  Excitons are neutral excitations that are composed of electrons and holes
bound together by their attractive Coulomb interaction. The electron and the
hole forming the exciton also interact with the underlying atomic lattice, and
this interaction can lead to a trapping potential that favors exciton
localization. The quasi-particle thus formed by the exciton and the surrounding
lattice distortion is called excitonic polaron. Excitonic polarons have long
been thought to exist in a variety of materials, and are often invoked to
explain the Stokes shift between the optical absorption edge and the
photo-luminescence peak. However, quantitative ab initio calculations of these
effects are exceedingly rare. In this manuscript, we present a theory of
excitonic polarons that is amenable to first-principles calculations. We first
apply this theory to model Hamiltonians for Wannier excitons experiencing
Fr\"{o}hlich or Holstein electron-phonon couplings. We find that, in the case
of Fr\"{o}hlich interactions, excitonic polarons only form when there is a
significant difference between electron and hole effective masses. Then, we
apply this theory to calculating excitonic polarons in lithium fluoride ab
initio. The key advantage of the present approach is that it does not require
supercells, therefore it can be used to study a variety of materials hosting
either small or large excitonic polarons. This work constitutes the first step
toward a complete ab initio many-body theory of excitonic polarons in real
materials.
