Molecular Symmetry and Spectroscopy

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.

Contenu

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