Validity of DFT+U band gaps in all its known functional forms

Abstract

The Density Functional Theory plus Hubbard $U$ (DFT+$U$) technique is one of the most widely used tools by condensed matter physicists and solid state chemists for the simulation of transition-metal and lanthanide bearing crystals, and increasingly of much more diverse chemistries. Although often synonymous with the corrective functionals of Dudarev et al. and Liechtenstein et al., there exists a wide variety of DFT+$U$-type functionals ready to be utilized, and no doubt yet to be developed. Since the earliest days, the gap in the DFT+$U$ single-particle eigenspectrum has been associated with the fundamental band gap, and the method has typically found more success for spectra than for total-energy derived properties. There has been some doubt, however, as to the conceptual validity of this association. Here, extending findings from recent years regarding local and semi-local functionals, we prove that the DFT+$U$ eigenspectrum gap is indeed valid, in the sense that it matches its own fundamental gap calculated using total-energy differences. This is true for pristine periodic systems with converged $k$-point sampling but not, however, for defective ones or isolated systems. We show that bandgap validity for solids holds in the presence of pseudopotentials and PAW potentials, when using hybrid functionals, and in DFT+$U$(+$J$) irrespective of the level of subspace projection onto the band-edge states. We survey every DFT+$U$-type functional known to have been published to date, within a unified notation. We verify analytically under which conditions the eigenvalue gap equals its fundamental gap for each functional, and analyze its effect on total energies and gaps for the hydrogen lattice in the Mott-Hubbard limit.