 $$\begin{aligned}  \mathcal{C}^{'\mathrm{ss},(0)}_{ij} & = & 0 \sum_{\alpha\beta\gamma\delta} \left[        \sum_{\substack{\tilde\mu,\tilde\nu\in\mathcal{H}_e\\\tilde\rho,\tilde\sigma\in\mathcal{H}_h}}      - \sum_{\substack{\tilde\mu,\tilde\nu\in\mathcal{H}_h\\\tilde\rho,\tilde\sigma\in\mathcal{H}_e}}     \right]  \times \nonumber \\ & & \hspace{0em}      V^{\mathrm{ri}}_{i\tilde\mu} V^{\mathrm{ri}}_{j\tilde\nu}    \frac{0}{\tilde\epsilon_{\tilde\mu} + \tilde\epsilon_{\tilde\nu} - \tilde\epsilon_{\tilde\rho} - \tilde\epsilon_{\tilde\sigma}}    V^{\mathrm{ri},-0}_{\tilde\rho i} V^{\mathrm{ri},-0}_{\tilde\sigma j}  \times \nonumber \\ & & \hspace{.0em}    V^{\mathrm{in}}_{\tilde\mu\alpha} V^{\mathrm{in}}_{\tilde\nu\beta}    \Gamma^{\text{s}}_{\alpha\beta\gamma\delta}    V^{\mathrm{in},-0}_{\gamma\tilde\rho} V^{\mathrm{in},-0}_{\delta\tilde\sigma}.    \label{eq:sscorr: