Density functional theory (DFT) is one of the most popular electronic structure methods due to its good balance between accuracy and speed. Other theoretical methods exist[efn_note]Franchini, C., Reticcioli, M., Setvin, M., & Diebold, U. (2021). Polarons in materials. Nature Reviews Materials, 0123456789.[/efn_note], but DFT is the most common method for atomistic material simulations, due to its balance of accuracy and computational time. A central problem in modeling polarons with DFT is the delocalization of electrons that may occur due to self-interaction errors. Self-interaction errors[efn_note]Pacchioni, Gianfranco. “Modeling doped and defective oxides in catalysis with density functional theory methods: Room for improvements.” The Journal of chemical physics 128.18 (2008): 182505.[/efn_note],[efn_note]Ganduglia-Pirovano, M. Veronica, Alexander Hofmann, and Joachim Sauer. “Oxygen vacancies in transition metal and rare earth oxides: Current state of understanding and remaining challenges.” Surface science reports 62.6 (2007): 219-270.[/efn_note],[efn_note]Huang, Patrick, and Emily A. Carter. “Advances in correlated electronic structure methods for solids, surfaces, and nanostructures.” Annu. Rev. Phys. Chem. 59 (2008): 261-290.[/efn_note] occur with many common exchange-correlation potentials, leading to the wrong description of polarons in many materials.

See for example below two different DFT solutions for TiO2. One solution (with self-interaction errors) leads to an unpaired electron being delocalized throughout the simulation cell. The proper solution has the electron localized on a Ti atom, forming a polaron state. If one is to model polarons, then the ‘correct’ method must be chosen in order to avoid an incorrect description of polarons.

A delocalized solution in TiO2, where an unpaired electron is spread throughout the simulation cell.
A localized solution in TiO2, where a polaron forms with an electron localized on a Ti atom.

While DFT often fails to properly model polarons, methods do exist which enable localization of polarons. The DFT+U method[efn_note]Dudarev, S. L., Botton, G. A., Savrasov, S. Y., Humphreys, C. J., & Sutton, A. P. (1998). Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study. Physical Review B, 57(3), 1505–1509.[/efn_note],[efn_note]Tolba, S. A., Gameel, K. M., Ali, B. A., Almossalami, H. A., & Allam, N. K. (2018). The DFT+U: Approaches, Accuracy, and Applications. Density Functional Calculations – Recent Progresses of Theory and Application, 3–30.[/efn_note],[efn_note]Capdevila-Cortada, M., Łodziana, Z., & López, N. (2016). Performance of DFT+ U Approaches in the Study of Catalytic Materials. ACS Catalysis, 6(12), 8370–8379. [/efn_note],[efn_note]Himmetoglu, B., Floris, A., de Gironcoli, S., & Cococcioni, M. (2014). Hubbard-corrected DFT energy functionals: The LDA+U description of correlated systems. International Journal of Quantum Chemistry, 114(1), 14–49.[/efn_note] is one of the common approaches to enabling charge localization. DFT+U adds a correction term (hence +U) to DFT calculations to help remove self-interaction errors. It requires little more computational time than standard DFT calculations, and is therefore one of the most popular methods for modeling polarons. A different approach uses hybrid functionals[efn_note]Deák, P., Aradi, B., & Frauenheim, T. (2011). Polaronic effects in TiO2 calculated by the HSE06 hybrid functional: Dopant passivation by carrier self-trapping. Physical Review B, 83(15), 155207. [/efn_note],[efn_note]Finazzi, E., Di Valentin, C., Pacchioni, G., & Selloni, A. (2008). Excess electron states in reduced bulk anatase TiO2: Comparison of standard GGA, GGA+U, and hybrid DFT calculations. The Journal of Chemical Physics, 129(15), 154113.[/efn_note],[efn_note]De Lile, J. R., Kang, S. G., Son, Y.-A., & Lee, S. G. (2019). Investigating Polaron Formation in Anatase and Brookite TiO2 by Density Functional Theory with Hybrid-Functional and DFT + U Methods. ACS Omega, 4(5), 8056–8064.[/efn_note]. Such functionals also enable localization of polarons, but often require considerable more computational time. In any case, an appropriate method must be chosen so that ‘correct’ localization of polarons occurs.

Even if a method is applied that overcomes self-interaction errors, polaron formation is not guaranteed. For a given level of theory there may be several stable or metastable solutions[efn_note]Meredig, B., Thompson, A., Hansen, H. A., Wolverton, C., & van de Walle, A. (2010). Method for locating low-energy solutions within DFT+U. Physical Review B, 82(19), 195128[/efn_note]. The initial structure and wavefunction may all influence whether polarons form. Key characteristics of a polaron are the distortion of bonds between the polaron site and neighboring atoms, as well as the occupancy (or vacancy for holes) of electronic orbitals. The polaron essentially breaks the symmetry of the crystal as the distortions and electrons are centered around the polaron site. Simply using default settings or perfect crystals may not enable polaron formation. The simulation input must be chosen so that initial geometries and/or electronic orbitals mimic a polaron to enable convergence to a stable polaron solution.

A sample polaron (d electron on Ti atom) in TiO2 modeled using DFT+U. Bond distortions occur, and the numbers are the percentage that a bond is elongated upon polaron formation. In this case all bond lengths become longer relative to the pure crystal (1.1 or 3.9 %).