Variational, V-representable, and variable-occupation-number perturbation theories

Abstract

Density-functional perturbation theory with variationally fitted Kohn-Sham (KS) potentials is described. Requiring the Fock matrix and density matrix to commute through each order of perturbation theory determines the off-diagonal elements of the density matrix, and thus the effect of changing occupation numbers in density-functional perturbation theory. At each order of perturbation theory, the change in occupation numbers at that order enters only the diagonal part of the density matrix. The theory contains no phases, and a limiting process relates the rest of the diagonal density matrix element, obtained from wave function perturbation theory, to the off-diagonal part, obtained by commutation. V-representable density-functional theory is most practical when the KS potential is expanded in a finite basis to create the Sambe-Felton (SF) potential of analytic density-functional theory. This reduces the dimensionality of perturbation theory from order N(2) in the orbital basis to order N in the SF basis. Computing the (occupied-virtual)(2), i.e., N(4), sum over states once at the end of a self-consistent-field molecular orbital calculation removes the orbitals from all higher orders of perturbation theory. The rank-N(2) iterative coupled-perturbed equations are replaced by rank-N matrix inversion, to fit variationally the perturbed SF potential at each order. As an example of the 2n+1 rule of perturbation theory, the variational, first-order potential is used to give precise second and third derivatives of the energy with respect to occupation number. The hardness and hyperhardness are computed for a standard set of molecules. Both are essentially independent of how the variational SF potential is constrained for four different constraint combinations. With variational fitting, the precision of derivatives and the fidelity of the fit to the SF potential are not related. Analytic derivatives are accurate to machine precision for any constraint and all fitting basis sets.