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

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Abstract
The expression Vee[Γ1]=(1/2)∑p≠q[npnqJpq−Ω(np,nq)Kpq], where {np} are the occupation numbers of natural spin orbitals, and {Jpq} and {Kpq} are the corresponding Coulomb and exchange integrals, respectively, generalizes both the Hartree–Fock approximation for the electron–electron repulsion energy Vee and the recently introduced Goedecker–Umrigar (GU) functional. Stringent constraints upon the form of the scaling function Ω(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<β<4/3 for any homogeneous Ω(x,y) of degree β [i.e., Ω(λx,λy)≡λβΩ(x,y)]. In addition, the Lieb–Oxford bound for Vee asserts that β⩾βcrit, where βcrit≈1.1130, for Ω(x,y)≡(xy)β/2. The GU functional, which corresponds to β=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 Ω(x,y) are also derived.