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How dielectric screening in two-dimensional crystals affects the convergence of excited-state calculations: Monolayer MoS$_2$

We present first-principles many-body calculations of the dielectric constant, quasiparticle band structure, and optical absorption spectrum of monolayer MoS$_2$ using a supercell approach. As the separation between the periodically repeated layers is increased, the dielectric function of the layer develops a strong $q$ dependence around $q=0$. This implies that denser $k$-point grids are required to converge the band gap and exciton binding energies when large supercells are used. In the limit of infinite layer separation, here obtained using a truncated Coulomb interaction, a $45\times45$ $k$-point grid is needed to converge the G$_0$W$_0$ band gap and exciton energy to within 0.1 eV. We provide an extensive comparison with previous studies and explain agreement and variations in the results. It is demonstrated that too coarse $k$-point sampling and the interactions between the repeated layers have opposite effects on the band gap and exciton energy, leading to a fortuitous error cancellation in the previously published results.