Equivalence of charged and neutral density functional formulations for correcting the many-body self-interaction of polarons (2511.02159v1)

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 $\beta$-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.

• The paper establishes that charged and neutral DFT formulations yield identical self-interaction corrected energies for polarons by enforcing piecewise linearity.
• It details derivations and component methods—including hybrid functionals, DFT+U, and pSIC—and validates results across materials like MgO and TiO2.
• The study emphasizes that proper treatment of finite-size corrections and screening effects is crucial for accurately modeling polaron formation and stability.

Equivalence of Charged and Neutral Density Functional Formulations for Correcting Many-Body Self-Interaction of Polarons

Introduction

This paper rigorously establishes the formal equivalence between charged and neutral density functional theory (DFT) formulations for correcting the many-body self-interaction error in polaronic systems. The self-interaction error, inherent in approximate exchange-correlation functionals, critically affects the localization and energetics of polarons—quasiparticles formed by a localized charge coupled to lattice distortions. The work connects piecewise-linear functionals (e.g., hybrid functionals with nonempirical tuning, DFT+$U$, $\gamma$DFT, $\mu$DFT), the polaron self-interaction correction (pSIC) method, and the unit-cell approach based on electron-phonon couplings, demonstrating that these methods yield the same formal expression for the self-interaction-corrected energy. The analysis is supported by extensive computational results for prototypical electron and hole polarons in MgO, $\beta$-Ga$_2$O$_3$, NaI, BiVO$_4$, and TiO$_2$.

Many-Body Self-Interaction in DFT and Its Correction

The many-body self-interaction error manifests as a deviation from piecewise linearity of the total energy with respect to fractional electron number. Standard semilocal functionals (e.g., PBE) exhibit a concave energy dependence, favoring charge delocalization and destabilizing localized polaronic states. Hybrid functionals and DFT+$U$ can suppress this error by enforcing piecewise linearity, typically via nonempirical tuning of parameters (e.g., Fock exchange fraction $\alpha$ or Hubbard $U$). The paper provides explicit expressions for the localizing potentials and energy corrections in these functionals, and details the procedure for parameter tuning to achieve piecewise linearity, as required by Janak's theorem.

The polaron formation energy is decomposed into the energy gain from charge localization (difference between polaron and band-edge levels) and the energy cost due to lattice distortions. When the functional is tuned to piecewise linearity, the formation energy expression becomes robust and independent of the specific functional form.

Unified Hybrid-Functional Formulation

A unified framework is presented, showing that the many-body self-interaction correction in hybrid functionals is related to the one-body correction by a factor involving the high-frequency dielectric constant $\varepsilon_\infty$. This formalism demonstrates that many-body corrections properly account for electronic screening, which is absent in one-body corrections. The superiority of the many-body approach is established both analytically and numerically.

Semilocal and Neutral Formulations

The paper derives a semilocal, parameter-free expression for the many-body self-interaction correction, enabling efficient polaron calculations at the PBE level. The key result is that, for the neutral polaronic structure, the corrected energy and forces can be obtained solely from PBE calculations, with the polaron level and band-edge level evaluated at $q=0$. This approach is shown to be formally equivalent to the pSIC method, which uses finite-difference derivatives of the energy and forces with respect to charge.

The unit-cell method, based on electron-phonon couplings in the undistorted cell, is also analyzed. While it neglects electronic screening effects, it yields the same formal expression for the formation energy in the harmonic limit, further supporting the equivalence of these approaches.

Finite-Size Effects and Their Treatment

Finite-size corrections are essential in supercell calculations of polarons due to spurious interactions with periodic images and background charge. The paper provides explicit formulas for correcting both charged and neutral polaronic states, highlighting that neglecting these corrections can lead to under- or overestimation of polaronic distortions depending on the method. The analysis shows that pSIC tends to underestimate distortions, while charged formulations ($\gamma$DFT, $\mu$DFT) tend to overestimate them in finite supercells. These discrepancies diminish with increasing supercell size and dielectric constant.

Numerical Results and Comparative Analysis

Extensive calculations are presented for representative polarons in MgO, $\beta$-Ga$_2$O$_3$, NaI, BiVO$_4$, and TiO$_2$. The results demonstrate excellent agreement in polaronic structures, densities, and formation energies among $\gamma$DFT, $\mu$DFT, and pSIC, with discrepancies typically below 0.07 Å in bond lengths and 0.13 eV in formation energies (except for weakly bound cases like NaI). The unit-cell method yields formation energies within 0.06 eV of pSIC for most systems, confirming the formal equivalence. The analysis also discusses the impact of finite-size corrections and the sensitivity of weakly bound polarons to the choice of functional and supercell size.

Implications and Future Directions

The formal equivalence established in this work has significant implications for the computational study of polarons. It enables the use of efficient semilocal functionals (e.g., $\gamma$DFT, $\mu$DFT, pSIC) for large-scale and molecular dynamics simulations without the need for expensive hybrid functional calculations or parameter tuning. The robustness of polaron properties across different piecewise-linear functionals is demonstrated, provided that finite-size and screening effects are properly treated.

The paper suggests that self-consistent finite-size corrections to forces and energies could further improve the accuracy of polaronic distortions and prevent spurious delocalization in small supercells. The integration of machine-learned functionals and neural network potentials is identified as a promising avenue for accelerating polaron simulations in complex and disordered systems.

Conclusion

This work provides a rigorous theoretical and computational foundation for the equivalence of charged and neutral DFT formulations in correcting the many-body self-interaction error for polarons. The results unify diverse approaches under a common formalism, demonstrate their practical agreement, and clarify the role of finite-size and screening effects. The findings facilitate efficient and accurate polaron modeling in materials science, with broad applicability to semiconductors, insulators, and molecular systems. Future developments may focus on self-consistent finite-size corrections and the integration of machine learning for large-scale polaron dynamics.