Irreducible representations ungerade

In the MuUiken notation, the subscripts u (ungerade = odd) and g (gerade = even) indicate whether an irreducible representation is symmetric (g) or anti-symmetric(M), in respect to the inversion operation (/). 

The correct combination must be two XI representations. At this point, any of these would be allowed because they each have character 1 under 2Coo in Table 5.21. However, under the ooo-v class the 11 (both gerade and ungerade cases) have -1. This means that the inclusion of these in our linear combination will lead to a total character of less than the required 2. So the set of irreducible representations we seek can only contain X1+ types. 

Next, looking at the inversion centre column, in F we find 0, and since gerade representations have character 1 under i and ungerade —1, the only possible combination of irreducible representations remaining is... 

Now we will consider each class of operations to narrow down the possible standard irreducible representations that can be present until we arrive at only one option. The 3 under the E class reminds us that there are three orbitals being represented, and so our combination must consist of either three E-type representations or one E and one doubly degenerate representation. Under the inversion centre i the total character is —3, and so all three orbitals must be reversed by the inversion. This means that any irreducible representation present must have ungerade symmetry if we assigned a gerade representation, then it would contribute positively under i. This eliminates all gerade representations from further consideration. 

For more complex molecules in point groups with the inversion, centre vibrations that are IR active will also have irreducible representations that are ungerade while Raman-active modes will be gerade. Since different vibrations will usually occur at different frequencies, it is unlikely that bands in the two spectra will appear to be coincident. For example, rra 5-l,2-dichloroethene has an inversion centre as it belongs to the point group C2h, the IR and Raman spectra for this molecule are compared in Figure 6.13a. This molecule has six atoms, and so 3x6 — 6 = 12 vibrational modes. The total number of bands in the Raman and IR spectra is fewer than 12 because some vibrations are too low frequency to be detected in the range shown. However, it can be seen that the Raman and IR frequencies are indeed different to one another. 

Table 6.6 Reduction of the reducible representation for C—H stretch modes of 1,4 difluorobenzene (a) gerade and (b) ungerade irreducible representations. Note that only classes with nonzero character for F from Table 6.5 are considered here.

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