TIGHT BOUND AND CONVEXITY CONSTRAINT ON THE EXCHANGE-CORRELATION-ENERGY FUNCTIONAL IN THE LOW-DENSITY LIMIT, AND OTHER FORMAL TESTS OF GENERALIZED-GRADIENT APPROXIMATIONS

For the purpose of approximating the exact density-functional exchange-correlation energy E(xc)[ n ], we previously established the coordinate scaling identity E(xc)alpha[n]=alpha2E(xc)[n(lambda)] for alpha=lambda-1, or E(xc)lambda[n(lambda)]=lambda2E(xc)[n], where E(xc)alpha[n] is the exchange-correlation energy functional for electronic charge square-root alphae and where n(lambda)(x,y,z)=lambd3n(lambdax,lambday,lambdaz). This identity is utilized here to derive the low-density limit lim(lambda-->0)lambda-1E(xc)[n(lambda)]=inf(PSI-->n) [PSI\V(ee)\PSI]-(e2/2)integral integral [n(r1)n(r2)/\r1-r2\d3r1d3r2, which allows us to express the Lieb-Oxford bound in the tighest-possible manner, namely, lim(lambda-->0)lambda-1E(xc)[n(lambda)] greater-than-or-equal-to - Ce2 integral n4/3(r)d3r, where 1.43 less-than-or-equal-to C less-than-or-equal-to 1.68. Meaningful adherences to and violations of the bound are presented to demonstrate that it is surprisingly tight and thus quite useful. Other key properties of lim(lambda-0)lambda-1E(xc)[n(lambda)] are found, including the observation that lim(lambda-->0)lambda-1E(xc)[n(lambda)]+(e2/2) integral integral [n(r1)n(r2)/\r1-r2\]d3r1d2r2 is convex, which is a severe constraint. We discuss these and other exact relationships as formal tests of generalized-gradient approximations (GGA's) for exchange and correlation. We find that the Perdew-Wang 1991 (PW91) GGA respects many of the known exact relationships, including those respected by the local-density approximation plus many others that are violated by the local-density approximation. We present a minor variant of the PW91 correlation-energy functional which additionally satisfies a strong lambda --> infinity (high-density) constraint. Finally, we show that atomic densities are much closer to the high-density than to the low-density limit.