Identifying non-zero integrals and selection rules in spectroscopy

2 Identifying non-zero integrals and selection rules in spectroscopy 

The importance of direct products becomes more obvious when we consider some of the integrals we encounter in quantum chemistry. Take the simple case of 

An example of I is the overlap integral with //, being an orbital on one atom and xfjj an orbital on a different atom. Integral I will vanish unless the integrand is invariant under all symmetry operations of the point group to which the molecule belongs. This condition is a generalization of the simple case of 

Returning to integral I, when we say that integrand V- / Ay is invariant to all symmetry operations, it means, in group theory language, the representation for ip fipj (denoted 71/) is or contains TVs - In order for T, - to be or contain TVs, 71 (representation for i/q) and 7 (representation for fy) must be one and the same. In chemical bonding language, for two atomic orbitals to have non-zero overlap, they must have the same symmetry. 

Another integral that appears often in quantum chemistry is the energy interaction integral between orbitals Vq and x/rj 

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