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 CH_{4} shown in Fig. 1, this leads to one of the three D_{2d} subgroups of T_{d}. It is also possible to orient the CH_{4} 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 F_{2} 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}, B_{2} 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}, A_{2}. 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 A_{1}, B_{1}.
E | 2S_{4} | C_{2} | 2C_{2}′ | 2σ_{d} | |||
---|---|---|---|---|---|---|---|
A_{1} | 1 | 1 | 1 | 1 | 1 | α_{xx} + α_{yy}, α_{zz} | |
A_{2} | 1 | 1 | 1 | −1 | −1 | R_{z} | |
B_{1} | 1 | −1 | 1 | 1 | −1 | α_{xx} − α_{yy} | |
B_{2} | 1 | −1 | 1 | −1 | 1 | T_{z} | α_{xy} |
E | 2 | 0 | −2 | 0 | 0 | (T_{x}, T_{y}) ; (R_{x}, R_{y}) | (α_{yz} , α_{zx}) |