  Anharmonic atomic motions can strongly influence the optoelectronic
properties of materials but how these effects are connected to the underlying
phonon band structure is not understood well. We investigate how the electronic
band gap is influenced by overdamped phonons, which occur in an intriguing
regime of phonon-phonon interactions where vibrational lifetimes fall below one
oscillation period. We contrast the anharmonic halide perovskite CsPbBr$_3$,
known to exhibit overdamped phonons in its cubic phase, with the anharmonic
oxide perovskite SrTiO$_3$ where the phonons are underdamped at sufficiently
high temperatures. Our results show that overdamped phonons strongly impact the
band gap and cause slow dynamic fluctuations of electronic levels that have
been implicated in the unique optoelectronic properties of halide perovskites.
This finding is enabled by developing augmented stochastic Monte Carlo methods
accounting for phonon renormalization and imaginary modes that are typically
neglected. Our work provides guidelines for capturing anharmonic effects in
theoretical calculations of materials.
