From 3a480b0adca84ebc4d6ced7640782517c0425895 Mon Sep 17 00:00:00 2001
From: Emmanuel Fromager <manu@Emmanuels-MacBook-Pro.local>
Date: Thu, 13 Feb 2020 20:18:16 +0100
Subject: [PATCH] Manu: (hopefully) found proper notations for density matrices
 and Coulomb-exchange integrals

---
 Manuscript/eDFT.tex | 32 +++++++++++++++++++++++++++-----
 1 file changed, 27 insertions(+), 5 deletions(-)

diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
index fdacc2d..47b31ce 100644
--- a/Manuscript/eDFT.tex
+++ b/Manuscript/eDFT.tex
@@ -415,7 +415,7 @@ integrals.
 \blue{
 \beq
 &&\dfrac{1}{2}\sum_{PQRS}\langle PQ\vert\vert
-RS\rangle\Gamma_{PR}\Gamma_{QS}
+RS\rangle\Gamma^{(K)}_{PR}\Gamma^{(L)}_{QS}
 \nonumber\\
 &&
 =\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}RS}
@@ -423,19 +423,41 @@ RS\rangle\Gamma_{PR}\Gamma_{QS}
 &&\Big(\langle p^\sigma\sigma q^\tau\tau\vert RS\rangle -\langle
 p^\sigma\sigma q^\tau\tau
 \vert SR\rangle
-\Big)\Gamma_{p^\sigma\sigma,R}\Gamma_{q^\tau\tau, S}
+\Big)\Gamma^{(K)}_{p^\sigma\sigma,R}\Gamma^{(L)}_{q^\tau\tau, S}
 \nonumber\\
 &&
 =\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
 \nonumber\\
 &&\Big(\sum_{r^\sigma s^\tau}\langle p^\sigma q^\tau\vert r^\sigma s^\tau\rangle
-\Gamma^\sigma_{p^\sigma r^\sigma}\Gamma^\tau_{q^\tau s^\tau}
+\Gamma^{(K)\sigma}_{p^\sigma r^\sigma}\Gamma^{(L)\tau}_{q^\tau s^\tau}
 \nonumber\\
 && -\sum_{s^\sigma r^\tau}\langle
 p^\sigma q^\tau
 \vert s^\sigma r^\tau\rangle
-\delta_{\sigma\tau}\Gamma^\sigma_{p^\sigma
-r^\sigma}\Gamma^\sigma_{q^\sigma s^\sigma}\Big)
+\delta_{\sigma\tau}\Gamma^{(K)\sigma}_{p^\sigma
+r^\sigma}\Gamma^{(L)\sigma}_{q^\sigma s^\sigma}\Big)
+\nonumber\\
+&&=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
+\nonumber\\
+&&\left(\langle p^\sigma q^\tau\vert p^\sigma q^\tau\rangle
+n_{p^\sigma}^{(K)\sigma}n_{q^\tau}^{(L)\tau}
+-\delta_{\sigma\tau}\langle p^\sigma q^\sigma\vert q^\sigma p^\sigma \rangle
+n_{p^\sigma}^{(K)\sigma}n_{q^\sigma}^{(L)\sigma}\right) 
+\nonumber\\
+&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
+\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
+-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\sigma}
+\Big)
+\nonumber\\
+&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
+-\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle
+\Big)
+\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
+\nonumber\\
+&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big[({\mu}{\nu}\vert{\lambda}{\omega})
+-\delta_{\sigma\tau}(\mu\omega\vert\lambda\nu)
+\Big]
+\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
 \eeq
 }
 %%%%