Analyzing Band Gaps in Ensemble Density Functional Theory using Thermodynamic Limits of Finite One-Dimensional Model Systems

TL;DR

This paper investigates the potential of Ensemble Density Functional Theory (EDFT) to accurately calculate band gaps in periodic systems by analyzing finite one-dimensional models and approaching the thermodynamic limit.

Contribution

It demonstrates that EDFT can approximate band gaps in periodic systems by studying finite models and suggests its promise for future development in this area.

Findings

Finite systems' Kohn-Sham gaps approach the periodic limit.

Proper identification of valence and conduction states is crucial.

EDFT provides a reasonable correction to band gaps in the periodic limit.

Abstract

Ensemble Density Functional Theory (EDFT) is a promising extension to Density Functional Theory (DFT) for calculating excited states. While Kohn-Sham eigenvalue differences underestimate gaps, EDFT has been shown to provide more accurate excitation energies in atoms, molecules and isolated model systems. However, it is unclear whether EDFT is capable of calculating band gaps of periodic systems -- and what an appropriate theoretical formulation would be to describe periodic systems. We explored how EDFT could calculate band gaps by estimating the thermodynamic limit with increasingly wide finite versions of the one-dimensional Kronig-Penney (KP) periodic model. We use Octopus, an ab initio, open-source, real-space DFT code, as in our previous work [R. J. Leano et al., Electron. Struct. 6, 035003 (2024)] in which we found with "particle in a box" models that EDFT can provide a reasonable effective mass correction for the homogeneous electron gas. Now, we use a periodic reference that is gapped. We find that the finite systems' Kohn-Sham gap approaches the same periodic limit for each of three ways of terminating the finite system, though the appropriate states corresponding to the valence band maximum and conduction band minimum have to be carefully identified in each case. Finally, our EDFT results, using a simple ensemblized LDA approximation, have a reasonable nonzero correction to the bandgap in the periodic limit. The results indicate that EDFT is promising for periodic systems, to motivate further work on developing a suitable formalism.