Static nonlinear optical susceptibilities: testing approximation schemes against exact results

The reliability of the approximations commonly adopted in the calculation of static optical (hyper) polarizabilities is tested against exact results obtained for an interesting toy-model. The model accounts for the principal features of typical nonlinear organic materials with mobile electrons strongly coupled to molecular vibrations. The approximations introduced in sum over states and finite field schemes are analyzed in detail. Both the Born–Oppenheimer and the clamped nucleus approximations turn out to be safe for molecules, whereas for donor–acceptor charge transfer complexes deviations from adiabaticity are expected. In the regime of low vibrational frequency, static susceptibilities are strongly dominated by the successive derivatives of the potential energy and large vibrational contributions to hyperpolarizabilities are found. In this regime anharmonic corrections to hyperpolarizabilities are very large, and the harmonic approximation, exact for the linear polarizability, turns out totally inadequate for nonlinear responses. With increasing phonon frequency the role of vibrations smoothly decreases, until, in the antiadiabatic (infinite vibrational frequency) regime, vibrations do not contribute anymore to static susceptibilities, and the purely electronic responses are regained.