Evaluation of alternative sum-over-states expressions for the first hyperpolarizability of push-pull π-conjugated systems

A dipolar-free sum-over-states expression for the diagonal components of the first hyperpolarizability (β) 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 β of four typical push-pull π-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 β 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 π-conjugated systems to estimate the static β, and the error in the value so obtained, by averaging the Kuzyk and traditional results.