Nonuniform coordinate scaling requirements for exchange-correlation energy.

摘要: 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 generally true ${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{x}}$]\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 ${\mathit{E}}_{\mathrm{xc}}$ exact exchange-correlation energy. A corresponding inequality also holds correlation component of when defined in one meaningful ways. In contrast, equalities always hold local-density approximations to these functionals. other words, exchange alone do not distinguish between nonuniform scaling along different coordinates.