Manu: done with self-teaching. Will clean the all thing after lunch

Manuscript/eDFT.tex

@@ -546,31 +546,122 @@ c}\left[n_{\bm\gamma^{\bw}}\right]
 \eeq
 % Manu's derivation %%%%
 \color{blue}
-Fock operator:\\
+I am teaching myself ...\\
 Stationarity condition
 \beq
-&&\sum_{t^\sigma}\Big(f_{p^\sigma\sigma,t^\sigma\sigma}\Gamma^{(K)\sigma}_{t^\sigma
+&&0=\sum_{K\geq 0}w_K\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}
+&&=\sum_{K\geq 0}w_K
+\Big(f_{p^\sigma\sigma,q^\sigma\sigma}n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}f_{p^\sigma\sigma,q^\sigma\sigma}\Big)
 \nonumber\\
-&&=
+&&
+=\sum_{\mu\nu}\sum_{K\geq 0}w_KF_{\mu\nu}^\sigma c^\sigma_{\mu
+p}c^\sigma_{\nu q}\left(n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}\right)
+\eeq
+thus leading to
+\beq
+&&0=\sum_{p^\sigma q^\sigma}c^\sigma_{\lambda
+p}c^\sigma_{\omega q}\left(\sum_{\mu\nu}\sum_{K\geq 0}w_KF_{\mu\nu}^\sigma c^\sigma_{\mu
+p}c^\sigma_{\nu q}\left(n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}\right)\right)
+\nonumber\\
+&&=\sum_{\mu\nu}\sum_{K\geq 0}w_K
+F_{\mu\nu}^\sigma\left(\Gamma^{(K)\sigma}_{\nu\omega}\sum_{p^\sigma}c^\sigma_{\lambda
+p}c^\sigma_{\mu
+p}-\Gamma^{(K)\sigma}_{\mu\lambda}\sum_{q^\sigma}c^\sigma_{\omega q}c^\sigma_{\nu q}\right)
+\nonumber\\
+\eeq
+If we denote $M^\sigma_{\lambda\mu}=\sum_{p^\sigma}c^\sigma_{\lambda
+p}c^\sigma_{\mu
+p}$ it comes 
+\beq
+S_{\mu\nu}=\sum_{\lambda\omega}S_{\mu\lambda}M^\sigma_{\lambda\omega}S_{\omega\nu}
+\eeq
+which simply means that
+\beq
+{\bm S}={\bm S}{\bm M}{\bm S}
+\eeq
+or, equivalently,
+\beq
+{\bm M}={\bm S}^{-1}.
+\eeq
+The stationarity condition simply reads
+\beq
+\sum_{\mu\nu}F_{\mu\nu}^\sigma\left(\Gamma^{\bw\sigma}_{\nu\omega}
+\left[{\bm S}^{-1}\right]_{\lambda\mu}
+-\Gamma^{\bw\sigma}_{\mu\lambda}\left[{\bm S}^{-1}\right]_{\omega\nu}\right)
+=0
+\eeq
+thus leading to
+\beq
+{\bm S}^{-1}{{\bm F}^\sigma}{\bm \Gamma}^{\bw\sigma}={\bm \Gamma}^{\bw\sigma}{{\bm F}^\sigma}{\bm S}^{-1}
+\eeq
+or, equivalently,
+\beq
+{{\bm F}^\sigma}{\bm \Gamma}^{\bw\sigma}{\bm S}={\bm S}{\bm
+\Gamma}^{\bw\sigma}{{\bm F}^\sigma}.
 \eeq
 %%%%%
+
+Fock operator:\\
 \beq
-&&f_{p^\sigma\sigma,q^\sigma\sigma}=\langle\varphi_p^\sigma\vert\hat{h}\vert\varphi_q^\sigma\rangle
+&&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}
+&&=\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}
+&&
+\Big(\langle p^\sigma r^\tau\vert
+q^\sigma s^\tau\rangle
 -\delta_{\sigma\tau}\langle p^\sigma r^\sigma\vert
-s^\sigma q^\sigma\rangle\Gamma^{(L)\tau}_{r^\tau
-s^\tau}
+s^\sigma q^\sigma\rangle
 \Big)
+\Gamma^{(L)\tau}_{r^\tau
+s^\tau}
+\nonumber\\
+&&
+=\sum_{L\geq 0}w_L\sum_{\tau}\sum_{r^\tau}\Big(\langle p^\sigma r^\tau\vert
+q^\sigma r^\tau\rangle
+-\delta_{\sigma\tau}\langle p^\sigma r^\tau\vert
+r^\tau q^\sigma\rangle
+\Big)
+n^{(L)\tau}_{r^\tau}
+\nonumber\\
+&&=\sum_{L\geq 0}w_L
+\sum_{\lambda\omega}\sum_{\tau}\Big[\langle
+p^\sigma\lambda\vert q^\sigma\omega\rangle
+-\delta_{\sigma\tau}
+\langle
+p^\sigma\lambda\vert \omega q^\sigma\rangle\Big]
+\Gamma^{(L)\tau}_{\lambda\omega}
+\nonumber\\
+&&=
+\sum_{\lambda\omega}\sum_{\tau}\Big[\langle
+p^\sigma\lambda\vert q^\sigma\omega\rangle
+-\delta_{\sigma\tau}
+\langle
+p^\sigma\lambda\vert \omega q^\sigma\rangle\Big]
+\Gamma^{\bw\tau}_{\lambda\omega}
+\nonumber\\
+&&=\sum_{\mu\nu\lambda\omega}\sum_{\tau}
+\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
+-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle
+\Big)\Gamma^{\bw\tau}_{\lambda\omega}c^\sigma_{\mu p}c^\sigma_{\nu q}
+\nonumber\\
+\eeq
+or, equivalently,
+\beq
+f_{p^\sigma\sigma,q^\sigma\sigma}=\sum_{\mu\nu}F_{\mu\nu}^\sigma c^\sigma_{\mu p}c^\sigma_{\nu q}
+\eeq
+where
+\beq
+F_{\mu\nu}^\sigma=h_{\mu\nu}+\sum_{\lambda\omega}\sum_\tau
+G_{\mu\nu\lambda\omega}^{\sigma\tau}\Gamma^{\bw\tau}_{\lambda\omega}
+\eeq
+and
+\beq
+G_{\mu\nu\lambda\omega}^{\sigma\tau}=({\mu}{\nu}\vert{\lambda}{\omega})
+-\delta_{\sigma\tau}(\mu\omega\vert\lambda\nu)
 \eeq
 \color{black}
 \\