General Approach to Analysing Vibrational Spectroscopy

We are now in a position to predict the bands that may exist in the vibrational spectra (IR or Raman) for a given molecule. A few examples have already been covered, but the steps that should be followed can be summarized as follows  

Decide on the point group of the molecule using the symmetry operations the atoms obey. 

Decide on a basis for the vibrations of interest. The complete basis for vibrational modes is a set of three vectors on each atom, but a simpler basis may be more convenient to give information on specific vibrations. 

Generate the reducible representation F of the basis by assigning characters for each basis vector according to the effect of an example operation from each class. If a basis vector is unaffected by the operation, then it contributes 1 to F if it is reversed, then it gives —1 and if it changes completely (e.g. if the atom on which the basis vector is sited moves), then a contribution of 0 results. Intermediate values occur for rotated basis vectors, as discussed in Section 4.7. F is then the sum of the characters obtained for the members of the basis set one summation for each class of operations in the point group. 

(a) For a nonlinear molecule, reduce the representation to its components using the 

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