In order to apply the full continuous pure rotation group to triply degenerate vibrational coordinates, it is convenient to express the latter in spherical polar form
$$ \left[ \begin{array}{c} Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array}\right] \equiv Q_s \left[ \begin{array}{l} \sin\theta_s \, \cos\phi_s\\ \sin\theta_s \, \sin\phi_s\\ \cos\theta_s \end{array}\right] ~ , $$
(eq. 38)
where Q_{s}^{2} = Q_{sx}^{2} + Q_{sy}^{2} + Q_{sz}^{2}. The vibrational angular momentum operators given in (eq. 36) become, after a tedious but well known calculation,
$$ \left[ \begin{array}{c} L_{sx} \\ L_{sy} \\ L_{sz} \end{array}\right] = \left[ \begin{array}{ccc} +\csc\theta_s \, \cos\phi_s & -\sin\phi_s & -\cot\theta_s \, \cos\phi_s \\ +\csc\theta_s \, \sin\phi_s & +\cos\phi_s & -\cot\theta_s \, \sin\phi_s \\ 0 & 0 & +1 \end{array}\right] ~ \left[ \begin{array}{c} 0 \\ P_{\theta_s} \\ P_{\phi_s} \end{array}\right] ~ , $$
(eq. 39)
where P_{θs} = −iℏ(∂/∂θ_{s}), etc., and s = 3 or 4. Note that entries in the first column of the 3 × 3 matrix on the right of (eq. 39) are arbitrary, since the first component of the column vector on the right is zero.
Comparison of (eq. 25) and (eq. 39) shows that the form of the operators L_{sx }, L_{sy }, L_{sz } is identical to the form of the operators J_{X}, J_{Y}, J_{Z} provided that we replace the momentum p_{χ} in (eq. 25) by zero. [Attempts to relabel φ_{s} as χ_{s }, and to write (eq. 39) in a fashion analogous to (eq. 23) fail because of difficulties with the minus signs.] We thus conclude, or prove by direct substitution, that eigenfunctions |L k_{L}⟩ of the form
$$|L \, k_L \rangle = [(2L+1)/4\pi]^{1/2} \, {\cal D}_{0k_L}^{(L)}\, (\{0\, \theta_s \phi_s \}) ~ , $$
(eq. 40)
belong to the eigenvalues L(L + 1)ℏ^{2} and k_{L}ℏ, respectively, of L^{2} and L_{z}.
The projection of the vibrational angular momentum along the molecule-fixed z axis is indicated by the second subscript in the D function in (eq. 40), whereas the projection of the total angular momentum along the molecule-fixed z axis is indicated by the first subscript in (eq. 26). This difference between (eq. 26) and (eq. 40) is related to the fact that the molecule-fixed components of J commute with the anomalous [30] sign of i, while the laboratory-fixed components of J and both the molecule-fixed and laboratory-fixed components of other angular momentum operators commute with the normal [29] sign of i, and to the fact that the Hamiltonian operator involves terms in the difference J − ζL of the two angular momenta, not the sum J + ζL. Further discussion of these two points will not be given, though further manifestations of these differences occur elsewhere in this section.
The normalization factors in (eq. 26) and (eq. 40) differ by (2π)^{1/2} because vibrational configuration space for a triply degenerate vibration is characterized by two angular variables (θ_{s}φ_{s}), whereas rotational configuration space is characterized by three (χθφ). The analog χ_{s} of the third Eulerian angle has been set to zero in (eq. 40). It can be given any value without altering D^{(j)}_{µ′ µ} functions having µ′ = 0.
The triply degenerate vibrational coordinates Q_{sx }, Q_{sy }, Q_{sz } transform under the pure rotation operators $$R_{\alpha\beta\gamma}$$ of the point group T_{d} as
$$R_{\alpha\beta\gamma} ~ \left[ \begin{array}{c} Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array}\right] = \left[ \begin{array}{c} ~\\ S(\alpha\beta\gamma) \\~ \end{array}\right] ~ \left[ \begin{array}{c} Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array}\right] ~ . $$
(eq. 41)
By extension, we require Q_{sx }, Q_{sy }, Q_{sz } to transform according to this equation under any pure rotation.
The question now arises of how the eigenfunctions |L k_{L}⟩ transform. This can be determined rather easily by making use of some slightly formalistic arguments, designed to make the transformation algebra of this section analogous to that used in previous sections.
Equation (eq. 41) can be rewritten in terms of the spherical polar forms of the vibrational variables, and recast somewhat to read
$$\begin{eqnarray*} R_{\alpha\beta\gamma}[Q_{sx}, Q_{sy}, Q_{sz}] &=& R_{\alpha\beta\gamma}[\sin\theta_s\cos\phi_s,\sin\theta_s\sin\phi_s,\cos\theta_s]\,Q_s\\ &=& R_{\alpha\beta\gamma}[0 \, 0 \, 1 ] \cdot S(\chi_s \theta_s \phi_s) \cdot Q_s\\ &=& [0 \, 0 \, 1 ] \cdot S(\chi_s \theta_s \phi_s) \cdot S^{\rm tr}(\alpha\beta\gamma) \cdot Q_s ~ . \end{eqnarray*}$$
(eq. 42)
In (eq. 42) we have introduced an extraneous variable χ_{s}, and generated with its help a complete 3 × 3 matrix of the form S(χ_{s}θ_{s}φ_{s}) from the row vector [sinθ_{s} cosφ_{s}, sinθ_{s} sinφ_{s}, cosθ_{s}]. Actually, however, because of the presence of the row vector [0 0 1], (eq. 42) makes a statement only about how the third row of the matrix S(χ_{s}θ_{s}φ_{s}) transforms and this third row does not involve the extraneous variable χ_{s}.
We now generalize (eq. 42) by omitting the row vector [0 0 1] and the scalar Q_{s}, to obtain
$$ R_{\alpha\beta\gamma}\,S(\chi_s\theta_s\phi_s) = S(\chi_s\theta_s\phi_s) \cdot S^{\rm tr}(\alpha\beta\gamma) ~ ,$$
(eq. 43)
It can be seen that this generalized equation introduces no inconsistencies with respect to the physically meaningful (eq. 42). Subjecting (eq. 43) to the similarity transformation defined by (eq. 30) and (eq. 31), making use of a property [22] of the D matrices
$${\cal D}_{\mu^\prime\mu}^{(j)} (\{\alpha\beta\gamma\}) = (-1)^{\mu-\mu^\prime} ~ {\cal D}_{-\mu^\prime-\mu}^{(j)*} (\{\alpha\beta\gamma\}) ~ ,$$
(eq. 44)
and extending, as in Sec. 8, validity of the transformed (eq. 43) from L = 1 to all L, we find that
$$R_{\alpha\beta\gamma}~ {\cal D}_{k^{\prime\prime}k_L}^{(L)} (\{\chi_s\theta_s\phi_s\})= \sum_{k_L^\prime}~{\cal D}_{k^{\prime\prime}k_L^\prime}^{(L)} (\{\chi_s\theta_s\phi_s\}) \, (-1)^{k_L^\prime-k_L} ~ {\cal D}_{-k_L-k_L^\prime}^{(L)} (\{\alpha\beta\gamma\}) ~ . $$
(eq. 45)
Specializing this to k″ = 0, and making use of (eq. 40) we obtain
$$R_{\alpha\beta\gamma}~ |L\, k_L\rangle = \sum_{k_L^\prime} \, (-1)^{k_L^\prime-k_L} ~ {\cal D}_{-k_L-k_L^\prime}^{(L)} (\{\alpha\beta\gamma\}) ~ |L\, k_L^\prime\rangle ~ . $$
(eq. 46)
The rather awkward looking transformation (eq. 46) can be made analogous to (eq. 33) by utilizing the concept of "reversed angular momenta," first introduced by Van Vleck [30] for this purpose. Quite simply, if one considers not [L_{x }, L_{y }, L_{z}], but instead
$$ \left[\tilde{L}_x, \tilde{L}_y, \tilde{L}_z \right] ≡ \left[-L_x,-L_y,-L_z \right] ~ , $$
then the
$$\tilde{L}$$
operators commute with the anomalous sign of i just as J_{x }, J_{y }, J_{z } do. A function belonging to the eigenvalue k_{L}ℏ of L_{z} and L(L + 1)ℏ^{2} of L^{2} clearly belongs to the eigenvalue
$$\tilde{k}_L \hbar = -k_L \hbar ~of ~ \tilde{L}_z$$
and
$$\tilde{L}(\tilde{L} + 1) \hbar^2 = L(L+1) \hbar^2 ~ of ~ \tilde{L}^2 ~. $$
Defining the eigenfunctions
$$|\tilde{L} ~ \tilde{k}_L \rangle ~ of ~ \tilde{L}^2 ~, ~ \tilde{L}_z$$
to be
$$|\tilde{L} \, \tilde{k}_L\rangle \equiv (-1)^{k_L} \, [(2L+1)/4\pi ]^{1/2} ~ {\cal D}_{0\,k_L}^{(L)} (\{0\, \theta_s\, \phi_s\}) = (-1)^{k_L} \, |L\, k_L\rangle ~ ,$$
(eq. 47)
where
$$L ~ = ~ \tilde{L}$$
and
$$k_L = -\tilde{L}_L$$
insures that matrix elements of the "reversed" ladder operators
$$\tilde{L}_x \mp i\tilde{L}_y$$
are real and positive. Equation (46) can then be rewritten, using the
$$|\tilde{L} - k \rangle ~, ~$$
functions rather than the |L +k⟩, as
$$R_{\alpha\beta\gamma}~ |\tilde{L} \, \tilde{k}_L \rangle = \sum_{\tilde{k}_L^\prime} ~ {\cal D}_{\tilde{k}_L \tilde{k}_L^\prime}^{(L)} \, (\{\alpha\beta\gamma\}) ~ |\tilde{L} \, \tilde{k}_L^\prime \rangle ~ ,$$
(eq. 48)
an expression analogous to (eq. 33).
Since the vibrational coordinates Q_{sx }, Q_{sy }, Q_{sz} transform according to the representation F_{2} of T_{d}, and since F_{2} correlates with the representation D_{u}^{(1)} of the full three-dimensional rotation-reflection group, it is convenient and consistent to require Q_{sx }, Q_{sy }, Q_{sz} to transform into their negatives under the molecule-fixed inversion operation i, even though i does not occur in the T_{d} point group. This transformation of Q_{sx }, Q_{sy }, Q_{sz} can be achieved by replacing the vibrational spherical polar coordinates θ_{s}φ_{s} in (eq. 38) by
$$\begin{eqnarray*} (\theta_s)_{\rm new} &=& \pi - \theta_s\\ (\phi_s)_{\rm new} &=& \pi + \phi_s ~ . \end{eqnarray*}$$
(eq. 49)
Direct substitution of (θ_{s})_{new} and (φ_{s})_{new} from (eq. 49) into Wigner's [22] D functions as used in (eq. 40) shows that the vibrational wave functions
$$| \tilde{L} \tilde{k}_L \rangle $$
transform into (−1)^{L} times themselves under the molecule-fixed inversion operation i, i.e., that the vibrational wave functions
$$| \tilde{L} \tilde{k}_L \rangle $$
for a triply-degenerate mode transform [14] as D_{g}^{(L)} for even L and as D_{u}^{(L)} for odd L.