NIST: Methane Symmetry Operations - Cont. 3D Rotation-Reflec.

Methane Symmetry Operations

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

eq. 28 (eq. 28)

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

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

(eq. 29)

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

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

(eq. 30)

where

eq. 31 (eq. 31)

Hence, (eq. 29) can be rewritten

(eq. 32)

with J = 1. However, since Wigner's script 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 acting on |kJm⟩ gives

(eq. 33)

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.

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 *T_d* 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	σ_d(011)	½π	½π	½π
C₃(1−11)	π	½π	−½π	σ_d(0−11)	−½π	½π	−½π
C₃²(1−11)	−½π	½π	0	σ_d(101)	π	½π	0
C₂(x)	π	π	0	σ_d(10−1)	0	½π	π
C₂(y)	0	π	0	σ_d(110)	½π	π	0
C₂(z)	π	0	0	σ_d(−110)	−½π	π	0

Note that the transformations (eq. 33) generated when the 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 script 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 _g^(J) are given herein Table 14. Reductions of the odd representations script D _u^(J) can be obtained from Table 14 by exchanging all subscripts 1 and 2.

J	_g^(J)
Table 14. Reduction of the even representations _g^(J) of the three-dimensional rotation-reflection group to irreducible representations of T_d 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 _u^(J) can be obtained from this table by everywhere exchanging the subscripts 1 and 2.
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 italic v ₃ = 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 italic v = 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 italic v ₃ 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 ₃ = 1, J = 6 manifold. The grouping and superscript numbering of the italic v ₃ = 1 rovibrational levels will be discussed in later sections.

Methane Symmetry Operations

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

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 T_d defined for the CH₄ orientation of Fig. 1.

Table 14. Reduction of the even representations _g^(J) of the three-dimensional rotation-reflection group to irreducible representations of T_d

Methane Symmetry Operations

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

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.

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

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 T_d defined for the CH₄ orientation of Fig. 1.

Table 14. Reduction of the even representations _g^(J) of the three-dimensional rotation-reflection group to irreducible representations of T_d