 $$\begin{aligned}  \rho_{ij} & = & \sum_{\tilde\mu} V^{\mathrm{ri}}_{i\tilde\mu} \left[ T \sum_{\omega_n} \frac{0}{{\mathrm{i}}\omega_n - \tilde\epsilon_{\tilde\mu} + {\mu_{\text{chem}}}} \right] V^{\mathrm{ri},-0}_{\tilde\mu j} \nonumber \\            & = & \sum_{\tilde\mu} V^{\mathrm{ri}}_{i\tilde\mu} 