Correlation-energy functional and its high-density limit obtained from a coupling-constant perturbation expansion - Görling, Andreas; Levy, Mel (Physical Review B, 1993)
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 Hα=T+αVee+tsumivα(ri). The external potential vα is constrained to keep the gound-state density n fixed for all α0. vα is given completely in terms of functional derivatives at full charge (α=1), from which Ec[nλ]=ec,2[n]+ λ-1ec,3[n]+λ-2ec,4[n]+..., where each ec,j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, nλ(x,y,x)=λ3n(λx,λy,λz) and λ=α-1. The identification of limλ→Ec[nλ], which is a high-density limit, as the second-order energy ec,2[n] allows one to compute bounds upon limλ→Ec[nλ]; the bounds imply that limλ→Ec[nλ]Ec[n] for a large class of small atoms and molecules, and suggest that limλ→Ec[nλ] should be of the same order of magnitude as Ec[n] in finite insulators and semiconductors. Approximations to Ec[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[nλ] 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. © 1993 The American Physical Society.