Successful modern generalized gradient approximations (GGA's) are biased toward atomic energies. Restoration of the first-principles gradient expansion for exchange over a wide range of density gradients eliminates this bias. We introduce PBEsol, a revised Perdew-Burke-Ernzerhof GGA that improves equilibrium properties of densely-packed solids and their surfaces.

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We present the case for the nonempirical construction of density functional approximations for the exchange-correlation energy by the traditional method of "constraint satisfaction" without fitting to data sets, and present evidence that this approach has been successful on the first three rungs of "Jacob's ladder" of density functional approximations [local spin-density approximation (LSD), generalized gradient approximation (GGA), and meta-GGA]. We expect that this approach will also prove successful on the fourth and fifth rungs (hyper-GGA or hybrid and generalized random-phase approximation). In particular, we argue for the theoretical and practical importance of recovering the correct uniform density limit, which many semiempirical functionals fail to do. Among the beyond-LSD functionals now available to users, we recommend the nonempirical Perdew-Burke-Ernzerhof (PBE) GGA and the nonempirical Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA, and their one-parameter hybrids with exact exchange. TPSS improvement over PBE is dramatic for atomization energies of molecules and surface energies of solids, and small or moderate for other properties. TPSS is now or soon will be available in standard codes such as GAUSSIAN, TURBOMOLE, NWCHEM, ADF, WIEN, VASP, etc. We also discuss old and new ideas to eliminate the self-interaction error that plagues the functionals on the first three rungs of the ladder, bring up other related issues, and close with a list of "do's and don't's" for software developers and users.

We assess the performance of recent density functionals for the exchange-correlation energy of a nonmolecular solid, by applying accurate calculations with the GAUSSIAN, BAND, and VASP codes to a test set of 24 solid metals and nonmetals. The functionals tested are the modified Perdew-Burke-Ernzerhof generalized gradient approximation ͑PBEsol GGA͒, the second-order GGA ͑SOGGA͒, and the Armiento-Mattsson 2005 ͑AM05͒ GGA. For completeness, we also test more standard functionals: the local density approximation, the original PBE GGA, and the Tao-Perdew-Staroverov-Scuseria meta-GGA. We find that the recent density functionals for solids reach a high accuracy for bulk properties ͑lattice constant and bulk modulus͒. For the cohesive energy, PBE is better than PBEsol overall, as expected, but PBEsol is actually better for the alkali metals and alkali halides. For fair comparison of calculated and experimental results, we consider the zeropoint phonon and finite-temperature effects ignored by many workers. We show how GAUSSIAN basis sets and inaccurate experimental reference data may affect the rating of the quality of the functionals. The results show that PBEsol and AM05 perform somewhat differently from each other for alkali metal, alkaline-earth metal, and alkali halide crystals ͑where the maximum value of the reduced density gradient is about 2͒, but perform very similarly for most of the other solids ͑where it is often about 1͒. Our explanation for this is consistent with the importance of exchange-correlation nonlocality in regions of core-valence overlap.

Semilocal density functional approximations for the exchange-correlation energy can improperly dissociate a neutral molecule XY (Y not =X) to fractionally charged fragments X(+q)...Y(-q) with an energy significantly lower than X0...Y0. For example, NaCl can dissociate to Na(+0.4)...Cl(-0.4). Generally, q is positive when the lowest-unoccupied orbital energy of atom Y0 lies below the highest-occupied orbital energy of atom X0. The first 24 open sp-shell atoms of the Periodic Table can form 276 distinct unlike pairs XY, and in the local spin density approximation 174 of these display fractional-charge dissociation. Finding these lowest-energy solutions with standard quantum chemistry codes, however, requires special care. Self-interaction-corrected (SIC) semilocal approximations are exact for one-electron systems and also reduce the spurious fractional charge q. The original SIC of Perdew and Zunger typically reduces q to 0. A scaled-down SIC with better equilibrium properties sometimes fails to reduce q all the way to 0. The desideratum of "many-electron self-interaction freedom" is introduced as a generalization of the one-electron concept.

Table I of this article, and Table SVII of its Supplementary Information, reported imprecise revTPSS meta-GGA cohesive energies for some of the 9 nontransition metals and 6 insulators.The revTPSS cohesive energies in the original Table SVII for the metals were computed from BAND, and those for some insulators were computed from VASP. LiCl, NaCl, and MgO were also calculated from BAND. Converged BAND and VASP calculations agreed rather well, but there were different technical difficulties for each code. For technical reasons, the nonself-consistent revTPSS values from VASP used PBE orbitals, while those from BAND used LSDA orbitals. Recently, we have discovered that the VASP calculations required improved PAWs for meta-GGA calculations and that the BAND free atomic energies were not low enough for many of the solids. These errors produced too-large cohesive energies in both VASP and BAND. Here we report improved BAND results, using LSDA orbitals.The ME and MAE values for revTPSS in Table I change to À0:03 and 0:12 eV=atom, respectively. This makes the cohesive energies slightly less accurate from revTPSS than from PBE, in comparison with the experimental values given in the Supplementary Information. While the present non-self-consistent BAND and self-consistent VASP cohesive energies agree well in most of the cases, relatively large (more than 0.1 eV) discrepancies can be observed for LiCl, NaCl, and MgO. We attribute these discrepancies to the too-high free atomic energies for Cl and O given by BAND. The more precise VASP results show slightly worse ME and MAE values for revTPSS than the BAND results do.

Semilocal density functionals for the exchange-correlation energy are needed for large electronic systems. The Tao-Perdew-Staroverov-Scuseria (TPSS) meta-generalized gradient approximation (meta-GGA) is semilocal and usefully accurate, but predicts too-long lattice constants. Recent "GGA's for solids" yield good lattice constants but poor atomization energies of molecules. We show that the construction principle for one of them (restoring the density gradient expansion for exchange over a wide range of densities) can be used to construct a "revised TPSS" meta-GGA with accurate lattice constants, surface energies, and atomization energies for ordinary matter.