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

Published

Journal Article

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] = 〈λ3N/2Φρλmin(λr1⋯λrN)|T̂(r1⋯rN) + Vee(r1⋯rN) λ3N/2Φρλmin(λ r1⋯λrN)〉,where λ3N/2Ωρλmin(λr1⋯λrN) is that antisymmetric function Ω which yields ρλ(r) = λ3ρ(λr) and simultaneously minimizes 〈Ω|T̂(r1⋯rN) + λVee(r1⋯rN)|Ω〉. The corresponding variational principle is EG.S.v= Infλ,ρ(r){∫drv(r)ρλ(r) + λ2T[ρ(r)] + λVee[ρ(r)]}, where EG.S.vis 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.

### Full Text

### Duke Authors

### Cited Authors

- Levy, M; Yang, W; Parr, RG

### Published Date

- January 1, 1985

### Published In

### Volume / Issue

- 83 / 5

### Start / End Page

- 2334 - 2336

### International Standard Serial Number (ISSN)

- 0021-9606

### Digital Object Identifier (DOI)

- 10.1063/1.449326

### Citation Source

- Scopus