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In exact density functional theory (DFT) the total ground-state energy is aseries of linear segments between integer electron points, a condition known as"piecewise linearity". Deviation from this condition is indicative of poorpredictive capabilities for electronic structure, in particular of ionizationenergies, fundamental gaps, and charge transfer. In this article, we take a newlook at the deviation from linearity (i.e., curvature) in the solid-state limitby considering two different ways of approaching it: a large finite system ofincreasing size and a crystal represented by an increasingly large referencecell with periodic boundary conditions. We show that the curvature approachesvanishing values in both limits, even for functionals which yield poorpredictions of electronic structure, and therefore can not be used as adiagnostic or constructive tool in solids. We find that the approach towardszero curvature is different in each of the two limits, owing to the presence ofa compensating background charge in the periodic case. Based on these findings,we present a new criterion for functional construction and evaluation, derivedfrom the size-dependence of the curvature, along with a practical method forevaluating this criterion. For large finite systems we further show that thecurvature is dominated by the self-interaction of the highest occupiedeigenstate. These findings are illustrated by computational studies of varioussolids, semiconductor nanocrystals, and long alkane chains.

Deviations from piecewise linearity in the solid-state limit with approximate density functionals