State energy functionals and variational equations in density functional theory

Abstract

To resolve certain physical and mathematical problems in the customary Levy–Lieb constrained search formulation of Density Functional Theory (DFT), especially its spin-polarized form, we have presented elsewhere a reformulation of both time-independent and time-dependent DFT. The analysis is based on the full space–spin density and some of its implications are most accessible via expansions in terms of General Spin Orbitals (GSOs). The approach defines a universal energy functional on equivalence classes of N-fermion states labelled by those full space–spin densities via the constrained search technique. Here we apply theorems from constrained optimization theory to show that such a universal functional is well-defined (not multi-valued) on paths of N-particle states as long as certain first and second derivative conditions are satisfied. These, we demonstrate, are equivalent to V-representability conditions. Two types of one-particle GSO equations are derived for the ground state density, one involving model states in the spirit of Kohn–Sham, the other not involving model states. The relationship between DFT based upon the space–spin density with the more common forms based on the spatial density alone is examined with respect to symmetry breaking solutions of the one-particle equations.