Allowed Overtone Bands in the Infrared

The calculation of the allowed overtones is performed in a manner similar to that used for combination bands, except that a degenerate type of vibration requires special consideration. An overtone of a vibration may be considered as a special case of a combination 

TABLE 4-VI. Selection Rules for Binary Combination Frequencies of a Molecule with Symmetry 

Number of times the vibration type appears in the character of the combination 

If the character of the overtone is comprised of the characters of several types of vibrations, then a reduction formula similar to (4-30) and (4-31) is used to determine the characters present. We can illustrate this for the 2 overtone calculated above, even though we have already shown that the character of the first 2 overtone is 1,1,1 (i.e., the same as that of an fundamental). 

To show that only the characters are present in the Xo2 overtone, we perform the following calculation  

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