Levy, M; Yang, W; Parr, RG, *A new functional with homogeneous coordinate scaling in density functional theory: F[ρ, λ]*,
The Journal of Chemical Physics, vol. 83 no. 5
(1985),
pp. 2334-2336 .
**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, 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 isF[ρ(r),λ] = 〈λ 3N/2Φρλmin(λr 1⋯λrN)|T̂(r1⋯r N) + Vee(r1⋯rN) λ3N/2Φρλmin(λ r1⋯λrN)〉,where λ3N/2 Ωρλmin(λr 1⋯λrN) is that antisymmetric function Ω which yields ρλ(r) = λ3ρ(λr) and simultaneously minimizes 〈Ω|T̂(r1⋯r N) + λVee(r1⋯r N)|Ω〉. The corresponding variational principle is E G.S.v = Infλ,ρ(r) {∫drv(r)ρλ(r) + λ2T[ρ(r)] + λV ee[ρ(r)]}, where EG.S.v is the ground-state energy for potential v(r). One is thus allowed to lower the energy and satisfy the virial theorem by optimum scaling just as if the naive scaling relations were correct for F[ρ]. © 1985 American Institute of Physics.*