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 T[ρλ]=λ2T[ρ] and V[ρλ]=λV[ρ], 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 F[ρ(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].

Levy, M., Yang, W., & Parr, R. G. (1985). A new functional with homogeneous coordinate scaling in density functional theory: F [ ρ,λ]. The Journal of Chemical Physics, 83(5), 2334–2336.