B^{\dagger}_{s,s',\alpha}(\vec{Q})$$ Here, $\vec{k}_{r}$ and $\vec{Q}$ equal $(m_{h}/m_{ex})\vec{k} + (m_{e}/m_{ex})\vec{k}'$ and $\vec{k}-\vec{k}'$ on the left hand side, respectively. Products of electron and hole creation and destruction operators can thus be expressed in terms of the exciton operators.

![The computed orbitals of the defect states in a MoS$_{2}$ monolayer corresponding to a sulfur vacancy are shown (from Noh et al. [@Noh14]). (a) and (b) show $A_{1}$ state and the two degenerate $E$ states, respectively.[]{data-label="fig:s_vac"}](./s_vac_b){width=".45\textwidth"}

Defect States
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TMDs ($MX_{2}$), and in particular Monolayer MoS$_{2}$, are known to have several different kinds of point defects, such as $M$ and $X$ vacancies and interstitials, impurity atoms, in addition to grain boundaries and dislocations [@Sofo04; @Komsa12; @Seifert13; @Kong13; @Noh14; @Guinea14; @Robertson13; @Hao13; @VanDerZande13]. The goal in this Section is not to give a detailed description of different defect states in TMDs, something well beyond the scope of this paper, but to capture the essential physics in a way that would enable us to obtain capture rates for electrons and holes and present the main ideas associated with the capture processes.

Since the Bloch states form a complete set, the wavefunction $\psi_{d}(\vec{r})$ of the electron in the defect state can be expanded in terms of the 