The electron self-interaction is a long-standing problem in density functional theory and is particularly critical in the description of polarons. Polarons are quasiparticles involving charge localization coupled with self-induced lattice distortions. Since their prediction by Landau almost a century ago, polarons have drawn a great deal of attention in physics, chemistry, and materials science. The polaron stability results from the competition between the energy gain associated with the charge localization and the energy cost of the involved lattice distortions. Therefore, the polaron localization and its formation energy are sensitively affected by the description of the electron self-interaction. Various competitive correction schemes based on either one-body or many-body descriptions of the self-interaction have been proposed to solve this longstanding problem. At present, it remains unclear which of these two descriptions of the self-interaction needs to be addressed in polaron physics.

In this thesis, we address the self-interaction problem in relation to polarons in density functional theory. First, we develop a scheme for correcting finite-size electrostatic effects involving the polaron charge density, which is crucial for achieving energetics of isolated polarons. Then, we study polarons with state-of-the-art hybrid functionals, highlighting the notion of formation energy for determining the polaron stability. Next, we develop a unified theoretical framework encompassing one-body and many-body forms of self-interaction, which confers superiority to the notion of many-body self-interaction over the notion of one-body self-interaction. Given the preeminence of the many-body self-interaction, we introduce an efficient semilocal scheme for localizing polarons based on the inclusion of a weak local potential in the semilocal Hamiltonian to suppress the many-body self-interaction. Taking advantage of these findings, we develop a selection criterion for the Hubbard interaction in Hubbard-corrected functionals. Finally, we apply our methodologies to the case of an anisotropic system, and use semilocal functionals free from many-body self-interaction to calculate polaron hopping rates. In this context, we demonstrate that polaron properties free from many-body self-interaction, including formation energies, hopping energy barriers, and hopping rates are robust upon variation of the functional. This supports the use of our semilocal scheme and of the Hubbard-corrected functional over computationally more expensive hybrid functionals.

This thesis advances the conceptual understanding of the self-interaction problem in density functional theory, and paves the way to efficient calculations of polarons in large systems, in systematic studies involving large sets of materials, in molecular dynamics evolving over long time periods, and in charge transfer mechanisms.