The Projection Operator and Degenerate Representations

The choice of the generating vector for a point group without degenerate states is quite straightforward. In the examples so far we have simply taken the first basis vector in the list and used this as the generating vector. In Problem 6.11 it was found that selecting any of the other vectors in the basis would give equivalent results. 

If the reduction formula has given degenerate representations then we expect two vibrations for each E representation and three for each T representation. But the projection of a single vector will give only a single mode, and so we must find an alternative generating function for the others. 

So the three basis vectors give rise to a vibration of Ai symmetry and two degenerate vibrations conforming to the E irreducible representation. From the character table we expect the A, and E modes to be both IR and Raman active. 

To find the linear combinations of the basis vectors for Aj and the first part of the E representation we can apply the projection method arbitrarily taking b (the N—Hi vector) as the generating vector. The results are summarized in Table 6.13, where the three mirror planes in the 3ay class are considered separately using A, B and C superscripts to refer to planes containing the N—Hj, N—Hj and N—Hj bonds respectively. 

For the Ai case we end up with a function in which the b vectors are present with the same sign and same weight. This represents a vibration of the molecule in which the three N—H bonds oscillate with the same amplitude and in phase with one another, akin to the symmetric stretch mode of H2O. The normalization factor for the Ai mode was found in Problem 6.10, so we know that this function with unit length would be 

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