(\[eq:Rec\_2\]), $G_{\alpha} + g_{s,s',\alpha}(\vec{r}=0)$, describes the enhancement in the probability of finding an electron and a hole close to each other as a result of the attractive Coulomb interactions. It is interesting to compare the electron capture rate in (\[eq:Rec\_2\]) with the result obtained assuming no electron-hole attractive interaction (i.e. $H_{eh}=0$), $$\begin{aligned}
R_{ec} & = & \frac{2\pi}{\hbar} n_{d} \left( 1-f_{d} \right) \nonumber \\  
& \times & \sum_{s,s'} D_{v,s'}(q_{o}) |V(q_{o})|^{2} |\chi_{d}(q_{o})|^{2}  |b_{c,s}|^{2} n_{s} p_{s'} \label{eq:Rec_3}\end{aligned}$$ where $q_{o}$ is approximately given by the relation, $E_{v,s'}(0) - E_{v,s'}(q_{o}) = E_{g} - E_{d}$. It can be seen that the capture rate in (\[eq:Rec\_2\]) is larger by the same enhancement factor. Assuming all the electrons and holes are in the lowest ($\alpha =1$) bound exciton state, values of $D_{v,s'}$ and $|b_{c,s}|$ are independent of the valley/spin indices, and the exciton density is $n_{ex} = \sum_{s,s'} n_{s,s',\alpha=1}$, the comparison between (\[eq:Rec\_2\]) and (\[eq:Rec\_3\]) shows that the enhancement of the electron capture rate in the case of excitons is roughly proportional to $G_{\alpha=1} + |\phi_{\alpha=1}(\vec{r}=0)|^{2}/n_{ex}$. Given that the radius of the lowest exciton state in monolayer MoS$_{2}$ is in the 7-10 $\AA$ range [@Changjian14], the enhancement, assuming an exciton density of $10^{12}$ cm$^{-2}$, is in the 72-138 