Methane Symmetry Operations - Symmetry Properties of Rotational Wave functions and Direction Cosines

7.   Symmetry Properties of Rotational Wave functions and Direction Cosines

It is in the determination of symmetry properties of functions of the Eulerian angles, and in particular in the question of how to apply sense-reversing point-group operations to these functions, that the principal differences arise in group-theoretical discussions of methane. The treatment given here follows from the discussion of Section 4. The two other most commonly followed treatments, due to Jahn [5−8] and Moret-Bailly [9−11], respectively, will be discussed briefly in Section 12.

7.1 Symmetric-top rotational basis functions

The molecule-fixed components J_x,J_y,J_z of the total angular momentum operator can be written as [4]

where p_χ = −iħ(∂/∂_χ), etc. [4, 28]. Since the transformation from molecule-fixed to laboratory-fixed vector components takes place via the matrix S in (eq. 10), the laboratory-fixed components J_X,J_Y,J_Z of the total angular momentum operator can be obtained from (eq. 23) and

Thus,

It can be shown, by direct application of the differential operators, that symmetric top rotational basis functions |kJm⟩ defined in terms of Wigner's $${\cal D}_{\mu^\prime\mu}^{(j)}\, (\{\alpha\beta\gamma\})$$ functions (equation (15.27) of [22]) as

are eigenfunctions of the operators J_z and J_Z belonging to the eigenvalues kħ and mħ, respectively, and give rise to real positive matrix elements for the laboratory-fixed [29] (J_X ± iJ_Y) and molecule-fixed [30] (J_x ‡ iJ_y) ladder operators. Substitution of the Eulerian angle transformations of Table 5 in the function |kJm⟩ leads to the transformations shown in Table 11.

If we further take sums and differences for K = | k | ≥ 1,

we find symmetry species for these functions under the D_2d symmetry operations as given in Table 12.

As mentioned earlier, functions belonging to a single symmetry species of the full point group T_d are rather less convenient to write down. In fact, it is necessary to introduce linear combinations of the symmetric top functions much more complicated than sums and differences [31]. These will not be given here, though transformation properties of the |kJm⟩ under operations of the full point group T_d are given in Section 8.

7.2 Direction cosines

The discussion of transformation properties and symmetry species is rather simple for the direction cosines. These quantities are just the nine elements of the 3 × 3 rotation matrix S(χθφ), and transformations of the Eulerian angles were in fact defined originally in Section 4.2 and Section 4.3 to insure certain transformation properties of the direction cosines themselves. Equations (eq. 14) and (eq. 18) prescribe the transformations of the direction cosine matrix when χ_new, θ_new, φ_new are substituted for χ,θ,φ. Since the matrices M in (eq. 14) must be taken from Table 2, we see that the three functions in any column of the direction cosine matrix S transform like functions of species F_2x,F_2y,F_2z as far as proper rotations are concerned. Since the matrices N in (eq. 18) must also be taken from Table 2, and since the transformation (eq. 18) involves −N, not +N, the columns of the direction cosine matrix do not transform like F_2x,F_2y,F_2z as far as improper rotations are concerned.

It happens, however, that the matrices in Table 3 are identical to those in Table 2 for proper rotations and equal to the negatives of those in Table 2 for improper rotations. Thus, the three functions in any column of the direction cosine matrix S(χθφ) transform like functions of species F_1x,F_1y,F_1z under the operations of the full molecular symmetry group T_d.