State energy functionals and variational equations in density functional theory

To resolve certain physical and mathematical problems in the customary Levy -Lieb constrained search formulation of Density Functional Theory (DFT), es pecially its spin-polarized form, we have presented elsewhere a reformulati on of both time-independent and time-dependent DFT. The analysis is based o n the full space-spin density and some of its implications are most accessi ble via expansions in terms of General Spin Orbitals (GSOs). The approach d efines a universal energy functional on equivalence classes of N-fermion st ates 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-repres entability conditions. Two types of one-particle GSO equations are derived for the ground state density, one involving model states in the spirit of K ohn-Sham, the other not involving model states. The relationship between DF T based upon the space-spin density with the more common forms based on the spatial density alone is examined with respect to symmetry breaking soluti ons of the one-particle equations. (C) 2000 Elsevier Science B.V. All right s reserved.