From 3b9c09ce2e6cfd935b7e2f86f5b3562b3a22133b Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Fri, 14 Feb 2020 10:03:34 +0100 Subject: [PATCH] Manu: saving work --- Manuscript/eDFT.tex | 31 +++++++++++++++++++++++++++++++ 1 file changed, 31 insertions(+) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 47b31ce..62bc222 100644 --- a/Manuscript/eDFT.tex +++ b/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