Methane Symmetry Operations - Subgroup and Full Group

5.   The Subgroup D_2dvs. the Full Group T_d

The spherical-top point group T_d is more complicated to treat than any of the symmetric-top or asymmetric-top point groups essentially because of the presence of more than one rotation or rotation-reflection axis of order greater than two. This complication manifests itself most markedly in trying to deal with rotational wave functions.

The basis-set rotational wave functions with which spectroscopists are most familiar are the symmetric top functions. Each such function is characterized by three quantum numbers: the total angular momentum J, its signed projection m along the laboratory-fixed Z axis, and its signed projection k along the molecule-fixed z axis. If the molecule-fixed z axis, which normally is associated with "its own" quantum number and corresponding Eulerian angle, is taken to coincide with the unique rotation or rotation-reflection axis of order greater than two in symmetric tops, then the transformations of the Eulerian angles and rotational functions are quite simple [12].

In methane there are four three-fold rotation axes and three four-fold rotation-reflection axes, none of which coincide. Because it is not possible to give each of these axes "its own" quantum number and Eulerian angle, the transformations of the Eulerian angles and rotational functions become rather complicated for methane.

It is possible, however, to single out the molecule-fixed z axis for special consideration in methane by considering only those point group operations of order greater than two which involve rotation about the z axis, together with whatever other operations are necessary to form a group. For the orientation of CH₄ shown in Fig. 1, this leads to one of the three D_2d subgroups of T_d. It is also possible to orient the CH₄ molecule differently, so that singling out the z axis leads to one of the four C_3v subgroups of T_d.

It is sometimes convenient, particularly in setting up numerical calculations involving rotational wave functions, to abandon the point group T_d in favor of one of the symmetric top subgroups [16, 24, 25]. The smallest loss of group-theoretical information occurs in this procedure if the largest symmetric top subgroup is used. Thus, in this article an orientation was chosen which leads naturally to the subgroup D_2d rather than C_3v. Numerical calculations can, of course, be carried out with no group theoretical help whatever. In any given situation, a balance must be struck between the amount of group-theoretical algebra required of the individual and the amount of numerical computation required of the computer.

The character table for D_2d is given in Table 6. The correlation table between symmetry species in T_d and those in D_2d is given in Table 7. Matrices, whose transposes form representations of D_2d, can be constructed easily from those given in Tables 2 to 4. Rows and columns of the matrices in Table 2 are labelled by the symmetry species F_2x, F_2y, F_2z if these matrices are considered to be the transposes of F₂ representation matrices for T_d. Rows and columns of the matrices in Table 2 corresponding to the D_2d operations listed in Table 5 are labelled by the symmetry species E_x, E_y, B₂ if they are considered to be transposes of (reducible) representation matrices for D_2d. Similarly, rows and columns in Table 3 are labelled by the T_d symmetry species F_1x, F_1y, F_1z and by the D_2d symmetry species +E_x, −E_y, A₂. Note that since the E_x, E_y matrices for D_2d were already defined in connection with Table 2, it is necessary to assign E_x and E_y labels in Table 3 in agreement with that definition. The shorthand notation +E_x, −E_y indicates that the T_d functions |F_1x⟩ and |F_1y⟩ must be taken to be the D_2d functions + |E_x⟩ and − |E_y⟩, respectively. Finally, rows and columns in Table 4 are labelled by the T_d symmetry species E_a, E_b and by the D_2d symmetry species A₁, B₁.

Table 7. Correlation between T_d and D_2d symmetry species