Bounds for the exchange and correlation potentials

Knowledge of bounds and equalities for the exact density-functional exchange-correlation potential δ${\mathit{E}}_{\mathrm{xc}}$[n]/δn(r) is necessary for its accurate approximation. With this in mind, it is shown, for λ→0, that ${\lambda }^{\mathrm{-}1}$ F ${\mathit{d}}^3$rn(r)δ${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }$]/ δn(r)≥2${\lambda }^{\mathrm{-}1}$ ${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }$] and Fn(r ’)‖r-r ’ ${\mathrm{\Vert }}^{\mathrm{-}1}$ ${\mathit{d}}^3$r’ +${\lambda }^{\mathrm{-}1}$δ${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }$]/δn(r)≥0, where ${\mathit{n}}_{\lambda }$(x,y,z)=${\lambda }^3$n(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 δ${\mathit{E}}_{\mathit{c}}$[n]/δn(r) satisfies ${\mathit{E}}_{\mathit{c}}$[n]-${\mathit{E}}_{\mathit{c}}$[n-Δ${\mathit{n}}_{\mathit{F}}$]≤Fδ${\mathit{E}}_{\mathit{c}}$[n]/ δn(r)Δ${\mathit{n}}_{\mathit{F}}$(r)${\mathit{d}}^3$ r, where Δ${\mathit{n}}_{\mathit{F}}$ is the density of the highest-occupied Kohn-Sham orbital of n. The corresponding inequality for the exact exchange potential δ${\mathit{E}}_{\mathit{x}}$[n]/δn(r) is in the opposite direction: ${\mathit{E}}_{\mathit{x}}$[n]-${\mathit{E}}_{\mathit{x}}$[n-Δ${\mathit{n}}_{\mathit{F}}$]≥Fδ