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

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.

💡 Research Summary

The paper investigates whether Ensemble Density Functional Theory (EDFT) can accurately predict band gaps in periodic systems by using a one‑dimensional Kronig‑Penney (KP) model as a testbed. The authors construct finite versions of the KP model with increasing numbers of unit cells, applying three distinct terminations: well‑centered (cut through a barrier), edge‑centered (cut at a well‑barrier interface), and barrier‑centered (cut through a well). For each termination they perform standard Kohn‑Sham (KS) density‑functional calculations using the real‑space code Octopus, and they identify the appropriate occupied and unoccupied single‑particle states that correspond to the valence‑band maximum (VBM) and conduction‑band minimum (CBM) in the infinite limit. The analysis shows that, despite differences in the ordering of states for small systems, all three terminations converge to the same KS band gap of approximately 6.78 eV as the system size grows, confirming that the finite‑size approach can recover the periodic limit.

To assess EDFT, the authors adopt the Gross‑Oliveira‑Kohn (GOK‑I) ensemble formalism. They construct ensembles of the lowest M I many‑electron states with monotonically decreasing weights, and they evaluate the ensemble‑generalized Hartree‑exchange‑correlation functional using a simple local‑density approximation (LDA) that is “ensemblized.” After a self‑consistent ground‑state calculation, they perform a one‑shot EDFT step that re‑weights the KS orbitals according to the ensemble prescription, thereby obtaining ensemble‑averaged densities and the corresponding HXC energies. By differentiating the total ensemble energy with respect to the ensemble weight, they extract the first excitation energy Ω₁, which serves as the EDFT correction to the KS gap.

The numerical results reveal two distinct behaviors. When the exchange‑correlation (XC) part of the functional is treated with broken symmetry (the “XC‑broken” case), the EDFT band gap approaches a value near 10 eV as the system size increases, representing a sizable upward correction of roughly 3 eV over the KS gap. This correction is of the same order as typical quasiparticle (GW) corrections for wide‑gap semiconductors, indicating that even a simple LDA‑based EDFT can capture a substantial portion of the missing gap. In contrast, when the full Hartree‑exchange‑correlation (HXC) functional is treated symmetrically, the convergence is poorer and the resulting gaps fluctuate, suggesting that the simple LDA form does not adequately describe the ensemble‑averaged exchange‑correlation energy for this model.

The authors also discuss practical aspects of the finite‑size calculations. The identification of the VBM and CBM states depends on the number of nodes in the wavefunctions rather than on the absolute state index; for the edge‑centered termination an “edge state” appears that must be skipped when forming the gap. Nevertheless, the node‑counting rule holds across all terminations and electron‑count patterns (4 m versus 4 m + 2), providing a robust criterion for extracting the bulk gap from finite systems.

In conclusion, the study demonstrates that (i) finite KP models with appropriate boundary conditions can faithfully reproduce the periodic KS band gap, (ii) EDFT with a simple ensemblized LDA functional yields a non‑zero, physically reasonable correction that brings the gap closer to the quasiparticle value, and (iii) the current LDA‑based EDFT formulation has limitations, especially in the treatment of the full HXC term, pointing to the need for more sophisticated ensemble‑compatible exchange‑correlation approximations. These findings support the promise of EDFT as a viable tool for band‑gap calculations in real periodic materials and motivate further methodological development for three‑dimensional solids.