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E=\int \mathcal{E}_{\textrm{edf}}(n,\nu,\ldots)\,d^3r+\sum_{\sigma}\int V_{\sigma}^{\textrm{(ext)}}(r)n_{\sigma}(r)\,d^3r\\-\frac{1}{2}\int\left(\Delta^{\textrm{(ext)}}(r)\nu^*(r)+\textrm{h.c.}\right)d^3r-\sum_{\sigma}\int \vec{v}_{\sigma}^{\textrm{(ext)}}(r)\cdot\vec{j}_{\sigma}(r)\,d^3r
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E=\int \mathcal{E}_{\textrm{edf}}(n,\nu,\ldots)\,d^3r+\sum_{\sigma}\int V_{\sigma}^{\textrm{(ext)}}(r)n_{\sigma}(r)\,d^3r\\-\int\left(\Delta^{\textrm{(ext)}}(r)\nu^*(r)+\textrm{h.c.}\right)d^3r-\sum_{\sigma}\int \vec{v}_{\sigma}^{\textrm{(ext)}}(r)\cdot\vec{j}_{\sigma}(r)\,d^3r
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