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Methane Symmetry Operations

J. T. Hougen
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 D2d vs. the Full Group Td

  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 Td 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 $\tilde{L}_s$ in italic vs = 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

      References


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