Constraints on density functionals and density functional derivatives

In order to improve approximations to density functionals such as the exchange–correlation functional, Exc[ρ], and the independent Fermion kinetic energy functional, Ts[ρ], it is essential to have information on the behavior of the exact functionals. Any approximation should then be made to satisfy these conditions. With this in mind, constraints on functionals and functional derivatives which reflect stability against translation, rotation and uniform scaling of the charge density for electrons moving in a fixed external potential, are derived from the minimal property of the energy functional F[ρ]=Ts[ρ]+U[ρ]+Exc[ρ] when evaluated at a v-representable density ρ(r).