range, and in the 644-1308 range if the exciton density is assumed to be $10^{11}$ cm$^{-2}$. Therefore, the correlations in the positions of the electrons and the holes as a result of the attractive Coulomb interaction make electrons and holes in tightly bound excitons in TMDs far more susceptible to capture by defects compared to uncorrelated free carriers. Interestingly, even when the exciton density $n_{s,s',\alpha}$ is zero the capture rate in (\[eq:Rec\_2\]) is enhanced by the factors $G_{\alpha}$ compared to the rate in (\[eq:Rec\_3\]) for uncorrelated electrons and holes. Therefore, Coulomb correlations in the positions of electrons and holes due to the attractive interaction between them enhances the Auger scattering rates even at the Hartree-Fock level.

Numerical Results and Discussion
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Carrier Capture Times at Low Exciton Densities {#sec:num1}
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For numerical computations, we consider monolayer MoS$_{2}$ on a quartz substrate, as is the case in many experiments. We first assume that the exciton density is small enough ($\le$$10^{12}$ cm$^{-2}$) to allow one to ignore phase-space filling effects [@Changjian14]. We use the wavevector-dependent dielectric constant $\epsilon(\vec{q})$ for monolayer MoS$_{2}$ on quartz given by Zhang et al. [@Changjian14]. The defect state wavefunction is given in (\[eq:defect2\]). The values of $|b_{c,s}|^{2}$ and 