Construction of a generalized gradient approximation by restoring the density-gradient expansion and enforcing a tight Lieb-Oxford bound

Recently, a generalized gradient approximation (GGA) to the density functional, called PBEsol, was optimized (one parameter) against the jellium-surface exchange-correlation energies, and this, in conjunction with changing another parameter to restore the first-principles gradient expansion for exchange, was sufficient to yield accurate lattice constants of solids. Here, we construct a new GGA that has no empirical parameters, that satisfies one more exact constraint than PBEsol, and that performs 20% better for the lattice constants of 18 previously studied solids, although it does not improve on PBEsol for molecular atomization energies (a property that neither functional was designed for). The new GGA is exact through second order, and it is called the second-order generalized gradient approximation (SOGGA). The SOGGA functional also differs from other GGAs in that it enforces a tighter Lieb-Oxford bound. SOGGA and other functionals are compared to a diverse set of lattice constants, bond distances, and energetic quantities for solids and molecules (this includes the first test of the M06-L meta-GGA for solid-state properties). We find that classifying density functionals in terms of the magnitude mu of the second-order coefficient of the density gradient expansion of the exchange functional not only correlates their behavior for predicting lattice constants of solids versus their behavior for predicting small-molecule atomization energies, as pointed out by Perdew and co-workers Phys. Rev. Lett. 100, 134606 (2008); Perdew ibid. 80, 891 (1998)], but also correlates their behavior for cohesive energies of solids, reaction barriers heights, and nonhydrogenic bond distances in small molecules.