From bf120b9af65550c8d9a9bc7f540c221ffe90ebb2 Mon Sep 17 00:00:00 2001
From: Emmanuel Fromager <manu@Emmanuels-MacBook-Pro.local>
Date: Tue, 5 May 2020 23:15:05 +0200
Subject: [PATCH] Manu: checking LIM formulas

---
 Manuscript/FarDFT.tex | 10 ++++++++++
 1 file changed, 10 insertions(+)

diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex
index a55bf50..f571e47 100644
--- a/Manuscript/FarDFT.tex
+++ b/Manuscript/FarDFT.tex
@@ -750,6 +750,16 @@ a pragmatic way of getting weight-independent excitation energies, defined as
 	\Ex{\LIM}{(2)} & = 3 \qty[\E{}{\bw{}=(1/3,1/3)} - \E{}{\bw{}=(1/2,0)}] + \frac{1}{2} \Ex{\LIM}{(1)}, \label{eq:LIM2}
 \end{align}
 \end{subequations}
+\manu{
+$\frac{1}{2}\Ex{\LIM}{(1)}=\frac{1}{2}\left(E_1-E_0\right)$\\
+$\E{}{\bw{}=(1/3,1/3)}=\frac{1}{3}\left(E_0+E_1+E_2\right)$\\
+$\E{}{\bw{}=(1/2,0)}=\frac{1}{2}\left(E_0+E_1\right)$\\
+$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
+\E{}{\bw{}=(1/2,0)}]=-\frac{1}{2}\left(E_0+E_1\right)+E_2$ 
+\\
+$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
+\E{}{\bw{}=(1/2,0)}]+\frac{1}{2} \Ex{\LIM}{(1)}=E_2-E_0$
+}\\
 which require three independent calculations, as well as the MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
 \begin{subequations}
 \begin{align}