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Methane Symmetry Operations - Subgroup and Full Group

5.   The Subgroup D2dvs. the Full Group Td

The spherical-top point group Td 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 D2d subgroups of Td. It is also possible to orient the CH4 molecule differently, so that singling out the z axis leads to one of the four C3v subgroups of Td.

It is sometimes convenient, particularly in setting up numerical calculations involving rotational wave functions, to abandon the point group Td in favor of one of the symmetric top subgroups [1624, 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 D2d rather than C3v. 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 D2d is given in Table 6. The correlation table between symmetry species in Td and those in D2d is given in Table 7. Matrices, whose transposes form representations of D2d, 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 F2x, F2y, F2z if these matrices are considered to be the transposes of F2 representation matrices for Td. Rows and columns of the matrices in Table 2 corresponding to the D2d operations listed in Table 5 are labelled by the symmetry species Ex, Ey, B2 if they are considered to be transposes of (reducible) representation matrices for D2d. Similarly, rows and columns in Table 3 are labelled by the Td symmetry species F1x, F1y, F1z and by the D2d symmetry species +Ex, −Ey, A2. Note that since the Ex, Ey matrices for D2d were already defined in connection with Table 2, it is necessary to assign Ex and Ey labels in Table 3 in agreement with that definition. The shorthand notation +Ex, −Ey indicates that the Td functions |F1x⟩ and |F1y⟩ must be taken to be the D2d functions + |Ex⟩ and − |Ey⟩, respectively. Finally, rows and columns in Table 4 are labelled by the Td symmetry species Ea, Eb and by the D2d symmetry species A1, B1.

Table 6. Character table for the point group D2d

  E 2S4 C2 2C2 d    
A1 1 1 1 1 1   αxx + αyy, αzz
A2 1 1 1 −1 −1 Rz  
B1 1 −1 1 1 −1   αxx − αyy
B2 1 −1 1 −1 1 Tz αxy
E 2 0 −2 0 0 (Tx, Ty) ; (Rx, Ry) yz , αzx)


Table 7. Correlation between Td and D2d symmetry species

Methane Symmetry Operations table 7


Created September 20, 2016, Updated November 15, 2019