Density-functional theory in insulators: Analytical model forΣxc,vxc, and the gap correction

This paper presents a tight-binding evaluation for an insulating system of the field-theoretic expressions for the self-energy ${\Sigma }_{\mathrm{xc}}$, the density-functional (DF) exchange-correlation potential $v_{\mathrm{xc}}$, and the band-gap correction to the one-particle DF eigenvalues. To this end an analytical, energy- and density-dependent model for the self-energy within the random-phase approximation is used. Across both valence and conduction bands the results of this simple self-energy model are within a few percent of recent first-principles, numerical results for diamond, Si, and LiCl. The ${\Sigma }_{\mathrm{xc}}$ model should therefore be very useful for a variety of applications, such as calculations of band offsets. The new potential $v_{\mathrm{xc}}$, which is obtained from an integral equation involving ${\Sigma }_{\mathrm{xc}}$, scales with the cube root of the density, with the proportionality factor α given by an explicit expression in terms of the plasma frequency and the average gap. For the valence electrons of Si, diamond, LiCl, and LiF α≃0.7. The band-gap correction due to the $v_{\mathrm{xc}}$ discontinuity is evaluated on the basis of the self-energy model. It is found to account for the DF underestimates both in the large-gap insulators LiCl and C and the small-gap semiconductor Si.