Using Symmetry in Integrals

Although equation 13.8 is somewhat general, it does not cover all cases. For example, in symmetry species that have E or T labels, multiplication of the irreducible representations yields a reducible representation that must be reduced using the great orthogonality theorem. In such cases, the integral is identically zero unless the reducible representation can be broken down into irreducible representations, one of which must be A (or whatever the totally symmetric representation is). Such a reducible representation is said to contain A. Mathematically, this is written as 

Determine whether the following integrals are exactly zero or might be nonzero strictly from symmetry considerations, f dr in a molecule 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

In each case, the characters of the irreducible representations must be multiplied together and the product evaluated for the presence of the totally symmetric representation of the respective point group. 

This is not an irreducible representation and so must be broken down using the GOT. One can show by applying equation 13.6 that 

---