A new functional with homogeneous coordinate scaling in density functional theory: F [ ρ,λ]

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Abstract
As previously shown [M. Levy and J. P. Perdew, Phys. Rev. A (in press)], the customary Hohenberg–Kohn density functional, based on the universal functional F[ ρ], does not exhibit naively expected scaling properties. Namely, if ρλ=λ3ρ(λr) is the scaled density corresponding to ρ(r), the expected scaling, n o t satisfied, is T[ρλ]=λ2 T[ρ] and V[ρλ]=λV[ρ], where T and V are the kinetic and potential energy components. By defining a new functional of ρ a n d λ, F[ ρ, λ], it is now shown how the naive scaling can be preserved. The definition is F[ρ(r), λ]=〈λ3N/2 Φmin ρλ (λr 1... λr N )|T̂(r 1...r N ) +V e e (r 1...r N )| λ3N/2Φmin ρλ(λr 1...λr N )〉, where λ3N/2 Φmin ρλ (λr 1... λr N ) is that antisymmetric function Φ which yields ρλ(r)=λ3ρ(λr) and simultaneously minimizes 〈Φ|T̂(r 1...r N ) +λV e e (r 1...r N )|Φ〉. The corresponding variational principle is E v G.S.=Infλ, ρ(r){∫ d r v(r) ρλ(r)+λ2 T[ ρ(r)] +λV ee[ ρ(r)]}, where E v G.S. is the ground‐state energy for potential v(r). One is thus allowed to satisfy the virial theorem by optimum scaling just as if the naive scaling relations were correct for F[ ρ].