Density-functional exchange correlation through coordinate scaling in adiabatic connection and correlation hole

Abstract

The exact exchange-correlation functional ${\mathit{E}}_{\mathrm{xc}}$[n] must be approximated in density-functional theory for the computation of electronic properties. By the coupling-constant integration (adiabatic-connection) formula we know that ${\mathit{E}}_{\mathrm{xc}}$[n]=${\mathcal{F}}_{0}^{1}$(${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n]-U[n])d\ensuremath{\alpha}, where ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n] is the electron-electron repulsion energy of ${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}^{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{\ensuremath{\alpha}}}$, which is that wave function that yields the density n and minimizes 〈T^+\ensuremath{\alpha}V${\mathrm{^}}_{\mathit{e}\mathit{e}}$〉. Here \ensuremath{\alpha} is the coupling constant. Consequently, knowledge of the behavior of ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n] as a function of \ensuremath{\alpha} ensures knowledge of ${\mathit{E}}_{\mathrm{xc}}$[n]. With this in mind and for the purpose of approximating ${\mathit{E}}_{\mathrm{xc}}$, it was previously established that (\ensuremath{\partial}${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\alpha})\ensuremath{\le}0. The present paper reveals that ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n]=\ensuremath{\alpha}${\mathit{V}}_{\mathit{e}\mathit{e}}^{1}$[${\mathit{n}}_{1/\mathrm{\ensuremath{\alpha}}}$], where ${\mathit{n}}_{\mathrm{\ensuremath{\beta}}}$(x,y,z)=${\mathrm{\ensuremath{\beta}}}^{3}$n(\ensuremath{\beta}x,\ensuremath{\beta}y,\ensuremath{\beta}z), and where \ensuremath{\beta} is a coordinate scale factor.In other words, knowledge of ${\mathit{V}}_{\mathit{e}\mathit{e}}^{1}$[n] implies knowledge of ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n] for all \ensuremath{\alpha}. Alternatively, knowledge of ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n] for some small \ensuremath{\alpha} implies knowledge of all of the ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n]. In any case, any viable approximation to ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n] should be made to satisfy the above displayed equality. Analogous conclusions hold for the second-order density matrix, the pair-correlation function, the exchange-correlation hole, and the correlation component of the exchange-correlation hole, etc. For example, ${\mathrm{\ensuremath{\rho}}}_{\mathrm{xc}}$([n,\ensuremath{\alpha}];${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$)=${\mathrm{\ensuremath{\alpha}}}^{3}$${\mathrm{\ensuremath{\rho}}}_{\mathrm{xc}}$([${\mathit{n}}_{1/\mathrm{\ensuremath{\alpha}}}$,1]; \ensuremath{\alpha}${\mathbf{r}}_{1}$,\ensuremath{\alpha}${\mathbf{r}}_{2}$), where ${\mathrm{\ensuremath{\rho}}}_{\mathrm{xc}}$([n,\ensuremath{\alpha}]; ${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$) is the exact exchange-correlation hole of ${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}^{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{\ensuremath{\alpha}}}$.(A corresponding expression holds for the correlation hole alone.) Further, when n belongs to a noninteracting ground state that is nondegenerate, then ${\mathrm{lim}}_{\mathrm{\ensuremath{\alpha}}\ensuremath{\rightarrow}0}$ ${\mathrm{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\ensuremath{\alpha}}}$[n]=A[n]+${\mathit{f}}_{\mathit{n}}$(\ensuremath{\alpha})B[n]+..., where ${\mathit{f}}_{\mathit{n}}$(\ensuremath{\alpha}) must vanish at least as rapidly as \ensuremath{\alpha}, and ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$ ${\mathrm{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]>-\ensuremath{\infty}, where ${\mathit{E}}_{\mathit{c}}$[n]=${\mathit{E}}_{\mathrm{xc}}$[n]- ${\mathrm{lim}}_{\ensuremath{\gamma}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$ ${\ensuremath{\gamma}}^{\mathrm{\ensuremath{-}}1}$${\mathit{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\gamma}}$], and where ${\mathit{E}}_{\mathit{c}}$ is a familiar exact density-functional ``correlation energy.'' In contrast, in the local-density approximation and in certain nonlocal approximations, ${\mathit{f}}_{\mathit{n}}$(\ensuremath{\alpha}) is replaced by a function that goes as \ensuremath{\alpha}[ln(${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}$)], \ensuremath{\alpha}\ensuremath{\rightarrow}0, and ${\mathit{E}}_{\mathit{c}}$ is replaced by a functional that is unbounded as \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}. Further, ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$ ${\mathrm{E}}_{\mathrm{xc}}$[${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{x}}$]>-\ensuremath{\infty} and ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$ ${\mathit{E}}_{\mathit{x}}$[${\mathit{n}}_{\ensuremath{\lambda}}^{\mathit{x}}$]>-\ensurema