Selection Rules for Fundamental Vibrational Transitions

The wave functions of normal modes can be written in the simplest possible way by using the normal coordinates as the variables. As is shown in many 

As shown in Section 5.1, the wave functions must form bases for irreducible representations of the symmetry group of the molecule, and the same holds, of course, for all kinds of wave functions, vibrational, rotational, electronic, and so on. Let us now see what representations are generated by the vibrational wave functions of the normal modes. Inserting Hn(Va4,) into 10.6-1, we obtain 

If Zb are normalized, ra and rb will be such that c a will also be normalized. We then have, because normal coordinates are orthogonal and normalized, 

in the degenerate case also,, (0) is invariant to all symmetry operations. 

All wave functions for normal vibrations in their ground states, y/, 0), are bases for the totally symmetric representation of the point group of the molecule. 

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