Assessing Exchange-Correlation\nFunctionals for Accurate\nDensities of Solids

The success of Kohn–Sham\ndensity functional theory\nin predicting\nelectronic properties from first-principles is key to its ubiquitous\npresence in condensed matter research. Central to this theory is the\nexchange-correlation functional, which can only be written in an approximate\nform using a handful of exact constraints. A recent criticism of these\napproximations is that they are designed to give an accurate description\nof the energy at the expense of a poor representation of the density,\nwhich is contrary to the spirit of density functional theory. These\nconclusions are drawn from studies of atoms or small molecules, where\nexact results are available. To shed light on this issue, we use the\nalmost exact densities and energies of three prototypical solids (a\nsemiconductor, silicon, an insulator, sodium chloride, and a metal,\ncopper) to compare the performance of exchange-correlation functionals\nfrom all rungs of Jacob’s ladder. By examining their errors\nin reproducing both energy and density, we show that several hybrids\nand semilocal functionals perform consistently well. Furthermore,\nfunctionals built to reproduce exact constraints tend to be among\nthe top performers for all tested material classes, strengthening\nthe argument for using these constraints in functional construction.\nOn average, functionals published up to the early 2000s simultaneously\nimprove the prediction of both densities and energies. This is often\nnot the case for more recent functionals, although errors in energy\nand density continue to evolve in a correlated manner.