state, made up of mostly the $d_{xz}$ and $d_{yz}$ orbitals of the Mo atoms adjacent to the missing S atom, with an energy few tenths of an eV above the valence band maxima, and (ii) two degenerate $E$ states, made up of mostly the $d_{z^{2}}$, $d_{x^{2}-y^{2}}$, and $d_{xy}$ orbitals of the Mo atoms adjacent to the missing S atom, with an energy 1.4-1.6 eV above the valence band maxima. All the defect states are spin-degenerate and correspond to the one ($A_{1}$) and two dimensional ($E$) representations of the trigonal symmetry group $C_{3v}$. The computed orbitals of these states are shown in Fig.\[fig:s\_vac\] (from Noh et al. [@Noh14]). A defect state can be an efficient center for non-radiative recombination due to Auger scattering only if it has good overlaps with the Bloch states of both the conduction and the valence bands. The $E$ states fit this criterion. The $E$ states can be described well by limiting the summation in the expression above to the Bloch states of the conduction and the valence band extrema at the $K$ and $K'$ points. Since all the orbitals forming the $E$ states have weights almost entirely on the Mo atoms adjacent to the missing S 