Differences between the exact sum-over-states and the canonical approximation for the calculation of static and dynamic hyperpolarizabilities

When molecules are subjected to static or dynamic uniform electric fields the effects are conveniently described in terms of dipole polarizabilities and hyperpolarizabilities. The fields act on both the electronic and vibrational motions simultaneously and, through perturbation theory, this leads to the exact sum-over-states (SOS) formulas for the (hyper) polarizabilities expressed in terms of vibronic states. As an approximation, the fields can be considered to act on the electronic motions first and subsequently, through the distorted electronic ground state potential surface, on the vibrational motions. This commonly used canonical or clamped nucleus approximation, leads to (hyper) polarizability expressions that are different from the SOS ones. In this paper we develop a compact first-order treatment of the differences and carry out a preliminary examination of the numerical repercussions. Our calculations also bear on a proposed theoretical relationship between the vibrational and electronic first hyperpolarizabilities in push-pull conjugated systems.