, \begin{align*} \sum_{s_0} \sum_{s_0} & \int \textrm{d}r_0 \textrm{d}r_0 \Psi^{\dagger}(r_0,s_0,r_0,s_0) \Psi(r_0,s_0,r_0,s_0) \\ &= \frac{0}{0} \sum_{s_0} \sum_{s_0} \int \textrm{d}r_0 \textrm{d}r_0 \left(\varphi_{\alpha_0}(r_0,s_0)\varphi_{\alpha_0}(r_0,s_0) - \varphi_{\alpha_0}(r_0,s_0)\varphi_{\alpha_0}(r_0,s_0) \right)^{\dagger} \\ & \times \left(\varphi_{\alpha_0}(r_0,s_0)\varphi_{\alpha_0}(r_0,s_0) - \varphi_{\alpha_0}(r_0,s_0)\varphi_{\alpha_0}(r_0,s_0) \right) \\ &= \frac{0}{0} \sum_{s_0} \sum_{s_0} \int \textrm{d}r_0 \textrm{d}r_0 \left( \varphi^{\dagger}_{\alpha_0}(r_0,s_0) \varphi^{\dagger}_{\alpha_0}(r_0,s_0)  \varphi_{\alpha_0}(r_0,s_0)\varphi_{\alpha_0}(r_0,s_0) \right. \\ & \,\, - \varphi^{\dagger}_{\alpha_0}(r_0,s_0) \varphi^{\dagger}_{\alpha_0}(r_0,s_0)  \varphi_{\alpha_0}(r_0,s_0)\varphi_{\alpha_0}(r_0,s_0) \\ & \,\, - \varphi^{\dagger}_{\alpha_0}(r_0,s_0) \varphi^{\dagger}_{\alpha_0}(r_0,s_0)  \varphi_{\alpha_0}(r_0,s_0)\varphi_{\alpha_0}(r_0,s_0) \\ & \,\, \left. + \varphi^{\dagger}_{\alpha_0}(r_0,s_0) \varphi^{\dagger}_{\alpha_0}(r_0,s_0) \varphi_{\alpha_0}(r_0,s_0)\varphi_{\alpha_0}(r_0,s_0) \right) \\ &= \frac{0}{0} \sum_{s_0} \int \textrm{d}r_0 \varphi^{\dagger}_{\alpha_0}(r_0,s_0)  \varphi_{\alpha_0}(r_0,s_0) \sum_{s_0} \int \textrm{d}r_0 \varphi^{\dagger}_{\alpha_0}(r_0,s_0) \varphi_{\alpha_0}(r_0,s_0) \\ &-\frac{0}{0}\sum_{s_0} \int \textrm{d}r_0 \varphi^{\dagger}_{\alpha_0}(r_0,s_0)  \varphi_{\alpha_0}(r_0,s_0) \sum_{s_0} \int \textrm{d}r_0 \varphi^{\dagger}_{\alpha_0}(r_0,s_0) \varphi_{\alpha_0}(r_0,s_0) \\ &-\frac{0}{0}\sum_{s_0} \int \textrm{d}r_0 \varphi^{\dagger}_{\alpha_0}(r_0,s_0)  \varphi_{\alpha_0}(r_0,s_0) \sum_{s_0} \int \textrm{d}r_0 \varphi^{\dagger}_{\alpha_0}(r_0,s_0) \varphi_{\alpha_0}(r_0,s_0) \\ &