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Detailed numerical models of chromosphere and corona are required to understand the heating of the solar atmosphere. An accurate treatment of the solar chromosphere is complicated by the effects arising from Non Local Thermodynamic Equilibrium (NLTE) radiative transfer. A small number of strong, highly scattering lines dominate the cooling and heating in the chromosphere. Additionally, the recombination times of ionised hydrogen are longer than the dynamical timescales, requiring a non-equilibrium (NE) treatment of hydrogen ionisation. The MURaM code is extended to include the physical process required for accurate simulation of the solar chromosphere, as implemented in the Bifrost code. This includes a time-dependent treatment of hydrogen ionisation, a scattering multi-group radiation transfer scheme and approximations for NLTE radiative cooling. The inclusion of NE and NLTE physics has a large impact on the structure of the chromosphere; the NE treatment of hydrogen ionisation leads to a higher ionisation fraction and enhanced populations in the first excited state throughout cold inter-shock regions of the chromosphere. Additionally this prevents hydrogen ioniation from buffering energy fluctuations, leading to hotter shocks and cooler inter-shock regions. The hydrogen populations in the ground and first excited state are enhanced by $10^2-10^3$ in the upper chromosphere and up to $10^9$ near the transition region. Including the necessary NLTE physics leads to significant differences in chromospheric structure and dynamics. The thermodynamics and hydrogen populations calculated using the extended version of the MURaM code are consistent with previous non-equilibrium simulations. The electron number and temperature calculated using the non-equilibrium treatment of the chromosphere are required to accurately synthesise chromospheric spectral lines.