Abstract
Consider the nonuniformly scaled electron density ${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{x}}$(x,y,z)=\ensuremath{\lambda}n(\ensuremath{\lambda}x,y,z), with analogous definitions for ${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{y}}$ and ${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{z}}$. It is shown that it is generally true that ${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{x}}$]\ensuremath{\ne}${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{y}}$]\ensuremath{\ne}${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{z}}$], 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.
Published Version
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