  In exact density functional theory (DFT) the total ground-state energy is a
series of linear segments between integer electron points, a condition known as
"piecewise linearity". Deviation from this condition is indicative of poor
predictive capabilities for electronic structure, in particular of ionization
energies, fundamental gaps, and charge transfer. In this article, we take a new
look at the deviation from linearity (i.e., curvature) in the solid-state limit
by considering two different ways of approaching it: a large finite system of
increasing size and a crystal represented by an increasingly large reference
cell with periodic boundary conditions. We show that the curvature approaches
vanishing values in both limits, even for functionals which yield poor
predictions of electronic structure, and therefore can not be used as a
diagnostic or constructive tool in solids. We find that the approach towards
zero curvature is different in each of the two limits, owing to the presence of
a compensating background charge in the periodic case. Based on these findings,
we present a new criterion for functional construction and evaluation, derived
from the size-dependence of the curvature, along with a practical method for
evaluating this criterion. For large finite systems we further show that the
curvature is dominated by the self-interaction of the highest occupied
eigenstate. These findings are illustrated by computational studies of various
solids, semiconductor nanocrystals, and long alkane chains.
