  We develop a theoretical and computational framework to study polarons in
semiconductors and insulators from first principles. Our approach provides the
formation energy, excitation energy, and wavefunction of both electron and hole
polarons, and takes into account the coupling of the electron or hole to all
phonons. An important feature of the present method is that it does not require
supercell calculations, and relies exclusively on electron band structures,
phonon dispersions, and electron-phonon matrix elements obtained from
calculations in the crystal unit cell. Starting from the Kohn-Sham (KS)
equations of density-functional theory, we formulate the polaron problem as a
variational minimization, and we obtain a nonlinear eigenvalue problem in the
basis of KS states and phonon eigenmodes. In our formalism the electronic
component of the polaron is expressed as a coherent superposition of KS states,
in close analogy with the solution of the Bethe-Salpeter equation for the
calculation of excitons. We demonstrate the power of the methodology by
studying polarons in LiF and Li2O2. We show that our method describes both
small and large polarons, and seamlessly captures Frohlich-type polar
electron-phonon coupling and non-Frohlich coupling to acoustic and optical
phonons. To analyze in quantitative terms the electron-phonon coupling
mechanisms leading to the formation of polarons, we introduce spectral
decompositions similar to the Eliashberg spectral function. We validate our
theory using both analytical results and direct calculations on large
supercells. This study constitutes a first step toward complete ab initio
many-body calculations of polarons in real materials.
