  We propose a new formalism and an effective computational framework to study
self-trapped excitons (STE) in insulators and semiconductors from first
principles. Using the many-body Bethe-Salpeter equation in combination with
perturbation theory, we are able to obtain the mode- and momentum-resolved
exciton-phonon coupling matrix element in a perturbative scheme, and explicitly
solve the real space localization of the electron (hole), as well as the
lattice distortion. Further, this method allows to compute the STE potential
energy surface and evaluate the STE formation energy and Stokes shift. We
demonstrate our approach using two-dimensional magnetic semiconductor chromium
trihalides and a wide-gap insulator BeO, the latter of which features dark
excitons, and make predictions of their Stokes shift and coherent phonon
generation which we hope to spark future experiments such as photoluminescence
and transient absorption studies.
