Since the functions |kJm⟩ and
$$| \tilde{L} ~ \tilde{k}_L \rangle$$
transform according to the analogous equations (eq. 33) and (eq. 48), it is possible to use standard vector coupling techniques [22] to obtain linear combinations of products of the form
$$| k'Jm ~ \rangle ~ | \tilde{L} ~ \tilde{k}'_L \rangle$$
which correspond to wave functions characterized by the quantum numbers
$$| J ~ m ~ \tilde{L} ~ R ~ k_R \rangle ~.$$
The two new quantum numbers R and k_{R}, which replace k and
$$\tilde{k}_L ~,$$
are eigenvalues of R^{2} and R_{z}. The operator R is defined by the equation
$$\mbox{$R = J - L$}_s = \mbox{$J + \tilde{L}$}_s $$
(eq. 50)
for s = 3 or 4. Molecule-fixed components of this equation are normally considered. It can be seen from (eq. 23) and (eq. 39) that all molecule-fixed components of J commute with all molecule-fixed components of L_{s}.
The rotational Hamiltonian for a spherical top molecule with zero or one quantum of a triply degenerate vibration excited can, to a first approximation, be written [4] as
$$\begin{eqnarray*} {\cal H}_{\rm r} &=& B[\mbox{ $J$} - \zeta_s\mbox{$L$}_s]^2\\ &=& B[\mbox{$J$} + \zeta_s\mbox{$\tilde L$}_s]^2\\ &=& B[\mbox{$J$}^2 + \zeta_s^2\mbox{$\tilde L$}_s^2 + 2\zeta_s \, \mbox{$J \cdot \tilde L$}_s] ~ , \\ s &=& 3 \, {\rm or} \, 4 \end{eqnarray*}$$
(eq. 51)
corresponding qualitatively to the fact that the rotational energy of a spherical top is given by the product of a rotational constant B, which is inversely proportional to the moment-of-inertia of the molecule, and the square of the purely rotational angular momentum
$$[\mbox{ $J$} + \zeta_s\tilde{L}_s]^2$$
Non-vanishing matrix elements of ℋ_{r} are diagonal in the basis set
$$| J m \tilde{L} R k_R \rangle ~,$$with values given by [6]
$$B \{J(J+1) + \tilde{L}_s(\tilde{L}_s +1) \, \zeta_s^2 + \zeta_s [R(R+1) - J(J+1) - \tilde{L}_s(\tilde{L}_s +1)\,]\,\} ~ . $$
(eq. 52)
When υ = 0,
$$\tilde {L} = 0$$
and R = J. When υ_{s} = 1,
$$\tilde{L}_s = 1$$
and R = J + 1, J, or J − 1. The Coriolis operator
$$+2B ~ \zeta_SJ ~· ~ \tilde{L}_S$$
in (eq. 51) thus gives rise to a natural grouping of υ_{s} = 1 rovibrational levels according to their R values. In Fig. 5 the υ_{3} = 1 levels are drawn in three stacks, corresponding to J_{R} = 6_{7}, 6_{6}, and 6_{5}. The relative energy positions of the stacks are correct for a positive value of ζ_{3}, but are not drawn to scale.
If laboratory-fixed components of the electric dipole moment operator are expanded as a power series in the vibrational coordinates Q_{s}, and if only terms linear in these coordinates for one value of s are considered, we find
$$\left[\begin{array}{c} \mu_X \\ \mu_Y \\ \mu_Z \end{array} \right] = (\partial\mu / \partial Q_s ) \left[\begin{array}{c} ~ \\ S^{-1} (\chi \theta \phi ) \\ ~\end{array} \right] \left[\begin{array}{c} Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array} \right] ~ , $$
(eq. 53)
where s = 3 or 4, and where (∂µ/∂Q_{s}) represents any one of the three dipole derivatives (∂µ_{x}/∂Q_{sx}), (∂µ_{y}/∂Q_{sy}), or (∂µ_{z}/∂Q_{sz}) evaluated at the equilibrium configuration. It can be shown by direct substitution that the right side of (eq. 53) commutes with the three components of the operator R given in (eq. 50). Consequently, to the approximation that only terms in the dipole moment expansion linear in the vibrational coordinates need be considered and to the approximation that R and k_{R} are good quantum numbers, we obtain the well known [2] selection rules
$$\begin{eqnarray*} \Delta v_3 ~ {\rm or} ~ \Delta v_4 &=& \pm 1 \\ \Delta R = \Delta k_R &=& 0 ~ . \end{eqnarray*}$$
(eq. 54)
These selection rules are valid for fundamental bands, but not for overtones and combination bands [59], whose intensity is governed by higher-order terms omitted in (eq. 53).
In Fig. 5, transitions indicated by solid lines obey the selection rule ΔR = 0 and are strong. The five P^{o}(7) transitions and the one P^{−}(7) transition, indicated by dashed lines, violate this rule and are weaker. The remaining weak vibration-rotation transition in Fig. 5, indicated by the dashed line among the P^{+}(7) transitions, does not violate the ΔR = 0 selection rule. It does, however, violate a selection rule requiring no change in the numerical counter superscript [14, 15]. This latter "selection rule" depends on the relative values of the various interactions giving rise to the splittings in the ground and first excited vibrational state [14,15], but has been found to hold rather well for the infrared-active vibrational fundamentals in CH_{4} [43−46, 61]. It is necessary, however, to "count" the υ_{3} = 1 rovibrational levels from highest energy to lowest when R = J ± 1, and from lowest energy to highest when R = J, to make this selection rule valid. (Readers should be cautioned that slight variations of the numerical counting scheme described here will be found in the methane literature, though all schemes have in common the selection rule that the counting index does not change for allowed transitions.)