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 evict density-functional exchange-correlation energy E sc [n], we previously established the coordinate scaling identity E a xc [n]=α 2 E xc [n λ ] for α=λ -1 , or E λ xc [n λ ]=λ 2 E xc [n], where E α xc [n] is the exchange-correlation energy functional for electronic charge √αe and where n λ (x,y,x)=λ 3 n(λx,λy,λz)