Townsend discharge theory is commonly used to describe and approximate the ionisation fraction growth rate in the very early phase of plasma initiation in tokamak devices via ohmic breakdown. The prediction of the ionisation fraction growth rate is done most commonly with continuum or kinetic models, which in turn boil down to the relation between the first Townsend's coefficient α, pressure p and electric field strength E (namely, α/p and E/p). To date there are few computational models that attempt to simulate the ionisation fraction growth rate via explicit modelling of each ionisation event through electron-neutral collisions. This is largely due to the challenge of addressing the exponential growth of charged particles from ionisation processes, combined with the high computational cost of N-body simulation. In this work, a new fully three-dimensional, first-principles model of a Townsend hydrogen discharge is demonstrated and benchmarked against prior experimental findings. These tests also include comparisons of three separate models for the scattering angle and their impact on the obtained α/p and mean electron drift velocity. It is found that isotropic scattering combined with restricting the freed electron's scattering angle along the incident electron's velocity vector during ionisation events gives the closest agreement of α/p compared to experimental measurements.