Calculation of static zero-point vibrational averaging corrections and other vibrational curvature contributions to polarizabilities and hyperpolarizabilities using field-induced coordinates

The zero-point static vibrational averaging (ZPVA) correction to the (hyper)polarizability is written in first-order as the sum of two contributions, one involving electric field derivatives and the other a sum over normal coordinate derivatives of the zero-point energy. It is shown that the sum over 3N-6 normal modes can be replaced by a single term using field-induced coordinates (FICs). A computational strategy that takes advantage of this simplification is presented and applied to a typical push-pull polyene NH₂-(CH = CH)₃-NO₂. From the dependence of the first-order ZPVA on the field-dependent equilibrium geometry, we also obtain other low-order static and dynamic vibrational curvature contributions to the (hyper)polarizabilities. The entire set of electronic and vibrational terms is partitioned into two different sequences, each of which exhibits rapid initial convergence for NH₂(CH = CH)₃NO₂ at the Hartree-Fock/6-31G+p level. Including electron correlation at the second-order Moller-Plesset (MP2) level, and the frequency dependence of the ZPVA correction is discussed.