$|b_{v,s}|^{2}$ are assumed to be independent of the valley/spin indices. This is a good approximation for many important cases. For example, in the case of the sulfur vacancy in MoS$_{2}$ discussed earlier, the $E$ states have a total weight of $\sim$0.25 on the $d_{z^{2}}$ orbitals of the Mo atoms adjacent to the missing sulfur atom [@Yong14]. Since the conduction band Bloch states of both $K$ and $K'$ valleys are made up of mostly the $d_{z^{2}}$ orbitals of Mo atoms, $|b_{c,s}|^{2}$ is the same for both the valleys. We approximate the envelope, $\chi_{d}(\vec{r})$, of the defect state wavefunction in (\[eq:defect2\]) by a Gaussian, $\chi_{d}(\vec{r}) = \sqrt{2/(\pi a_{d}^{2})}e^{-r^{2}/a_{d}^{2}}$, where $a_{d} \approx 3$ $\AA$ (see Fig.\[fig:s\_vac\]). Note that the in-plane S-Mo bound length in MoS$_{2}$ is $\sim$1.83 $\AA$. Fig.\[fig:capture1\] plots the computed capture times of electrons ($\tau_{ec}$) and holes ($\tau_{hc}$) of excitons assuming that all the excitons are in the lowest state ($\alpha=1$). In the low exciton density limit considered here these capture times are independent of the exciton density. The defect density $n_{d}$ is assumed to be $2\times 10^{11}$ cm$^{-2}$. The capture times for electrons and holes shown in Fig.\[fig:capture1\] have been normalized by multiplying them by $|b_{c}|^{2}$ and $|b_{v}|^{2}$, respectively, given 