  A member of the RMn$_6$Sn$_6$ rare-earth family materials, TbMn$_6$Sn$_6$,
recently showed experimental signatures of the realization of a quantum-limit
Chern magnet. In this work, we use quantum Monte Carlo (QMC) and density
functional theory with Hubbard $U$ (DFT$+U$) calculations to examine the
electronic structure of TbMn$_6$Sn$_6$. To do so, we optimize accurate,
correlation-consistent pseudopotentials for Tb and Sn using coupled-cluster and
configuration-interaction (CI) methods. We find that DFT$+U$ and
single-reference QMC calculations suffer from the same overestimation of the
magnetic moments as meta-GGA and hybrid density functional approximations. Our
findings point to the need for improved orbitals/wavefunctions for this class
of materials, such as natural orbitals from CI, or for the inclusion of
multi-reference effects that capture the static correlations for an accurate
prediction of magnetic properties. DFT$+U$ with Mn magnetic moments adjusted to
experiment predict the Dirac crossing in bulk to be close to the Fermi level,
within $\sim 120$ meV, in agreement with the experiments. Our
non-stoichiometric slab calculations show that the Dirac crossing approaches
even closer to the Fermi level, suggesting the possible realization of Chern
magnetism in this limit.
