Analyzing Band Gaps in Ensemble Density Functional Theory using Thermodynamic Limits of Finite One-Dimensional Model Systems
Gregory G. V. Kenning, Remi J. Leano, David A. Strubbe
TL;DR
This work tests whether Ensemble DFT (EDFT) can predict band gaps in periodic systems by studying the thermodynamic limit of finite one-dimensional Kronig–Penney models. It combines KS and EDFT calculations within a GOK ensemble framework, using the excitation expression $Ω_I = E_I - E_0 + \left. \frac{\partial E_{\rm Hxc}^w [\rho]}{\partial w} \right|_{\rho=\rho^w}$ to extract gaps and analyzes three finite-termination schemes to identify bulk band edges. The results show that KS gaps of finite KP systems converge to the periodic gap across terminations when the correct edge states are selected, and that EDFT provides a nonzero band-gap correction in the periodic limit (approximately from $6.8$ eV to $10$ eV) with a simple XC approximation. Overall, the findings indicate that EDFT is promising for periodic semiconductors and motivate the development of a rigorous periodic EDFT formalism for wider applicability.
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.