  We describe how to simulate charge diffusion in organic semiconductors using
a recently introduced mixed quantum-classical method, the mapping approach to
surface hopping. In contrast to standard fewest-switches surface hopping, this
method propagates the classical degrees of freedom deterministically on the
most populated adiabatic electronic state. This correctly preserves the
equilibrium distribution of a quantum charge coupled to classical phonons,
allowing one to time-average along trajectories to improve the statistical
convergence of the calculation. We illustrate the method with an application to
a standard model for the charge transport in the direction of maximum mobility
in crystalline rubrene. Because of its consistency with the quantum-classical
equilibrium distribution, the present method gives a time-dependent diffusion
coefficient that plateaus correctly to a long-time limiting value. We end by
comparing our results for the mobility and optical conductivity to those of the
widely used relaxation time approximation, which uses a phenomenological
relaxation parameter to obtain a non-zero diffusion coefficient from a
calculation with static phonon disorder. We find that this approximation
generally underestimates the charge mobility when compared with our method,
which is parameter free and directly simulates the thermal equilibrium
diffusion of a quantum charge coupled to classical phonons.
