Methane Symmetry Operations - Application of the Continuous Three-Dimensional Rotation-Reflection Group to Symmetric Top Rotational Functions

8.   Application of the Continuous Three-Dimensional Rotation-Reflection Group to Symmetric Top Rotational Functions

Up to this point we have determined explicitly how to transform Eulerian angles and rotational basis functions only under the eight symmetry operations belonging to the D_2d group given in Table 5. We shall now determine transformations corresponding to the remaining 16 operations of the T_d group, and in fact shall determine transformations corresponding to all rotations of the continuous three-dimensional pure rotation group, and to all rotation-reflections of the (piecewise continuous) three-dimensional rotation-reflection group [22, 27].

Let us characterize a rotation in three dimensions by the angles αβγ, such that a particle originally at position x,y,z occupies a position x_new,y_new,z_new after subjecting it to the rotation R_αβγ, where the new and old positions are related by

The matrix S(αβγ) can be obtained from equation (eq. 10).

If R_αβγ belonged to the molecular point group, we would require (see Section 4.2) the direction cosine matrix to transform as follows

By extension, we thus require the direction cosine matrix to transform according to (eq. 29) for any three-dimensional pure rotation R_αβγ.

Wigner's matrix [22] D⁽¹⁾ can be obtained from the S matrix by carrying out the unitary transformation

where

Hence, (eq. 29) can be rewritten

with J = 1. However, since Wigner's D^(J) matrices are representations of the three-dimensional pure rotation group, (eq. 3.32) must hold for all J if it holds for J = 1.

We see from (eq. 26) that the rotation R_αβγ acting on |kJm⟩ gives

which is the desired transformation equation for the symmetric top rotational basis functions.

Values of αβγ corresponding to T_d point-group operations for CH₄ oriented as in Fig. 1 are given in Table 13. Rotations corresponding to the sense-reversing operations, as defined in (eq. 18), are also given. Table 13 is similar to Table I of [31], but not identical with it because of some differences in approach.

Note that the transformations (eq. 33) generated when the R_αβγ considered in this section act on a given |kJm⟩ function consist of linear combinations of functions with different k quantum numbers but the same m quantum number. This is just opposite to the situation to be encountered in Section 15, where rigid-body rotations of the methane molecule in free space are considered.

Even though we shall not give linear combinations of symmetric top basis functions corresponding to definite symmetry species in the point group T_d, it is relatively easy to give the number of such species occurring in the manifold of all (2J + 1) functions |kJm⟩ having different values of k but a fixed value of J and of m. Since, from Section 4.3, the effect of the molecule-fixed inversion operation i on the Eulerian angles is the same as i · i = E, we see from (eq. 33) that the functions |kJm⟩ with fixed J and m transform according to the representation D_g^(J) of the continuous three-dimensional rotation-reflection group. Reductions of these representations to irreducible representations of the point group T_d are given in many places [6]. Reductions of the even representations D_g^(J) are given herein Table 14. Reductions of the odd representations D_u^(J) can be obtained from Table  14 by exchanging all subscripts 1 and 2.

Figure 5 depicts the J = 7 rotational levels of the ground vibrational state of methane and the J = 6 rotational levels of the υ₃ = 1 vibrational state, and will be used to illustrate various points throughout this article. Preceding results permit us to verify the number and kind of symmetry species occurring in this diagram.

From Table 14 we obtain directly the symmetry species of the υ = 0, J = 7 levels. Because a given symmetry species may occur more than once in a given rotational manifold, we follow a convention introduced by Jahn [6] and number ground state levels of identical J value and symmetry species with right superscripts (1), (2), etc., beginning with the member of each set at lowest energy. From Table 8 we find that υ₃ is of species F₂, from Table 14 we obtain symmetry species for J = 6 rotational functions, and from Table 10 we obtain the direct products of these latter species with F₂, leading to the 2A₁+A₂+3E+5F₁+5F₂ rovibrational levels occurring in the υ₃ = 1, J = 6 manifold. The grouping and superscript numbering of the υ₃ = 1 rovibrational levels will be discussed in later sections.