Bloch states from all the bands [@Landsberg92]. In most cases of practical interest, only Bloch states in the vicinity of certain points, $\vec{K}_{s}$, in the Brillouin zone, such as $\Gamma$, $M$, $K$ and $K'$ in the case of 2D-TMDs, need to be included in the expansion and therefore one may write, $$\psi_{d}(\vec{r}) = \frac{1}{\sqrt{A}} \sum_{n,\vec{k},s} c_{n,s}(\vec{k}) \frac{e^{i(\vec{K}_{s} + \vec{k}).\vec{r}}}{\sqrt{A}} u_{n,\vec{k},s}(\vec{r}) \label{eq:defect1}$$ In the expression above, $u_{n,\vec{k},s}(\vec{r})$ are the periodic parts of the Bloch functions. The sum over $n$ runs over all the energy bands. Whereas shallow defect levels can usually be described well by limiting the summation above to a single band, deep mid-gap defect levels generally have contributions from multiple bands [@Landsberg92; @Landsberg80]. The above expression can usually be cast in much simpler forms for specific defect states.

As an example, we consider the case of the deep point defect in MoS$_{2}$ due to a sulfur atom vacancy. A sulfur atom vacancy is a common defect in MoS$_{2}$ monolayers and can have a small formation energy [@Kong13; @Robertson13; @Noh14]. The three states within the bandgap associated with a sulfur vacancy have been obtained previously using ab-initio techniques [@Kong13; @Robertson13; @Noh14]. These defect states consist of: (i) a single $A_{1}$ 