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

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### 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 D2d group given in Table 5. We shall now determine transformations corresponding to the remaining 16 operations of the Td 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 xnew,ynew,znew after subjecting it to the rotation Rαβγ, where the new and old positions are related by

$$\left[\begin{array}{c} x_{\rm new} \\ y_{\rm new} \\ z_{\rm new} \end{array}\right] = [ S(\alpha\beta\gamma) ] \left[\begin{array}{c} x\\ y\\ z \end{array}\right] ~ .$$

(eq. 28)

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

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

$$S(\chi_{\rm new},\theta_{\rm new},\phi_{\rm new}) = S(\alpha\beta\gamma) \cdot S(\chi\theta\phi) ~ .$$

(eq. 29)

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

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

$${\cal D}^{(1)}(\{\chi\theta\phi\}) = U\cdot S(\chi\theta\phi) \cdot U^{-1} ~ ,$$

(eq. 30)

where

$$U = 2^{-1/2} ~ \left[ \begin{array}{rrr} 1 & +{\rm i} & 0 \\ 0 & 0 & \sqrt{2} \\ -1 & +{\rm i} & 0 \end{array} \right] ~ .$$

(eq. 31)

Hence, (eq. 29) can be rewritten

$${\cal D}^{(J)}(\{\chi_{\rm new},\theta_{\rm new},\phi_{\rm new}\}) = {\cal D}^{(J)}(\{\alpha\beta\gamma\}) \cdot {\cal D}^{(J)}(\{\chi\theta\phi\}) ~ ,$$

(eq. 32)

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 Rαβγ acting on |kJm⟩ gives

$$R_{\alpha\beta\gamma} \, |kJm\rangle = \sum_{k^\prime}~ {\cal D}^{(J)}_{kk^\prime} \, (\{\alpha\beta\gamma\}) \, |k^\prime Jm\rangle ~ ,$$

(eq. 33)

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

Values of αβγ corresponding to Td 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.

#### Table 13. Values of αβγ which make (eq. 29) coincide with (eq. 14) for proper rotations or (eq. 18) for improper rotations of the point group Td defined for the CH4 orientation of Fig. 1.

Proper rotations                 Improper rotations
Operation α β γ Operation α β γ
E 0 0 0 S4(x) −½π ½π ½π
C3(111) ½π ½π 0 S43(x) ½π ½π −½π
C32(111) π ½π ½π S4(y) π ½π π
C3(−111) 0 ½π ½π S43(y) 0 ½π 0
C32(−111) ½π ½π π S4(z) −½π 0 0
C3(−1−11) −½π ½π π S43(z) ½π 0 0
C32(−1−11) 0 ½π −½π σd(011) ½π ½π ½π
C3(1−11) π ½π −½π σd(0−11) −½π ½π −½π
C32(1−11) −½π ½π 0 σd(101) π ½π 0
C2(x) π π 0 σd(10−1) 0 ½π π
C2(y) 0 π 0 σd(110) ½π π 0
C2(z) π 0 0 σd(−110) −½π π 0

Note that the transformations (eq. 33) generated when the Rαβγ 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 Td, 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 Dg(J) of the continuous three-dimensional rotation-reflection group. Reductions of these representations to irreducible representations of the point group Td are given in many places [6]. Reductions of the even representations Dg(J) are given herein Table 14. Reductions of the odd representations Du(J) can be obtained from Table  14 by exchanging all subscripts 1 and 2.

#### Table 14. Reduction of the even representations Dg(J) of the three-dimensional rotation-reflection group to irreducible representations of Td

These reductions give the species of the (2J + 1) symmetric top rotational basis functions |kJm⟩ with fixed J and m. Reductions of the odd representations Du(J) can be obtained from this table by everywhere exchanging the subscripts 1 and 2.
J Dg(J)
12p p(A1+A2+2E+3F1+3F2) + A1
12p + 1 p(A1+A2+2E+3F1+3F2) + F1
12p + 2 p(A1+A2+2E+3F1+3F2) + E+F2
12p + 3 p(A1+A2+2E+3F1+3F2) + A2+F1+F2
12p + 4 p(A1+A2+2E+3F1+3F2) + A1+E+F1+F2
12p + 5 p(A1+A2+2E+3F1+3F2) + E+2F1+F2
12p + 6 p(A1+A2+2E+3F1+3F2) + A1+A2+E+F1+2F2
12p + 7 p(A1+A2+2E+3F1+3F2) + A2+E+2F1+2F2
12p + 8 p(A1+A2+2E+3F1+3F2) + A1+2E+2F1+2F2
12p + 9 p(A1+A2+2E+3F1+3F2) + A1+A2+E+3F1+2F2
12p + 10 p(A1+A2+2E+3F1+3F2) + A1+A2+2E+2F1+3F2
12p + 11 p(A1+A2+2E+3F1+3F2) + A2+2E+3F1+3F2

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.

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Created September 21, 2016, Updated March 1, 2023