From 61c368748d833eb4e28f18ed9d605b8e8ababec5 Mon Sep 17 00:00:00 2001
From: Emmanuel Fromager <manu@Emmanuels-MacBook-Pro.local>
Date: Mon, 9 Mar 2020 18:43:37 +0100
Subject: [PATCH] Manu: saving work in the theory section.

---
 Manuscript/eDFT.tex | 33 ++++++++++++++++++++++++++++++++-
 1 file changed, 32 insertions(+), 1 deletion(-)

diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
index 15097a9..e92df62 100644
--- a/Manuscript/eDFT.tex
+++ b/Manuscript/eDFT.tex
@@ -919,13 +919,44 @@ of Eq.~\eqref{eq:EI-eLDA}.
 \Big)
 d\br{}
 \\
-&
+&=\int 
+\Big(\be{c}{(I)}(\n{\bGam{\bw}}{}(\br{}))
+-
+\e{c}{\bw}(\n{\bGam{\bw}}{}(\br{}))
+\Big)\,\n{\bGam{\bw}}{}(\br{})
+ d\br{}
 %\sum_{K>0}\delta_{IK}\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})}
 \end{split}
 \eeq
+thus leading to the following Taylor expansion
 \beq
+\begin{split}
+&
 \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
       \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
+\\
+&=-\int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
+\\
+&+\int \be{c}{(I)}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
+\\
+&+\int \Bigg[
+\n{\bGam{(I)}}{}(\br{}) 
+\left.\left(
+\pdv{\be{c}{{(I)}}(\n{}{})}{\n{}{}} 
+-
+\pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} 
+\right)\right|_{\n{}{} =
+\n{\bGam{(I)}}{}(\br{})}
+\\
+&+\be{c}{(I)}(\n{\bGam{(I)}}{}(\br{}))
+-
+\e{c}{\bw}(\n{\bGam{(I)}}{}(\br{}))\Bigg]\times
+\Big(\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})\Big)
+d\br{}
+\\
+&
++\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right).
+\end{split}
 \eeq
 }