Projection Operator Applied to the C—H Stretches of 1,4-Difluorobenzene

The relative motions can be more complex when we have a larger set of basis vectors however, the idea of using the basis to build up the fundamental modes using the projection operator can still be used. Each mode will be described by a symmetry adapted linear combination (SALC) of the basis vectors used in the symmetry analysis. 

As an example application of the projection operator to a slightly more complex case, we will return to the C—H stretch vibrations of 1,4-difluorobenzene that were first met in Section 6.3.1. There, the basis vectors were labelled fei to for the C—Hi to C—H4 

To apply the projection operator, we arbitrarily select bi as the generating vector and then consider the result of each operation of the point group in turn. This is laid out in Table 6.11 along the row labelled Tjbi. Tj is used in the projection operator to represent the th operation of the group. So here, J = 1 would mean the E operation, y = 2 the C2(Z) rotation, etc. This first row of the table lists the transformed bi vector in terms of the original basis in each case. So, for example, reflection in the a XT) would send b to the 2 position. 

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