Comparison between first-principles supercell calculations of polarons and the ab initio polaron equations

Polarons are composite quasiparticles formed by excess charges and the accompanying lattice distortions in solids and play a critical role in the transport, optical, and catalytic properties of semiconductors and insulators. The standard approach for calculating polarons from first principles relies on density functional theory and periodic supercells. An alternative approach consists of recasting the calculation of polaron wave function, lattice distortion, and energy as a coupled nonlinear eigenvalue problem, using the band structure, phonon dispersions, and the electron-phonon matrix elements as obtained from density functional perturbation theory. Here, we revisit the formal connection between these two approaches, with an emphasis on the handling of self-interaction correction, and we establish a compact formal link between them. We perform a quantitative comparison of these methods for the case of small polarons in the prototypical insulators TiO2, MgO, and LiF. We find that the polaron wave functions and lattice distortions obtained from these methods are nearly indistinguishable in all cases, and the formation energies are in good (TiO2) to fair (LiF) agreement. We show that the residual deviations can be ascribed to the neglect of higher-order electron-phonon couplings in the density functional perturbation theory approach.