Bounds for the exchange and correlation potentials - Levy, Mel; Görling, Andreas (Physical Review A, 1995)
Knowledge of bounds and equalities for the exact density-functional exchange-correlation potential δExc[n]/δn(r) is necessary for its accurate approximation. With this in mind, it is shown, for λ→0, that λ-1Fd3rn(r)δExc[nλ]/ δn(r)2λ-1Exc[nλ] and Fn(r')r-r'-1d3r' +λ-1δExc[nλ]/δn(r)0, where nλ(x,y,z)=λ3n(λx,λy,λz). The local-density approximation satisfies the former inequality but violates the latter one. Moreover, with respect to the Fermi level, it is shown that the exact correlation potential δEc[n]/δn(r) satisfies Ec[n]-Ec[n-ΔnF]≤FδEc[n]/ δn(r)ΔnF(r)d3r, where ΔnF is the density of the highest-occupied Kohn-Sham orbital of n. The corresponding inequality for the exact exchange potential δEx[n]/δn(r) is in the opposite direction: Ex[n]-Ex[n-ΔnF]FδEx[n]/ δn(r)ΔnF(r)d3r. It is a difficult challenge for an approximate exchange-correlation functional to simultaneously satisfy both inequalities. For instance, the local-density approximation does not. © 1995 The American Physical Society.