  Charge transport is well understood in both highly ordered materials (band
conduction) or highly disordered ones (hopping conduction). In moderately
disordered materials -- including many organic semiconductors -- the
approximations valid in either extreme break down, making it difficult to
accurately model the conduction. In particular, describing wavefunction
delocalisation requires a quantum treatment, which is difficult in disordered
materials that lack periodicity. Here, we present the first three-dimensional
model of partially delocalised charge and exciton transport in materials in the
intermediate disorder regime. Our approach is based on polaron-transformed
Redfield theory, but overcomes several computational roadblocks by mapping the
quantum-mechanical techniques onto kinetic Monte Carlo. Our theory, delocalised
kinetic Monte Carlo (dKMC), shows that the fundamental physics of transport in
moderately disordered materials is that of charges hopping between partially
delocalised electronic states. Our results reveal why standard kinetic Monte
Carlo can dramatically underestimate mobilities even in disordered organic
semiconductors, where even a little delocalisation can substantially enhance
mobilities, as well as showing that three-dimensional calculations capture
important delocalisation effects neglected in lower-dimensional approximations.
