NIST: Methane Symmetry Operations

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 CH₄ 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.