  Using the ab initio computed Potential Energy Surface (PES) for the
electronic interaction of the MgH$^+$ ($^1\Sigma$) ion with the He($^1$S) atom,
we calculate the relevant state-changing rotationally inelastic collision cross
sections from a quantum treatment of the multichannel scattering problem. We
focus on the quantum dynamics at the translationally low energies for the
present partners discussed in the earlier, cold ion trap experiments (see
below) which we wish to model in detail. The corresponding state-changing rates
computed between the lower rotational states of the molecular ion are employed
to describe the time-evolution kinetics followed by recent experiments on
Coulomb-crystalized MgH$^+$ ($^1\Sigma$), where the ions are rotationally
cooled by micromotion tuning after the uploading into the trap of He as a
buffer gas. The present computational modeling of the final ions' rotational
temperatures in the experiments turns out to agree very well with their
observations and points at a fast equilibration between rotational and thermal
temperatures of the ions.
