Hyperpolarizability residues

The first-order hyperpolarizability has both single and double residues. From Eq. (53), we see that one of the first-order residues becomes... 

I/) = 1 ), we choose the expression in Eq. (56) for the hyperpolarizability and where the secular divergences have been removed. This particular double residue will then become... 

It is also possible to have g) = f) in the double residue. However, this situation corresponds to a secular singularity in the second-order hyperpolarizability and it is therefore appropriate to use Eq. (66) as a starting point ... 

From, for instance, Bloembergen (1965) and Stevens et al. (1963), we see that the quadratic response function is the r , r , r "th component of the dynamic hyperpolarizability tensor, as it should be according to standard electric response theory. The residues of the quadratic response functions provide information about transition moments between two excited states, i.e. neither of the two states is the reference state 0>. More specifically we see from... 

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