Efficient calculation of the density response function from generalized polarizabilities

Abstract

We present a method to calculate the density response function (DRF), also known as the density response kernel or the polarization propagator in the static limit. Our method uses generalized polarizabilities (GPs) which are second derivatives of the molecular energy with respect to two arbitrary perturbations of the external electrostatic potential. They are generalizations of the common multipole polarizabilities. The latter use solid spherical harmonics as perturbing potentials, while GPs can use any function. We use a sine function expansion of the electrostatic potential. Generalized polarizabilities were originally introduced for a different project, ultrafast quantum/molecular mechanics calculations. By transforming the GPs to a (discretized) direct space representation, we obtain the DRF in the static limit. The method has been implemented for single-determinant (Hartree–Fock and density functional theory) wavefunctions, but can be generalized to more accurate wavefunctions. The number of coupled-perturbed self-consistent field (CP-SCF) calculations in our method is proportional to the molecular volume at a given spatial resolution, i.e., scales linearly with the system size, and is independent of the basis set size. The best previous method has the number of CP-SCF calculations proportional to the product of the number occupied and virtual orbitals, and therefore scaling quadratically with the system size at constant basis set quality. The diagonal elements of the DRF (local polarizabilities) of water and butadiene in the molecular plane are displayed. An example of non-diagonal DRF is presented for butadiene 0.77 Å above the molecular plane, with one point fixed over the midpoint of a C=C bond. We obtain a fairly localized DRF, even in this conjugated system. The values become quite small if the distance of the two local perturbations exceeds a bond length. By using frequency-dependent polarizabilities, one could readily calculate the dynamic DRF.