Consequences for finite electronic systems of homogeneity properties of density-functional energy components

For finite electronic systems, the proposal is explored that the functional expansions of the universal Hohenberg–Kohn functionals T[ρ] and Vee[ρ] in terms of their own functional derivatives truncate at the first- and second-order levels, respectively. It is shown that then the universal functional F[ρ]=T[ρ]+Vee[ρ] can be expressed as the sum of functionals that are homogeneous of degrees one and two in the density, and the variational principle, which determines the ground-state density and total electronic energy E0 for an N-electron system having external potential v0, takes the form 〈ρv0〉+〈ρδF/δρ〉−1/2〈ρρδ2F/δρδρ〉≥E0. Knowledge of δF[ρ]/δρ(r) and δ2F[ρ]/δρ(r)δρ(r′), without knowledge of F[ρ] itself, accordingly may suffice for the variational determination of the electron density and total energy. A simple energy formula is derived if v0 is Coulombic, namely, (3N−1)E=(N−1)∑ϵi+NVne[ρ], where the ϵi are Kohn–Sham orbital energies and Vne is the nuclear–electron attraction energy. Numerical tests are reported for atoms, and a small correction to the formula is suggested.