where U [ρ] = 1/2 ρ(r)v H (r)dr is the Hartree energy. By Eq. 10, MRF reduces approximating E xc [ρ] by finding σ λ i . The underlying idea is that is much easier to approximate σ λ (r) than the corresponding energy densities as the former is bounded in between -1 and 1, changes little locally in a given system and across systems, and can be represented through very simple mathematical forms. For example, setting σ λ=1 i = 0, already gives accurate atomic energy densities at the full coupling strength. In Refs. the following form for σ λ=1 i has been considered: