Methane Symmetry Operations

Version History

J. T. Hougen
Optical Technology Division

Gloria Wiersma, HTML conversion
Office of Electronic Commerce in Scientific and Engineering Data
National Institute of Standards and Technology
Gaithersburg, Maryland USA 20899

Abstract
Catalog of explicit symmetry operations for the vibrational, rotational and nuclear spin functions of methane, and the use of these operations in determining symmetry labels and selection rules.

Table of Contents

1. Introduction
2. Familiar Point Group Ideas
   2.1 Geometric operations
   2.2 Algebraic operations
   2.3 Representation matrices
3. Permutation-Inversion Group Ideas
4. Relation of Point-Group Operations to Permutation-Inversion Group Operations
   4.1 Algebraic equation relating laboratory-fixed Cartesian coordinates to rotational angles and vibrational displacement vectors
   4.2 Proper rotations
   4.3 Improper rotations
   4.4 Explicit Eulerian-Angle transformations
5. The Subgroup D_2dvs. the Full Group T_d
6. Symmetry Properties of Vibrational Functions
7. Symmetry Properties of Rotational Wave Functions and Direction Cosines
   7.1 Symmetric-top rotational basis Functions
   7.2 Direction cosines
8. Application of the Continuous Three-Dimensional Rotation-Reflection Group to Symmetric Top Rotational Functions
9. Symmetry Properties of Laboratory-Fixed Nuclear Spin Functions, Nuclear Spin Statistics, and Parities
   9.1 Symmetry properties of the nuclear spin functions
   9.2 Nuclear spin statistics and overall parities
10. Symmetry Properties of Selected Quantum Mechanical Operators
    10.1 The Hamiltonian
    10.2 The electric dipole-moment operator
    10.3 The total angular momentum operator
    10.4 The proton spin operators
    10.5 The vibrational angular momentum operator
11. Applications of the T_d Symmetry Species
    11.1 Dipole transition selection rules
    11.2 Perturbations and perturbation operators
12. Comparison of the Present Treatment with Those of Jahn and Moret-Bailly
    12.1 Jahn's treatment
    12.2 Moret-Bailly's treatment
13. Application of the Continuous Three-Dimensional Rotation-Reflection Group to Triply Degenerate Vibrational Wave Functions
    13.1 Spherical polar vibrational coordinates and angular momentum operators
    13.2 Comparison of vibrational and rotational angular momentum eigenfunctions
    13.3 Transformation properties of the vibrational angular momentum eigenfunctions
    13.4 Van Vleck's reversed angular momenta
    13.5 The molecule-fixed inversion operation i
14. Coupling of J and L ~_s in υ_s = 1 States
15. Laboratory-Fixed Rotations of CH₄ Considered as an Arbitrary Three-Dimensional Body
16. Irreducible Tensorial Sets
    16.1 Vector coupling coefficients
    16.2 Coupling of vibrational and rotational basis functions
    16.3 Coupling of nuclear spin basis functions
    16.4 Vibration-rotation operators in the transformed Hamiltonian
    16.5 Operators involving the nuclear spin angular momentum
    16.6 Reduced matrix elements: the Wigner-Eckart theorem
17. The Compound Operation Time-Reversal x Hermitian-Conjugation
18. References