Nonuniform coordinate scaling requirements for exchange-correlation energy

Abstract

Consider the nonuniformly scaled electron density ${\mathit{n}}_{\lambda }^{\mathit{x}}$(x,y,z)=λn(λx,y,z), with analogous definitions for ${\mathit{n}}_{\lambda }^{\mathit{y}}$ and ${\mathit{n}}_{\lambda }^{\mathit{z}}$. It is shown that it is generally true that ${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }^{\mathit{x}}$]≠${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }^{\mathit{y}}$]≠${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\lambda }^{\mathit{z}}$], where ${\mathit{E}}_{\mathrm{xc}}$ is the exact exchange-correlation energy. A corresponding inequality also holds for the correlation component of ${\mathit{E}}_{\mathrm{xc}}$ when the correlation component is defined in one of the meaningful ways. In contrast, equalities always hold for the local-density approximations to these exact functionals. In other words, the local-density approximations for exchange correlation and for correlation alone do not distinguish between nonuniform scaling along different coordinates.