Number of Allowed Infrared Fundamentals

In the previous sections the allowed fundamentals, overtones, and combination bands for a C3y molecule were calculated. The number of fundamentals is determined by the number of atoms in the molecule, and for some symmetries the number of atoms also determines whether the allowed vibration exists for the particular molecule in question. For example, a vibration, although an allowed infrared band for larger C2V molecules, does not exist for a triatomic C2V configuration. 

We shall show here the calculations used to determine the number of and e fundamentals of the molecule methylacetylene. 

We must first calculate the quantity 0, which is given as follows  

All the above quantities except have been defined earlier. For C v molecules the identity / and rotations C3 are proper rotations, and the reflections are improper rotations. We define Ur as the number of nuclei unchanged by the symmetry operation. For methylacetylene the identity operation leaves all seven atoms unchanged for the rotation C3, four atoms remain unchanged and for the reflection Cy, five atoms remain unchanged. Thus, Ur is 7, 4, 5 for the three classes of operation. 0 can now be calculated for each operation from the above equations and is found to be 15, 0, 5. 

To determine the number of times each type of vibration appears, a reduction formula is used  

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