Doubly degenerate species

As a second, slightly more complicated example, consider the four stretches and six bending and atj coordinates of methane (Fig. 6-4). Only the a set is involved in the doubly degenerate species, so that it is permissible to construct the symmetry coordinate according to the formula... 

The formula for the character of an overtone of a doubly degenerate species as found in Appendix XIV is... 

When the product rule is applied to a single factor of the secular equation, the character tables may be used to indicate how many powers of or which ratios TUlx), if any, will appear, thus if Tt and appear in the given symmetry species, M /M) I Jh) will appear in the product rule expression. If a doubly degenerate species is under consideration and Rx and 74 are involved, (I JIx) = would be used... 

For certain doubly degenerate species of the dihedral groups, namely, those which represent translation in the plane perpendicular to the principal symmetry a.xis (2T and T ), the transformation of (f> is readily... 

Exactly the s.amc arguments apply to doubly degenerate species other than that which expresses the transformation of the cartesian coordinates X and /, except that the angle an is no longer visualizable in a direct fashion. As shown in Appendix X, the transformation coefliciorits (for... 

The second difference is the appearance of a doubly degenerate E symmetry species whose characters are not always either the - - 1 or — 1 that we have encountered in nondegenerate point groups. 

The doubly degenerate single electron MOs are designated by the symbol e and the triply degenerate by t. Corresponding molecular symmetry species are termed E and T, respectively. Other important symmetry groups are Td and >e. ... 

For an unsymmetrical linear molecule there are N—1 non-degenerate vibrational normal coordinates belonging to the symmetry species 2+ (see Table 7-8.1) and they are of the longitudinal type and N—2 pairs of vibrational normal coordinates (doubly degenerate), each pair belonging to the II symmetry species and they are of the transverse type. 

Benzene thus has 10 nondegenerate and 20 doubly degenerate normal modes. The convention8 is to number the vibrational frequencies according to the order their symmetry species are listed in the character table modes of the same symmetry species are numbered in order of decreasing frequency. Thus the a2u frequency is called p4, the frequency of the two degenerate elg modes is called vn, the lower of the two e2u frequencies is called p20. 

For a doubly degenerate normal mode, both components must be used together as the basis of a two-dimensional irreducible representation. For example, the operations C2 and ctv on the two normal vibrations that constitute the i>6 mode lead to the character (sum of the diagonal elements of the corresponding 2x2 matrix) of -2 and 0, respectively, as illustrated below. Working through the remaining symmetry operations, the symmetry species of can be identified as Eu. 

---