J. Cioslowski et K. Pernal, Constraints upon natural spin orbital functionals imposed by properties ofa homogeneous electron gas, J CHEM PHYS, 111(8), 1999, pp. 3396-3400
The expression V-ee[Gamma(1)] = (1/2)Sigma(p not equal q)[n(p)n(q)J(pq)-Ome
ga(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 exchang
e 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 homogene
ous electron gas. The stability and N-representability of the 1-matrix dema
nd 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 addit
ion, 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) equi
valent to (xy)(beta/2). The GU functional, which corresponds to beta = 1, d
oes not give rise to admissible solutions of the Euler equation describing
a spin-unpolarized homogeneous electron gas of any density. Inequalities va
lid for more general forms of Omega(x,y) are also derived. (C) 1999 America
n Institute of Physics. [S0021-9606(99)30831-X].