Quadratic response theory of frequency-dependent first hyperpolarizability. Calculations in the dipole length and mixed-velocity formalisms

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Abstract
The quadratic response function (QRF) is evaluated within the random phase approximation (RPA), to compute frequency-dependent first hyperpolarizabilities β(ω,ω). The method treats electron correlation consistent through first order, so the computed values are equivalent to coupled-perturbed Hartree–Fock (CPHF) results. The QRF is obtained by solving systems of linear equations, thus circumventing the RPA eigenvalue problem. The QRF equation of motion is used to develop hyperpolarizability identities in the dipole length and mixed-velocity representations. The two forms of β are equivalent at the RPA level, and provide a useful measure of completeness of basis. The method is applied to the hyperpolarizability of HF and H2O. It is found that basis sets used in previous studies were not saturated for all β components, and that basis sets which satisfy length–velocity sum rules for linear response properties are not sufficient for agreement of quadratic response properties. The calculated dispersion ratios are in good agreement with experimental measurement, indicating that dispersion effects are properly described by frequency-dependent calculations in the RPA at field energies which are small compared to vertical excitation energies.