atom, one may write $c_{{c \atop v},s}(\vec{k}) \approx \chi_{d}(\vec{k}) e^{i\gamma\tau\phi_{\vec{k}}/2} b_{{c \atop v},s}$. Since $e^{i\gamma\tau\phi_{\vec{k}}/2}  u_{{c \atop v},\vec{k},s}(\vec{r})$ does not vary much with $\vec{k}$ near the band extrema, the sum in (\[eq:defect1\]) can be rearranged to give, $$\psi_{d}(\vec{r}) =  \chi_{d}(\vec{r}) \sum_{{n=c,v \atop s}} b_{n,s} e^{i\vec{K}_{s}.\vec{r}} e^{i\gamma\tau\phi_{\underline{\vec{k}}}/2} u_{n,\underline{\vec{k}},s}(\vec{r}) \label{eq:defect2}$$ Here, the line under $\vec{k}$ means that any wavevector near the band extrema can be chosen. The function $\chi_{d}(\vec{r})$ is expected to be localized at the defect, becoming very small at the second nearest Mo atom near the defect site.

Hamiltonian for the Capture of Holes and Electrons {#sec:Hc}
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Consider process (b) in Fig.\[fig:auger\_1\] in which a hole scatters off an electron and is captured by a defect and the electron is scattered to a higher energy. The relevant term in the Coulomb interaction Hamiltonian that describes the hole capture process in Fig.\[fig:auger\_1\](b) can be written as, $$H_{hc} =  \frac{1}{A} \sum_{\vec{k},\vec{k}',\vec{q},s,s'} V(\vec{q}) M_{s,s'}(\vec{k},\vec{k}',\vec{q}) c^{\dagger}_{\vec{k}+\vec{q},s} b^{\dagger}_{\vec{k}',s'} d_{\sigma'} c_{\vec{k},s}   + h.c.$$ The overlap factor $M_{s,s'}(\vec{k},\vec{k}',\vec{q})$ equals, $$\begin{aligned}
& & M_{s,s'}(\vec{k},\vec{k}',\vec{q})  =  \langle v_{c,\vec{k}+\vec{q},s} | v_{c,\vec{k},s} \rangle \nonumber \\
& & \times  \sum_{n=c,v} b_{n,s'} e^{i\gamma\tau'\phi_{\underline{\vec{k}}}/2} \langle v_{v,\vec{k}',s'} | v_{n,\underline{\vec{k}},s'} \rangle \nonumber \\
& & \times \frac{1}{\sqrt{A}} \int d^{2}\vec{r} \, \chi_{d}(\vec{r}) \, e^{-i(\vec{k}' + \vec{q}).\vec{r}} 