Spin-Phonon Coupling in Transition Metal Insulators: General Computational Approach and Application to MnPSe$_3$

Spin-phonon coupling underlies a number of diverse range of phenomena of recent interest, particularly in transition metal insulators with strong spin-orbit effects, where it may give rise to hybrid magnetoelastic excitations, and the controversial phonon thermal Hall effect. In this work, we describe a general approach to the first principles estimation of generic spin-orbit-phonon couplings suitable for investigating these diverse effects. The method is demonstrated by application to study the temperature-dependent optical phonon lineshapes in MnPSe$_3$, for which we find quantitative agreement with recent experiments.