Let us now apply (eq. 59) to two examples. First consider the coupling of vibrational and rotational angular momenta already discussed in Section 14. The functions | k J m⟩ are eigenfunctions of the operators J^{2}, J_{z }, and the functions
$$| ~ \tilde{L} ~ \tilde{k}_L ~ \rangle$$
are eigenfunctions of the operators
$$\tilde{L}_S^2 ~,~ \tilde{L}_{SZ} ~,$$
with s = 3 or 4. Furthermore, these two sets of eigenfunctions transform, as given in (eq. 33) and (eq. 48), according to the irreducible representations D^{(J)} and D^{(L)} of the continuous three-dimensional rotation group (or according to D_{g}^{(J)} and D_{u}^{(L)}, for L = 1, of the continuous three-dimensional rotation-reflection group).
We wish to construct eigenfunctions of the operators
$$J^2 ~,~\tilde{L}_S^2~,~R^2 ~\equiv~(J + \tilde{L}_S)^2$$
and R_{z }. According to (eq. 59) we have
$$| J\,m\,\tilde{L}\,R\,k_R \rangle = \sum_{k\,\tilde{k}_L} ~ |k\,J\,m\rangle \, | \tilde{L}\,\tilde{k}_L \rangle \, (J \tilde{L}\,k\,\tilde{k}_L \, | \, J \tilde{L}\,R\,k_R) ~ . $$
(eq. 60)
Eigenfunctions on the left with fixed J, m,
$$\tilde{L},$$
and R transform according to the irreducible representation D^{(R)} of the continuous three-dimensional rotation group when both rotational and vibrational variables are subjected to "point-group-type" rotations.
Two interesting points arise here. First, the quantum number R represents an approximation to the purely rotational angular momentum, since R is the difference between the total angular momentum and 1/ζ_{s} times the vibrational angular momentum.
The second point is something of an anomaly. The quantum number
$$\tilde{L}$$
determines the amount of vibrational angular momentum and the transformation properties of the vibrational wave functions. On the other hand, the quantum number J determines the amount of total (vibration-rotation) angular momentum and the transformation properties of the rotational wave functions, while the quantum number R gives an approximation to the amount of rotational angular momentum and determines the transformation properties of the total (vibration-rotation) wave functions.
Since the eigenfunctions
$$| ~ J~m~\tilde{L} ~R~k_R~\rangle$$
with
$$\tilde{L}~=~1$$
transform according to the representation D_{u}^{(R)} of the continuous three-dimensional rotation-reflection group, Table 14 immediately gives the T_{d} symmetry species which can occur for given values of J,
$$\tilde{L}~=~1~and~R.$$
The three groupings of υ_{3} = 1 symmetry species in Figure 5 correspond to T_{d} species for D_{u}^{(R)} with R = 7, 6, 5.
Consider now the introduction of nuclear spin. We can couple the laboratory-fixed projections m_{I} of the nuclear spin functions | ΓI m_{I} ⟩ with the laboratory-fixed projections m of the rovibrational functions
$$|~J~m~\tilde{L}~R~k_R~\rangle$$
to obtain eigenfunctions of the operators F^{2} ≡ (J + I)^{2} and F_{Z}. Since the laboratory-fixed components of both J and I commute with the ordinary [29] sign of i, atomic coupling formalism can be used immediately. We obtain
$$| J\,\tilde{L}\,R\,k_R\,\Gamma\,I\,F\,m_F \rangle = \sum_{m\,m_I} ~ |J\,m\,\tilde{L}\,R\,k_R\,\rangle ~ | \Gamma\,I\,m_I \rangle \, (J\,I\,m\,m_I \, | \, J\,I\,F\,m_F\,) ~ . $$
(eq. 61)
Furthermore, since the coupling in (eq. 61) involves projections along a laboratory-fixed axis, the continuous three-dimensional rotation-reflection group is the appropriate symmetry group to consider, and no reduction of D^{(F)} into species of the group T_{d} is to be performed. Also, the coupling in (eq. 61) is not a recoupling in the usual sense; summation is over an as yet unused projection m of one of the already coupled angular momenta J, a situation which does not arise in atomic problems.
There remain some considerations associated with the point group T_{d} which have yet to be taken into account. From an examination of transformation properties under point-group rotations and under laboratory-fixed rotations, it can be shown that quantum numbers of the functions |k J m⟩ and |Γ I m_{I }⟩ are analogous in pairs: J and I determine the magnitude of angular momentum involved; m and m_{I} determine its projection along the laboratory-fixed Z axis as well as transformation properties of the functions under laboratory-fixed rotations; and k and Γ determine transformation properties under point-group rotations. Unfortunately, as can be seen by evaluating the appropriate commutation relations, it is not possible simultaneously to quantize nuclear spin projections along one laboratory-fixed axis and one molecule-fixed axis, as is done for the total angular momentum J. Thus, the pairwise analogy is not quite exact.
In any case, the nuclear spin functions transform as indicated in Table 15 and Table 16 under the T_{d} point-group operations. The rovibrational functions transform as indicated in (eq. 33) and Table 13, with J and k replaced by R and k_{R}, under these same operations. It is necessary to couple the value of k_{R} with the symmetry species of the nuclear spin functions (e.g., Γ = E_{a }, E_{b }, F_{2x }, etc.) to obtain overall wave functions which transform according to irreducible representations of the point group T_{d} and which obey the correct statistics (see Section 9.2).
The coupling of k_{R} with the nuclear spin species is a true recoupling, since k_{R} is itself a quantum number obtained by coupling k and
$$\tilde{k}_L~.$$
This recoupling can be carried out using atomic formalism when the symmetry species of the nuclear spin functions of given total spin I correspond to the reduction of one of the species D_{g ,u}^{(j)} of the three-dimensional rotation-reflection group into species of T_{d} (e.g., F_{2} → D_{u}^{(1)}, but E ↛ D_{g ,u}^{(j)} for CH_{4}). Such recoupling is probably best carried out in general, however, using irreducible tensor techniques specifically adapted for tetrahedral molecules [63]. Here, of course (as indeed at any point along the way), a reader doing numerical calculations can retreat to the point group D_{2d }, where nuclear-spin coupling becomes almost trivial [25].