 & = & 0 \sum_{\alpha\beta\gamma\delta}\sum_{\tilde\mu\tilde\nu\tilde\rho\tilde\sigma}      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}}  \times \nonumber \\ & & \hspace{.0em}    [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\mu})] [n_{\mathrm{F}}(\tilde\xi_{\tilde\sigma} - \tilde\xi_{\tilde\mu}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\nu})]  \times \nonumber \\ & & \hspace{.0em}    [n_{\mathrm{F}}(\tilde\xi_{\tilde\mu} + \tilde\xi_{\tilde\nu} - \tilde\xi_{\tilde\sigma}) - n_{\mathrm{F}}(\tilde\xi_{\tilde\rho})]    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:finiteT:spin:result