  Halide perovskites are strongly influenced by large amplitude anharmonic
lattice fluctuations at room temperature. We develop a tight binding model for
dynamically disordered MAPbI$_3$ based on density functional theory (DFT)
calculations to calculate electronic structure for finite temperature crystal
structures at the length scale of thermal disorder and carrier localization.
The model predicts individual Hamiltonian matrix elements and band structures
with high accuracy, owing to the inclusion of additional matrix elements and
descriptors for non-Coulombic interactions. We apply this model to electronic
structure at length and time scales inaccessible to first principles methods,
finding an increase in band gap, carrier mass, and the sub-picosecond
fluctuations in these quantities with increasing temperature as well as the
onset of carrier localization in large supercells induced by thermal disorder
at 300 K. We identify the length scale $L^*= 5$ nm as the onset of localization
in the electronic structure, associated with associated with decreasing band
edge fluctuations, increasing carrier mass, and Rashba splitting approaching
zero.
