  We present a first-principle numerical study of charge transport in a
realistic two-dimensional tight-binding model of molecular organic
semiconductors. We use the Hybrid Monte Carlo (HMC) algorithm to simulate the
full quantum dynamics of phonons and either a single or multiple charge
carriers without any tunable parameters. We introduce a number of algorithmic
improvements, including efficient Metropolis updates for phonon fields based on
analytic insights, which lead to negligible autocorrelation times and allow to
reach sub-permille precisions at small computational cost of $O(1)$ CPU-hour.
In the experimentally relevant range of Hamiltonian parameters and temperatures
the HMC simulations produce charge mobility estimates that are in good
agreement with experiment and with the popular Transient Localisation approach.
