In exact density functional theory, 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 cannot 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.
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For spin unpolarized (polarized) KS systems the value of the occupation number fi is equal to if εi < εH, equal to 0 if εi > εH and if εi = εH, where εH, the highest occupied eigenvalue, is determined such that is equal to the total number of electrons . The lowest eigenvalue for which fi = 0 is referred to as εL.
Where the interacting-electron system is mapped into a partially interacting electron gas that is still represented by a single Slater determinant.31
In this paper all systems are treated strictly in the closed shell spin-unpolarized ensemble, so any removal or addition of small amounts of electronic charge preserves the unpolarized spin nature of the system.
Obviously the lowest-unoccupied eigenstate becomes the highest-occupied one upon charge addition.
Alternatively, the bulk limit can be approached using the concept of k-point sampling of the unit cell.82 For now, we do not pursue this alternative, but we discuss it extensively below.
The same dependence on N can be obtained by considering the curvature directly as the second derivative of the energy, i.e., , because ERC and QRC are both extensive quanti ties and therefore proportional to N.