Expansions of the correlation-energy density functional Ec[ρ] and its kinetic-energy component Tc[ρ] in terms of homogeneous functionals

Abstract

Based on the constrained-search formulation of a coupled Hamiltonian, new formulas are given involving the correlation-energy density functional ${\mathit{E}}_{\mathit{c}}$[ρ] and its kinetic component ${\mathit{T}}_{\mathit{c}}$[ρ], starting from two relations of Levy and Perdew. Consequences are examined of assuming (following earlier authors) the existence of a Taylor series expansion of ${\mathit{E}}_{\mathit{c}}^{\lambda }$[ρ] in the coupling parameter λ. If one truncates the series at the linear term, one finds that ${\mathit{E}}_{\mathit{c}}$[ρ] and ${\mathit{T}}_{\mathit{c}}$[ρ] are homogeneous of degree zero with respect to coordinate scaling, and if local, they are homogeneous of degree one in the density scaling. More generally, it is shown that ${\mathit{E}}_{\mathit{c}}$[ρ] and ${\mathit{T}}_{\mathit{c}}$[ρ] are linear combinations of homogeneous functionals of different specific degrees in coordinate scaling: 0,-1,-2,-3,...,(1-n),... . If the functionals also are local, both ${\mathit{E}}_{\mathit{c}}$[ρ] and ${\mathit{T}}_{\mathit{c}}$[ρ] are combinations of functionals 〈${\mathrm{\rho }}^{\mathit{k}}$〉 homogeneous in ρ of degrees k=1,2/3,1/3,0,...,(4-n)/3,... . For atoms and molecules, k≥0, and so ${\mathit{E}}_{\mathit{c}}$ and ${\mathit{T}}_{\mathit{c}}$ take the form ${\mathit{X}}_{\mathit{c}}$[ρ]=aN+b∫${\mathrm{\rho }}^{2/3}$(r)${\mathit{d}}^3$r+c∫${\mathrm{\rho }}^{1/3}$(r) ${\mathrm{d}}^3$r+d, where a, b, c, and d are constants to be determined. Numerical tests are given that demonstrate the effectiveness of such series of local functionals. We also give definitions of density scaling, coordinate scaling, and homogeneities, and relations among them. © 1996 The American Physical Society.