Allowed Combination Bands in the Infrared

To determine the combination bands that are allowed in the infrared, it is necessary to form the direct product of the characters of the vibrations making up the combination. For example, if a combination band is due to the sum of fundamentals and V2, the combination band will have a character made up of the direct product of the characters of and v 2. Let us consider a combination band which is due to an and an e fundamental. This com- 

Since the combination band has the same character as an e fundamental, it will be allowed in the infrared. 

Often the character obtained for a combination band is not simply that of a fundamental. In this case, we can use a reduction formula to determine which species of vibrations have characters present in the combination band. For example, let us again consider the combination band + v its character is 2, —1, 0. These values can be reduced by the following formula  

All the terms of this equation were used and defined in connection with the previous reduction formula (4-30), except Xai e which is the character of the combination band. If we substitute into (4-31), the following results are obtained  

We see that only e vibrations make up the character of the combination + Vg. Thus, since e vibrations are allowed in the infrared, the combination band is also allowed. 

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