Evaluation of alternative sum-over-states expressions for the first hyperpolarizability of push-pull pi-conjugated...

A dipolar-free sum-over-states expression for the diagonal components of the first hyperpolarizability (beta) tensor has been proposed by Kuzyk [Phys. Rev. A 72, 053819 (2005)] as an alternative to the traditional expression. We examine both alternatives for the longitudinal beta of four typical push-pull pi-conjugated systems using the ab initio CIS and CIS(D) schemes to approximate the excited state properties. Since they are each evaluated approximately the two SOS expressions yield different values for beta and it is found that (i) they evolve symmetrically as the number of excited states is increased so that their average is nearly constant; (ii) in the static limit, the two values agree better with one another when their average is close to the "exact" correlated result; and (iii) frequency dispersion can affect the agreement between the alternative expressions. On the basis of (i) and (ii) it appears best for typical push-pull pi-conjugated systems to estimate the static beta, and the error in the value so obtained, by averaging the Kuzyk and traditional results.