## Methane Symmetry Operations

### 1. Introduction

The goal of this article, a more detailed title for which is "a catalogue
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," is to provide a consistent,
pedagogically convenient and explicit treatment of symmetry properties in the
CH_{4} molecule, including a unified discussion of the
permutation-inversion molecular symmetry group ideas of Longuet-Higgins
[1] and the more traditional crystallographic point
group symmetry ideas [2-4]. Consistency requires
careful attention to sign conventions. It is hoped that this article will
provide an acceptable set of ready-made conventions for workers who do not wish
to establish a sign convention system of their own.
At least three *T*_{d} symmetry classification systems are widely
used at present in the methane literature [5-13].
The three systems differ sufficiently from each other to lead to different
symmetry labels for the same state of methane and appropriately different
selection rules for the same electric dipole transitions. Even though all
systems appear to lead to the same physical results, the present author
believes that rather compelling arguments, as outlined in
Section 12, can be made favoring the system
treated in detail here.

Various controversies exist, even in the non-methane literature, over the
application of group theory to some of the more subtle molecular spectroscopic
problems. Many of these controversies could be quickly resolved if arguments
were transferred from the domain of *geometric intuition* (where
admittedly our final "understanding" often lies) to the domain of
more easily verified and agreed upon *algebraic proofs*. In the present
article many required algebraic proofs are briefly outlined.

It is common [6, 10,
14,15] to approach problems of symmetry in a molecule
belonging to the point group *T*_{d} by considering first symmetry
properties of the full continuous three-dimensional rotation-reflection group,
and then treating *T*_{d} as a finite subgroup of that larger
non-denumerably infinite group. This procedure is characterized by extreme
power and elegance. The present article takes an alternative approach
[16] and attempts initially to discuss the symmetry
properties associated with the spherical-top point group *T*_{d}
by drawing on the more familiar symmetry properties associated with the
*D*_{2d} symmetric-top subgroup of *T*_{d}.
It is hoped that this alternative approach may help make available to a greater
number of molecular spectroscopists than before an understanding of the
theoretical bases for symmetry arguments in the methane molecule.

The references cited in this article do not represent an exhaustive compilation
of the extensive methane literature. An attempt was made, however, to include
illustrative references from the various schools of thought on theoretical
matters, and from the different schools of experimental work on
vibration-rotation spectra, pure rotational spectra, laser double-resonance
spectra, and molecular-beam hyperfine spectra. Much of the theoretical material
presented here has been discussed previously by workers associated with the
various schools of theoretical thought, though changes in phase conventions,
notational differences and genuine disagreements over the mathematical
development and physical interpretation of the formalism make a detailed
comparison difficult.