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

A perturbation theory is developed for the correlation energy Ec[n], of a finite-density system, with respect to the coupling constant {α} which multiplies the electron-electron repulsion operator in H^α=T^+{α}V^ₑₑ+tsumᵢ{v}_{{{α}}}$(${r}ᵢ$). The external potential ${v}_{{{α}}}is constrained to keep the gound-state density n fixed for all α≥0.{v}_{{{α}}}is given completely in terms of functional derivatives at full charge (α=1), from which{E}c$[${n}_{{λ}}$]=${e}c,2$[n]+ ${{λ}}^{{{-}}1}ec,3[n]+λ^-2{e}c,4$[n]+..., where each ${e}c,$j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, ${n}_{{λ}}$(x,y,x)=${{λ}}³n(λx,λy,λz) and λ={{{α}}}^{{{-}}1}$. The identification of ${lim}_{{λ}{→}{{∞}}}Ec[n_λ], which is a high-density limit, as the second-order energy ec,2[n] allows one to compute bounds upon lim_λ→∞{E}c$[${n}_{{λ}}$]; the bounds imply that ${lim}_{{λ}{→}{{∞}}}Ec[n_λ]{}Ec[n] for a large class of small atoms and molecules, and suggest that lim_λ→∞{E}c$[${n}_{{λ}}$] should be of the same order of magnitude as ${E}c$[n] in finite insulators and semiconductors.Approximations to ${E}c$[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 Ec[n_λ] 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.