Virial fragments and the Hohenberg–Kohn functional

Starting from the Hohenberg‐Kohn functional we show that when the energy density is given as a function of ρ and ∇_ρ, i.e., ξ = ξ(ρ, ∇_ρ), the condition ∇_ρ · n = 0 (which was found by Bader et al. to define virial fragments), appears as a natural boundary condition for the variation of this functional. We also show that when the energy density includes second order derivatives (∇²_ρ) this condition is necessary but not sufficient to guarantee the vanishing of the variation. The implications of these results are discussed in the context of a density functional theory for virial fragments.