Manu: saving work

Manuscript/eDFT.tex

@@ -544,6 +544,37 @@ c}\left[n_{\bm\gamma^{\bw}}\right]
 \Big\}.
 \nonumber\\
 \eeq
+% Manu's derivation %%%%
+\color{blue}
+Fock operator:\\
+Stationarity condition
+\beq
+&&\sum_{t^\sigma}\Big(f_{p^\sigma\sigma,t^\sigma\sigma}\Gamma^{(K)\sigma}_{t^\sigma
+q^\sigma}-\Gamma^{(K)\sigma}_{p^\sigma
+t^\sigma}f_{t^\sigma\sigma,q^\sigma\sigma}\Big)
+\nonumber\\
+&&=
+f_{p^\sigma\sigma,q^\sigma\sigma}n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}f_{p^\sigma\sigma,q^\sigma\sigma}
+\nonumber\\
+&&=
+\eeq
+%%%%%
+\beq
+&&f_{p^\sigma\sigma,q^\sigma\sigma}=\langle\varphi_p^\sigma\vert\hat{h}\vert\varphi_q^\sigma\rangle
+\nonumber\\
+&&+\sum_{L\geq 0}w_L\sum_{\tau}\sum_{r^\tau s^\tau}
+\nonumber\\
+&&\Big(\langle p^\sigma r^\tau\vert
+q^\sigma s^\tau\rangle\Gamma^{(L)\tau}_{r^\tau
+s^\tau}
+-\delta_{\sigma\tau}\langle p^\sigma r^\sigma\vert
+s^\sigma q^\sigma\rangle\Gamma^{(L)\tau}_{r^\tau
+s^\tau}
+\Big)
+\eeq
+\color{black}
+\\
+%%%%%
 Note that this approximation, where the ensemble density matrix is
 optimized from a non-local exchange potential [rather than a local one,
 as expected from Eq.~(\ref{eq:var_ener_gokdft})] is applicable to real