Equivalence of charged and neutral density functional formulations for correcting the many-body self-interaction of polarons

TL;DR

The paper tackles the electron self-interaction problem in polaron systems and demonstrates the formal equivalence between charged and neutral density functional corrections. By starting from a unified hybrid-functional framework and deriving a parameter-free semilocal mb self-interaction functional, it links γDFT, μDFT, pSIC, and the unit-cell method through the neutral-polaron energetics. For prototypical hole and electron polarons, the ground-state properties and formation energies are in excellent agreement across these approaches once finite-size corrections are applied; residual differences arise mainly from polaron distortions and screening treatment. The work provides a unified, efficient route to accurate polaron energetics, enabling large-scale simulations and potential integration with molecular dynamics and machine-learning aids.

Abstract

The electron self-interaction problem in density functional theory affects the accurate modeling of polarons, particularly their localization and formation energy. Charged and neutral density functional formulations have been developed to address this issue, yet their relationship remains unclear. Here, we demonstrate their equivalence in treating the many-body self-interaction of the polaron state. In particular, we connect with each other piecewise-linear functionals based on adding an extra charge to the supercell, the pSIC approach derived from the energetics of the neutral defect with polaronic distortions in a supercell, and the unit-cell method for polarons based on electron-phonon couplings. We show that these approaches lead to the same formal expression of the self-interaction corrected energy, which is fully defined by the energetics of the neutral charge state of the charged polaronic structure. Residual differences between these methods solely arise from the achieved polaronic structure, which is affected by different treatments of electron-screening and finite-size effects. We apply these methods to a set of prototypical small hole and electron polarons, including the hole polaron in MgO, the hole polaron in $β$-Ga$_2$O$_3$, the $V_\text{k}$ center in NaI, the electron polaron in BiVO$_4$, and the electron polaron in TiO$_2$. We show that the ground-state properties of polarons obtained using charged and neutral density functional formulations are in excellent agreement.

Figures (4)

• Figure 1: Different treatment of finite-size effects on polaron energy levels when using $\gamma$DFT at $\gamma_\text{k}$, $\mu$DFT at $\mu_\text{k}$, and pSIC for electron and hole polarons in a finite-size supercell. The finite-size corrections on the polaron energy levels are given in Eqs. \ref{['eq:eps_fs_integerq']} and \ref{['eq:eps_fs_zeroq']}. Occupied levels are given by solid lines, unoccupied levels by dashed lines. Without applying finite-size corrections, the energy gain due to charge localization is underestimated in pSIC and overestimated in $\gamma$DFT and $\mu$DFT. This may lead to underdistorted polaronic structures in pSIC, and overdistorted polaronic structures in $\gamma$DFT and $\mu$DFT. These differences are reduced when increasing the size of the supercell.
• Figure 2: Polaron isodensity surface at 5% of its maximum calculated at $q=0$ for the hole polaron in MgO (216-atom supercell), the hole polaron in $\beta$-Ga$_2$O$_3$ (120-atom supercell), the $V_\text{k}$ center in NaI (216-atom supercell), the electron polaron in BiVO$_4$ (96-atom supercell), and the electron polaron in TiO$_2$ (216-atom supercell). Mg in pink, O in red, Ga in grey, Na in green, I in violet, Bi in orange, V in cyan, and Ti in blue. The vertical axis is the $z$ axis.
• Figure 3: Charged and neutral polaron energy levels as a function of (a) $\gamma$ in $\gamma$DFT calculations and (b) $\mu$ in $\mu$DFT calculations for the hole polaron in MgO (216-atom supercell). The polaron levels are identified by their respective polaron charge states, namely +1 for the charged and 0 for the neutral state. The geometry is kept fixed at that obtained for $\gamma = \gamma_\text{k}$ and $\mu = \mu_\text{k}$, respectively.
• Figure 4: Polaron densities integrated over $xy$ planes obtained with $\gamma$DFT, $\mu$DFT, and pSIC for the various polarons considered in this work. The polaron densities are calculated for the neutral charge state $(q=0)$ in the presence of polaronic distortions, and are averaged over planes according to Eq. \ref{['eq:average_density']}.