Effect of spatial nonlocality on the density functional band gap

We show that for a large class of exchange-correlation functionals the local exchange-correlation potential obtained within an optimized effective potential severely underestimates the band gap. On the other hand, the corresponding nonlocal potential obtained from a generalized Kohn-Sham scheme provides a much better description of the band gap, in good agreement with experiments. These results strongly indicate that a local exchange-correlation potential, however good the exchange-correlation approximation, cannot capture the delicate interplay between correlation effects and spatial localization in the KS band structure, unless the (cumbersome) contribution from the derivative discontinuity of the exchange-correlation energy functional is considered.

Figure 1

Direct band gap at $\Gamma$ for Si, $\mathrm{LiF}$, and Ar evaluated from the OEP-KS and the GKS eigenvalue differences using the $\mathrm{CHSX}+$ approximation. The striped bar is ${\Delta }_{\mathrm{XC}}^{\mathrm{orb}}$ contribution to the gap evaluated at the first order perturbation theory [Eq. (14)]. The continuous (dashed) line indicates the experimental (Refs. 30, 31, 32) (LDA) band gap.

Figure 2

Same as in Fig. 1 using an exchange-only energy functional (GKS reduces to the Hartree-Fock approximation). The striped bar corresponds to the first-order ${\Delta }_x$ contribution to the gap (see text). The continuous line indicates the experimental value (Refs. 30, 31, 32).