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

The expression V-ee[Gamma(1)] = (1/2)Sigma(p not equal q)[n(p)n(q)J(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 Omega(x,y) of degree beta [i.e., Omega(lambda x,lambda y) equivalent to lambda (beta)Omega(x,y)]. In addition, the Lieb-Oxford bound for V-ee asserts that beta greater than or equal to beta(crit), where beta(crit) approximate to 1.1130, for Omega(x,y) equivalent 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. (C) 1999 American Institute of Physics. [S0021-9606(99)30831-X].