\omega)$ and $(j, E-\hbar \omega)$. Within the Born approximation the in-scattering self-energy into a fully empty state is [@Mahan87] $$\Sigma_{\text{S}}^{\text{in}} (E) = D_{0} [ n_{\text{B}} G^{\text{n}} (E-\hbar \omega) + (n_{\text{B}} + 1) G^{\text{n}} ( E + \hbar \omega) ]  .
\label{eq8}$$ where $D_0$ represents the e-ph scattering strength at grid point $j$, $n_{\text{B}}$ is the Bose-Einstein distribution function for phonons of energy $\hbar \omega$, and $G^{n} (E-\hbar \omega)$ is the electron density at $E-\hbar \omega$. The first and second terms in Eq. (\[eq8\]) represent in-scattering of electrons from $E - \hbar \omega$ (phonon absorption) and $E + \hbar \omega$ (phonon emission) to $E$, respectively. The out-scattering self energy, $\Sigma_{\text{S}}^{\text{out}} (E)$, from a fully filled state at energy $E$ is given by [@Mahan87] $$\Sigma_{\text{S}}^{\text{out}} (E) = D_{0} [ (n_{\text{B}} + 1) G^{\text{p}} (E-\hbar \omega) + n_{\text{B}} G^{\text{p}} ( E + \hbar \omega) ]  ,
\label{eq9}$$ where $G^{\text{p}} (E-\hbar \omega) $ and $G^{\text{p}} (E+\hbar \omega)$ are the densities of unoccupied states at $E-\hbar \omega$ and $E+\hbar \omega$. The first and second terms in Eq. (\[eq9\]) represent out-scattering of electrons from $E$ to $E-\hbar \omega$ (phonon emission) and $E+\hbar \omega$ (phonon absorption), respectively. In the case of acoustic phonons, $\hbar \omega \rightarrow 0$, 