  By means of $ab~initio$ computation schemes, we examine the low-energy
electronic structure of monolayer TaS$_2$ in its low-temperature commensurate
charge-density-wave structure. We estimate and take into account both local and
non-local Coulomb correlations within cRPA (constrained random phase
approximation) and GW+EDMFT (GW plus extended dynamical mean-field theory)
method. Mott nature of its insulating phase is clearly identified. By
increasing the level of nonlocal approximation from DMFT ($V=0$) to EDMFT and
GW+EDMFT, a systematic change of charge screening effects is clearly observed
while its quantitative effect on the electronic structure is small in the
realistic Mott state.
