Correlation-energy functional and its high-density limit obtained from a coupling-constant perturbation expansion

Abstract

A perturbation theory is developed for the correlation energy ${\mathit{E}}_{\mathit{c}}$[n], of a finite-density system, with respect to the coupling constant \ensuremath{\alpha} which multiplies the electron-electron repulsion operator in ${\mathit{H}}^{\mathrm{\ensuremath{\alpha}}}$=T^+\ensuremath{\alpha}V${\mathrm{^}}_{\mathit{e}\mathit{e}}$+${\mathit{tsum}}_{\mathit{i}}$${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$(${\mathbf{r}}_{\mathit{i}}$). The external potential ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is constrained to keep the gound-state density n fixed for all \ensuremath{\alpha}\ensuremath{\ge}0. ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is given completely in terms of functional derivatives at full charge (\ensuremath{\alpha}=1), from which ${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]=${\mathit{e}}_{\mathit{c},2}$[n]+ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$${\mathit{e}}_{\mathit{c},3}$[n]+${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}2}$${\mathit{e}}_{\mathit{c},4}$[n]+..., where each ${\mathit{e}}_{\mathit{c},}$j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, ${\mathit{n}}_{\ensuremath{\lambda}}$(x,y,x)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z) and \ensuremath{\lambda}=${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}$. The identification of ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$], which is a high-density limit, as the second-order energy ${\mathit{e}}_{\mathit{c},2}$[n] allows one to compute bounds upon ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]; the bounds imply that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]\ensuremath{\simeq}${\mathit{E}}_{\mathit{c}}$[n] for a large class of small atoms and molecules, and suggest that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$] should be of the same order of magnitude as ${\mathit{E}}_{\mathit{c}}$[n] in finite insulators and semiconductors.Approximations to ${\mathit{E}}_{\mathit{c}}$[n] should reflect all this. In contrast, perhaps the well-known overbinding of the local-density approximation (LDA) in molecules and solids is due, in part, to the fact that the LDA correlation energy is too sensitive to a coordinate scaling of n. Indeed, the LDA for ${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$] diverges when \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} because of the presence of the -ln(\ensuremath{\lambda}) term in the Gell-Mann and Brueckner high-density expression for the correlation energy, per particle, of a homogeneous density, which is infinite. In a sense, the derived perturbation expansion transforms the Gell-Mann and Brueckner expression into one that applies specifically to an inhomogeneous density which integrates to a finite number of electrons.