Localization and Delocalization Errors in Density Functional Theory and Implications for Band-Gap Prediction

The band-gap problem and other systematic failures of approximate exchange-correlation functionals are explained from an analysis of total energy for fractional charges. The deviation from the correct intrinsic linear behavior in finite systems leads to delocalization and localization errors in large and bulk systems. Functionals whose energy is convex for fractional charges such as the local density approximation display an incorrect apparent linearity in the bulk limit, due to the delocalization error. Concave functionals also have an incorrect apparent linearity in the bulk calculation, due to the localization error and imposed symmetry. This resolves an apparent paradox and identifies the physical nature of the error to be addressed to obtain accurate band gaps from density functional theory.

Figure 1

Energy versus number of electron curve for a finite system with three hypothetical functionals with exact straight line behavior, $\delta N$-convex behavior, and $\delta N$-concave behavior all which gave the same $I$ and $A$.

Figure 2

The same as Fig. 1 except taken to the bulk limit. An additional curve is shown where crystal symmetry is imposed on a $\delta N$-concave functional calculation.

Figure 3

Energy (in $E_h$) versus numbers of electrons for $({\mathrm{H}}_{16}{)}_M$.

Figure 4

Density difference for the $({\mathrm{H}}_{16}{)}_M$ system with HF and LDA.