Gradient corrections to the local spin density (LSD) approximation for the exchangecorrelation energy are making density functional theory as useful in quantum chemistry as it is in solid-state physics. But which of the many gradient-corrected density functionals should be preferred a priori? We make a graphical comparison of the gradient dependencies of some popular approximations, discussing the exact formal conditions which each obeys and identifying which conditions seem most important. For the exchange energy, there is little formal or practical reason to choose among the Perdew-Wang 86, Becke 88, or Perdew-Wang 91 functionals. But, for the correlation energy, the best formal properties are displayed by the nonempirical rw91 correlation functional. Furthermore, the real-space foundation of rw91 yields an insight into the character of the gradient expansion which suggests that rw91 should work especially well for solids. Indeed, while improving dissociation energies over LSD, rw91 remains the most "local" of the gradient-corrected exchange-correlation functionals and, thus, the least likely to overcorrect the subtle errors of LSD for solids. To show that our analysis of spin-unpolarized functionals is sufficient, we also compute spin-polarization energies for atoms, finding ~w 9 1 values only slightly more negative than SD values. Wiley & Sons, Inc. 0 1996 John exchange-correlation energy Ex,[ n , n , I, a functional of the up-and down-spin electron densities, and its functional derivative, the exchange-correlation potential, vxc, ,(r) = 6Ex,/6n,(r) [2-41. They o h and Sham [l] proved that the exact also proposed the local spin density (LSD) approxiground-state density and energy of a many-mation, E , L S ~[ n , , n , I = / d 3 r n ( r ) ~, , ( n , ( r ) , n (r)), (1)
electron system may be found through the selfconsistent solution of a single-particle problem like that of Hartree-Fock (HF) theory, but with the exchange energy and potential replaced by the where n = n , + n l and E,,(n,, n , ) is the H (1s t)' 1 -0.98 -1.18 Li (2s t)' 1 -0.24 -0.27 Be+' (2s f ) ' 1 -0.49 -0.53 B (2p T I ' 1 -0.26 -0.28 WP t)3(2p 1)' 1 -0.38 -0.39 sc (4s)'(3d f ) ' 1 -0.18 -0.19 C(2P f)' 4 -1.22 -1.28 o (2p t13(2p 1)' 4 -1.43 -1.48 N (2p ?I3 9 -3.09 -3.14 sc (4s f)'(3d r)' 5 -0.99 -1.10 Note that Aβ¬,, -EP:, where P, is the net number of electron spins in a shell with principal quantum number n.