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 α which multiplies the electron-electron repulsion operator in ${\mathit{H}}^{\mathrm{\alpha }}$=T^+αV${\mathrm{^}}_{\mathit{e}\mathit{e}}$+${\mathit{tsum}}_{\mathit{i}}$${\mathit{v}}_{\mathrm{\alpha }}$(${\mathbf{r}}_{\mathit{i}}$). The external potential ${\mathit{v}}_{\mathrm{\alpha }}$ is constrained to keep the gound-state density n fixed for all α≥0. ${\mathit{v}}_{\mathrm{\alpha }}$ is given completely in terms of functional derivatives at full charge (α=1), from which ${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\lambda }$]=${\mathit{e}}_{\mathit{c},2\mathrm{}}$[n]+ ${\lambda }^{\mathrm{-}1}$${\mathit{e}}_{\mathit{c},3\mathrm{}}$[n]+${\lambda }^{\mathrm{-}2}$${\mathit{e}}_{\mathit{c},4\mathrm{}}$[n]+..., where each ${\mathit{e}}_{\mathit{c},}$j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, ${\mathit{n}}_{\lambda }$(x,y,x)=${\lambda }^3$n(λx,λy,λz) and λ=${\mathrm{\alpha }}^{\mathrm{-}1}$. The identification of ${\lim }_{\lambda \to \mathrm{\infty }}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\lambda }$], which is a high-density limit, as the second-order energy ${\mathit{e}}_{\mathit{c},2\mathrm{}}$[n] allows one to compute bounds upon ${\lim }_{\lambda \to \mathrm{\infty }}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\lambda }$]; the bounds imply that ${\lim }_{\lambda \to \mathrm{\infty }}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\lambda }$]≃${\mathit{E}}_{\mathit{c}}$[n] for a large class of small atoms and molecules, and suggest that ${\lim }_{\lambda \to \mathrm{\infty }}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\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 ${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\lambda }$] diverges when λ→∞ because of the presence of the -ln(λ) 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.

• Received 30 November 1992

DOI:https://doi.org/10.1103/PhysRevB.47.13105