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 threedimensional pure rotation group, and to all rotationreflections of the (piecewise continuous) threedimensional rotationreflection 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
$$\left[\begin{array}{c} x_{\rm new} \\ y_{\rm new} \\ z_{\rm new} \end{array}\right] = [ S(\alpha\beta\gamma) ] \left[\begin{array}{c} x\\ y\\ z \end{array}\right] ~ .$$
(eq. 28)
The matrix S(αβγ) can be obtained from equation (eq. 10).
If R_{αβγ} 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 threedimensional pure rotation R_{αβγ}.
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 threedimensional 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
$$R_{\alpha\beta\gamma} \, kJm\rangle = \sum_{k^\prime}~ {\cal D}^{(J)}_{kk^\prime} \, (\{\alpha\beta\gamma\}) \, k^\prime Jm\rangle ~ ,$$
(eq. 33)
which is the desired transformation equation for the symmetric top rotational basis functions.
Values of αβγ corresponding to T_{d} pointgroup operations for CH_{4} oriented as in Fig. 1 are given in Table 13. Rotations corresponding to the sensereversing 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.
Proper rotations  Improper rotations  

Operation  α  β  γ  Operation  α  β  γ  
E  0  0  0  S_{4}(x)  −½π  ½π  ½π  
C_{3}(111)  ½π  ½π  0  S_{4}^{3}(x)  ½π  ½π  −½π  
C_{3}^{2}(111)  π  ½π  ½π  S_{4}(y)  π  ½π  π  
C_{3}(−111)  0  ½π  ½π  S_{4}^{3}(y)  0  ½π  0  
C_{3}^{2}(−111)  ½π  ½π  π  S_{4}(z)  −½π  0  0  
C_{3}(−1−11)  −½π  ½π  π  S_{4}^{3}(z)  ½π  0  0  
C_{3}^{2}(−1−11)  0  ½π  −½π  σ_{d}(011)  ½π  ½π  ½π  
C_{3}(1−11)  π  ½π  −½π  σ_{d}(0−11)  −½π  ½π  −½π  
C_{3}^{2}(1−11)  −½π  ½π  0  σ_{d}(101)  π  ½π  0  
C_{2}(x)  π  π  0  σ_{d}(10−1)  0  ½π  π  
C_{2}(y)  0  π  0  σ_{d}(110)  ½π  π  0  
C_{2}(z)  π  0  0  σ_{d}(−110)  −½π  π  0 
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 rigidbody 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 moleculefixed 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 threedimensional rotationreflection 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.
Table 14. Reduction of the even representations D_{g}^{(J)} of the threedimensional rotationreflection 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 D_{u}^{(J)} can be obtained from this table by everywhere exchanging the subscripts 1 and 2. 

J  D_{g}^{(J)} 

12p  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + A_{1} 
12p + 1  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + F_{1} 
12p + 2  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + E+F_{2} 
12p + 3  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + A_{2}+F_{1}+F_{2} 
12p + 4  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + A_{1}+E+F_{1}+F_{2} 
12p + 5  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + E+2F_{1}+F_{2} 
12p + 6  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + A_{1}+A_{2}+E+F_{1}+2F_{2} 
12p + 7  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + A_{2}+E+2F_{1}+2F_{2} 
12p + 8  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + A_{1}+2E+2F_{1}+2F_{2} 
12p + 9  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + A_{1}+A_{2}+E+3F_{1}+2F_{2} 
12p + 10  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + A_{1}+A_{2}+2E+2F_{1}+3F_{2} 
12p + 11  p(A_{1}+A_{2}+2E+3F_{1}+3F_{2}) + A_{2}+2E+3F_{1}+3F_{2} 
Figure 5 depicts the J = 7 rotational levels of the ground vibrational state of methane and the J = 6 rotational levels of the υ_{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 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 υ_{3} is of species F_{2}, 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 F_{2}, leading to the 2A_{1}+A_{2}+3E+5F_{1}+5F_{2} rovibrational levels occurring in the υ_{3} = 1, J = 6 manifold. The grouping and superscript numbering of the υ_{3} = 1 rovibrational levels will be discussed in later sections.