Resonant-convergent second-order nonlinear response functions at the levels of Hartree-Fock and Kohn-Sham density functional theory

The second-order nonlinear (or cubic) response function is derived from the Ehrenfest theorem with inclusion made of the finite lifetimes of the excited states, representing the extension of the derivation of the quadratic response function in the same framework [P. Norman et al., J. Chem. Phys. 123, 194103 (2005)]. The resulting damped response functions are physically sound and converging also in near-resonance and resonance regions of the spectrum. Being an accurate approximation for small complex frequencies (defined as the sum of an optical frequency and an imaginary damping parameter), the polynomial expansion of the complex cubic response function in terms of the said frequencies is presented and used to validate the program implementation. In terms of approximate state theory, the computationally tractable expressions of the damped cubic response function are derived and implemented at the levels of Hartree-Fock and Kohn-Sham density functional theory. Numerical examples are provided in terms of studies of the intensity-dependent refractive index of para-nitroaniline and the two-photon absorption cross section of neon. For the latter property, a numerical comparison is made against calculations of the square of two-photon matrix elements that are identified from a residue analysis of the resonance-divergent quadratic response function.