Optimal transition states for polaron hopping transport without supercells

Polaron formation localizes charge carriers and drives a crossover from band-like to hopping transport in materials. Hopping dynamics can be obtained from DFT supercell calculations of transition states, but these suffer from polaron self-interaction, spurious electrostatics, and poor scaling with polaron size. We introduce a supercell-free framework for \textit{ab initio} polaron hopping transport based on variational polaron equations and the string method. The approach optimizes transition states between self-trapped polaron states directly in reciprocal space and provides the polaron configurations along the path, enabling evaluation of adiabatic hopping rates and mobilities. We apply the method to LiF and rutile TiO$_2$, revealing multi-step and anisotropic hopping mechanisms. In rutile TiO$_2$, the computed electron-polaron mobility agrees with experiment, whereas band-like Boltzmann transport substantially overestimates the mobility. Our results establish a scalable route to first-principles polaron-hopping dynamics in materials in which charge motion is governed by self-trapping.