:**440

## Description

Molecular Symmetry and Spectroscopy deals with the use of group theory in quantum mechanics in relation to problems in molecular spectroscopy. It discusses the use of the molecular symmetry group, whose elements consist of permutations of identical nuclei with or without inversion. After reviewing the permutation groups, inversion operation, point groups, and representation of groups, the book describes the use of representations for labeling molecular energy. The text explains an approximate time independent Schrödinger equation for a molecule, as well as the effect of a nuclear permutation or the inversion of E* on such equation. The book also examines the expression for the complete molecular Hamiltonian and the several groups of operations commuting with the Hamiltonian. The energy levels of the Hamiltonian can then be symmetrically labeled by the investigator using the irreducible representations of these groups. The text explains the two techniques to change coordinates in a Schrödinger equation, namely, (1) by using a diatomic molecule in the rovibronic Schrödinger equation, and (2) by a rigid nonlinear polyatomic molecule. The book also explains that using true symmetry, basis symmetry, near symmetry, and near quantum numbers, the investigator can label molecular energy levels. The text can benefit students of molecular spectroscopy, academicians, and investigators of molecular chemistry or quantum mechanics.

## Table of Contents

Preface

Acknowledgments

Introduction

Bibliographical Notes

1. Permutations and Permutation Groups

Permutations

The Successive Application of Permutations

Permutation Groups

The Complete Nuclear Permutation Group of a Molecule

Bibliographical Notes

2. The Inversion Operation and Permutation Inversion Groups

The Inversion Operation and Parity

Combining Permutations with the Inversion

The Detailed Effects of P and P* Operations

Summary

3. Rotation Groups and Point Groups

Rotational Symmetry and the Rotation Group

Reflection Symmetry and the Point Group

The Point Group Symmetry of Molecules

The Rotation Group Symmetry of Molecules

Discussion

Bibliographical Notes

4. Representations of Groups

Matrices and Matrix Groups

Isomorphism and Faithful Representations

Homomorphism and Unfaithful Representations

Equivalent and Irreducible Representations

Reduction of a Representation

Conjugate Elements and Classes

Bibliographical Notes

5. The Use of Representations for Labeling Molecular Energy Levels

A Molecular Schrödinger Equation in (X, Y, Z) Coordinates

The Effects of Nuclear Permutations and the Inversion on the Schrödinger Equation

The Symmetry of a Product

The Use of Symmetry Labels and the Vanishing Integral Rule

Diagonalizing the Hamiltonian Matrix

Appendix 5-1: Proof That the Matrices D[R] Generated in Eq. (5-49) Are Representations

Appendix 5-2: Projection Operators

Appendix 5-3: Addendum to Problem 5-2

Bibliographical Notes

6. The Molecular Hamiltonian and its True Symmetry

The Molecular Hamiltonian

The Full Symmetry Group of the Molecular Hamiltonian

Basis Functions and Basis Function Symmetry

Discussion

Bibliographical Notes

7. The Coordinates in the Rovibronic Schrödinger Equation

The Rovibronic Schrödinger Equation

Two Methods for Changing Coordinates in a Schrödinger Equation

Introduction to the Molecule Fixed Axis System

The Diatomic Molecule

Rigid Nonlinear Polyatomic Molecules

Bibliographical Notes

8. The Rovibronic Wavefunctions

The Born-Oppenheimer Approximation

The Electronic Wavefunctions

The Rotation-Vibration Schrödinger Equation

The Rigid Rotor Schrödinger Equation

The Harmonic Oscillator Schrödinger Equation

Summary

Bibliographical Notes

9. The Definition of the Molecular Symmetry Group

The Complete Nuclear Permutation Inversion Group

The Molecular Symmetry Group

The Character Tables and Correlation Tables of MS Groups

The MS Group for Levels of More Than One Electronic State

Summary

Bibliographical Note

10. The Classification of Molecular Wavefunctions in the Molecular Symmetry Group

The Classification of the Complete Internal Wavefunction

The Classification of the Nuclear Spin Wavefunctions and the Determination of Nuclear Spin Statistical Weights

The Classification of the Rotational Wavefunctions

The Classification of the Vibrational Wavefunctions

The Classification of the Electronic Orbital Wavefunctions

The Classification of the Electron Spin Wavefunctions

The Classification of Rotational Wavefunctions Having Half-Integral J

Summary

Bibliographical Notes

11. Near Symmetry, Perturbations, and Optical Selections Rules

Near Symmetry

Near Quantum Numbers

Nonvanishing Perturbation Terms

Perturbations between States

Optical Selection Rules and Forbidden Transitions

Magnetic Dipole and Electric Quadrupole Transitions

Multiphoton Processes and the Raman Effect

The Zeeman Effect

The Stark Effect

Summary

Bibliographical Notes

12. Linear Molecules and Nonrigid Molecules

Linear Molecules

Nonrigid Molecules

Discussion and Summary

Appendix A. The Character Tables

Appendix B. The Correlation Tables

References

Index

## Details

- No. of pages:
- 440

- Language:
- English

- Copyright:
- © Academic Press 1979

- Published:
- 28th March 1979

- Imprint:
- Academic Press

- eBook ISBN:
- 9780323150255