Methane Symmetry Operations - Comparison of the Present Treatment

12.   Comparison of the Present Treatment with Those of Jahn and Moret-Bailly

As mentioned earlier, the three most commonly followed discussions of the symmetry properties of methane differ rather markedly in their treatments of the symmetries of the rotational functions. Qualitatively, the problem arises because of difficulties in applying sense-reversing operations to the Eulerian angles. Every set of Eulerian angles corresponds to some proper rotation of one axis system with respect to another. No set of Eulerian angles corresponds to an inversion of one of the axis systems with respect to the other. As a consequence, it is not intuitively obvious which transformations of Eulerian angles to associate with sense-reversing point-group operations.

This difficulty was overcome in the present treatment by introducing the minus sign in (eq. 18). Physically speaking, the minus sign causes every point-group operation to be associated with a proper rotation of the Eulerian angles: if the point-group operation is itself a proper rotation, that same rotation is applied to the Eulerian angles; if the point-group operation is a sense-reversing operation, the proper rotation obtained by multiplying the sense-reversing operation by the molecule-fixed inversion operation is applied to the Eulerian angles. The latter prescription is followed whether or not the molecule-fixed inversion operation or the resultant proper rotation belongs to the point group. The minus sign in (eq. 18) leads ultimately to the minus sign in (eq. 19), and to the correlation of sense-reversing point-group operations with permutation-inversion operations. As pointed out earlier, however, these operations do not actually change the sense of the CH₄ molecular framework or the sense of the molecule-fixed axis system.  

12.1   Jahn's treatment

Jahn [6] overcame the inversion difficulty by deducing that "the rotational wave functions u^J_MK of the spherical top transform, for even and odd values of J respectively, according to the representations D^g_J, D^u_J of the group Dⁱ_∞ of all rotations and reflexions about axes fixed in the molecule." The present treatment corresponds, in that language, to requiring all rotational wave functions u^J_MK to transform according to D^g_J, since the transformation for each u^J_MK associated with the molecule-fixed inversion operation i is that corresponding to the product of i with itself, i.e., that corresponding to the identity operation.

Jahn's scheme suffers from two principal inconveniences. First, it is not actually possible mathematically to satisfy his postulated transformation properties for all spherical-top rotational wave functions simultaneously [13]. This is most easily seen by considering a trigonometric identity in the rotational wave functions, which for a choice of phase factors implied by setting u^J_MK = |kJm⟩ in (eq. 26), can be written

It can be seen that an application of the molecule-fixed inversion operation i destroys this identity if the J = 1 functions are changed into their negatives while the J = 0 function remains unchanged.

As it happens, Jahn [6] was considering only vibration-rotation energies in the absence of nuclear spin interactions. Thus, he quite naturally chose to work with functions characterized by a given value of m, namely the simplest value m = 0. No inconsistencies need arise if consideration is restricted to m = 0 functions, since no identity analogous to (eq. 37) can be written involving only m = 0 functions. It is clear, however, that any treatment of the symmetry properties of hyperfine levels would be awkward in Jahn's scheme, since such a treatment involves "coupling" values for the quantum numbers m and m_I to give a total quantum number F and projection m_F.

The second inconvenience in Jahn's scheme concerns the impossibility of assigning a unique symmetry species to the laboratory-fixed components of the electric dipole-moment operator. Jahn's selection rules are such that the laboratory-fixed Z component of the dipole moment operator must be considered to be of species A₂ when ΔJ = 0 transitions are to be calculated, but of species A₁ when ΔJ = ±1 transitions are to be calculated. Again, for a molecule in free space and in the absence of nuclear spin interactions, no inconsistencies need ever arise, since J is rigorously a good quantum number. On the other hand, a discussion of allowed transitions including nuclear spin effects or involving perturbations by a magnetic field would be awkward in Jahn's scheme.  

12.2   Moret-Bailly's treatment

Moret-Bailly [9-11] has developed a treatment of symmetry properties in CH₄ having selection rules such that both vibration-rotation interactions caused by the Hamiltonian operator and also electric dipole transitions caused by a radiation field are allowed only between levels having the same T_d symmetry species. This treatment thus differs significantly from the present one, in which perturbations take place between levels of the same symmetry, whereas electric-dipole transitions take place between levels of different symmetry.

The two schemes actually lead to allowed transitions between the same sets of rovibrational levels for the υ₃ and υ₄ fundamental transitions, since Moret-Bailly's symmetry labels for the ground vibrational state of CH₄ agree with those of the present treatment, while his labels for the υ₃ = 1 or υ₄ = 1 states differ from those of the present treatment by an exchange of subscripts 1 and 2 for all A₁, A₂ and F₁, F₂ species. While Moret-Bailly's scheme need lead to no inconsistencies, provided it is applied only to situations for which it was intended, it has two principal inconveniences.

The first inconvenience concerns selection rules for electric-dipole transitions. By exchanging all subscripts 1 and 2 in the υ₃ = 1 and υ₄ = 1 states, while leaving the υ = 0 state unchanged (with respect to the present treatment), allowed electric dipole transitions between υ = 0 rovibrational levels and υ₃ = 1 or υ₄ = 1 rovibrational levels occur only between states of the same symmetry. However, as can be seen from Figure 5, electric dipole transitions within the υ = 0 state [33-35, 48, 49], or within the υ₃ = 1 state [47, 58, 59] must still occur between states of different symmetry whose direct product contains A₂. Thus, a generalization of Moret-Bailly's treatment to include a discussion of the recently observed pure rotational transitions in methane will be awkward, leading to different rovibrational selection rules for Δυ = 0 transitions than for Δυ = 1 transitions. Complications of this nature may also arise in discussions of an extended set of overtone, combination, and difference bands.

The second inconvenience concerns selection rules for perturbations caused by the Hamiltonian operator. An interchange of subscripts 1 and 2 for one vibrational state, while leaving those for another vibrational state unchanged, does not affect the selection rule that only states of the same symmetry can perturb each other, provided that only interactions within a given vibrational state are considered. When calculating vibration-rotation energies of CH₄ using a transformed Hamiltonian approach [60], as Moret-Bailly did, such Δυ = 0 interactions are the only ones which need be considered. Brief reflection indicates, however, that interactions between rovibrational levels of different vibrational states may require different selection rules. For example, levels belonging to υ = 0 and levels belonging to υ₃ = 1 can perturb each other in Moret-Bailly's scheme only if they have different species, i.e., only if their direct product contains A₂.

In the author's opinion, the treatment presented in this article, which is based conceptually on clearly specified algebraic transformations of variables and which has neither the two inconveniences associated with Jahn's treatment nor the two associated with Moret-Bailly's treatment is to be strongly preferred over the latter two alternatives, even though adoption of the present treatment will require some changes in the future publications of many authors on the spectrum of the methane molecule.