Constraints upon natural spin orbital functionals imposed by properties of a homogeneous electron gas

摘要

The expression V_(ee)[#GAMMA#_1]=(1/2)#SIGMA#_(p not= q)[n_pn_qJ_(pq)-#OMEGA#(n_p,n_q)K_(pq)], where {n_p} are the occupation numbers of natural spin orbitals, and {J_(pq)} and {K_(pq)} are the corresponding Coulomb and exchange integrals, respectively, generalizes both the Hartree-Fock approximation for the electron-electron repulsion energy V_(ee) and the recently introduced Goedecker-Umrigar (GU) functional. Stringent constraints upon the form of the scaling function #OMEGA#(x,y) are imposed by the properties of a homogeneous electron gas. The stability and N-representability of the 1-matrix demand that 2/3<#beta#<4/3 for any homogeneous #GAMMA#(x,y) of degree #beta#[i.e., #OMEGA#(#lambda#x,#lambda#y)ident to #lambda# ~#beta##OMEGA#(x,y)]. In addition, the Lieb-Oxford bound for V_(ee) asserts that #beta#>=#beta#_(crit), where #beta#_(crit) approx= 1.1130, for #OMEGA#(x,y) indet to (xy)~(#beta#/2). The GU functional, which corresponds to #beta#=1, does not give rise to admissible solutions of the Euler equation describing a spin-unpolarized homogeneous electron gas of any density. Inequalities valid for more general forms of #OMEGA#(x,y) are also derived.