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

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, not satisfied, is Tρλ=λ2Tρ and Vρλ=λVρ, where T and V are the kinetic and potential energy components. By defining a new functional of ρ and λ, F ρ, λ, it is now shown how the naive scaling can be preserved. The definition is Fρ (r), λ=〈λ3N/2 Φminρλ (λr1. . . λrN) |T̂ (r1. . . rN) +Vee (r1. . . rN) | λ3N/2Φminρλ (λr1. . . λrN) 〉, where λ3N/2 Φminρλ (λr1. . . λrN) is that antisymmetric function Φ which yields ρλ (r) =λ3ρ (λr) and simultaneously minimizes 〈Φ|T̂ (r1. . . rN) +λVee (r1. . . rN) |Φ〉. The corresponding variational principle is EvG. S. =Infλ, ρ (r) ∫ drv (r) ρλ (r) +λ2T ρ (r) +λVee ρ (r), where EvG. 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.