  Exciton-phonon coupling (ExPC) is crucial for energy relaxation in
semiconductors, yet the first-principles calculation of such coupling remains
challenging, especially for low-dimensional systems. Here, an accurate
algorithm for calculating ExPC is developed and applied in exciton relaxation
problems in monolayer WSe2. Considering the interplay between the exciton wave
functions and electron-phonon coupling (EPC) matrix elements, we find that ExPC
shows distinct selection rules from the ones of EPC. By employing the Wannier
exciton model, we generalize these selection rules, which state that the
angular quantum numbers of the exciton must match the winding numbers of the
EPC matrix elements for the ExPC to be allowed. To verify our theory and
algorithm, we calculate inter-valley exciton relaxation pathways, which agrees
well with a recent experiment.
