From 4c1a44454639246d8f7b497a543f33994d200e9c Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Thu, 28 Nov 2019 12:16:46 +0100
Subject: [PATCH] Clotilde discovered a typo

---
 Manuscript/FarDFT.tex | 37 ++++++++++++-------------------------
 1 file changed, 12 insertions(+), 25 deletions(-)

diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex
index f62366f..9f18f9b 100644
--- a/Manuscript/FarDFT.tex
+++ b/Manuscript/FarDFT.tex
@@ -700,17 +700,10 @@ For eLDA, the ensemble energy can be decomposed as
 \end{equation}
 which \textit{could} be linear with respect to $\ew{}$ if the weight-dependent xc functional compensates exactly the quadratic terms in the Hartree term.
 This would be, for example, the case with the exact xc functional.
-\end{widetext}
 
 Extracting excitation energies from Eqs.~\eqref{eq:bEwHF}, \eqref{eq:bEwLDA} and \eqref{eq:bEweLDA} is more tricky.
 To do so, we will employ Eq.~\eqref{eq:dEdw}.
-The two first terms are simply
-\begin{align}
-	\Eps{0}{\ew{}} & = 2 \eps{1}{\ew{}},
-	&
-	\Eps{1}{\ew{}} & = 2 \eps{2}{\ew{}},
-\end{align}
-and the HF, LDA and eLDA weight-dependent orbital energies are
+The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew{}} = 2 \eps{2}{\ew{}}$, and the HF, LDA and eLDA weight-dependent orbital energies are
 \begin{subequations}
 \begin{align}
 	\eps{1}{\ew{},\HF} 
@@ -723,42 +716,36 @@ and the HF, LDA and eLDA weight-dependent orbital energies are
 
 \begin{subequations}
 \begin{align}
-\begin{split}
 	\eps{1}{\ew{},\LDA} 
 	& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
+	+ \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) 
+	+ \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
 	\\
-	& + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
-\end{split}
-	\\
-\begin{split}
 	\eps{2}{\ew{},\LDA} 
 	& = \eHc{2} +  2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22}
-	\\
-	& + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
-\end{split}
+	+ \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) 
+	+ \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
 \end{align}
 \end{subequations}
 
 \begin{subequations}
 \begin{align}
-\begin{split}
 	\eps{1}{\ew{},\eLDA} 
 	& = \eHc{1} + (1-\ew{})(2\eJ{11} - \eK{11}) + \ew{}(2\eJ{12} - \eK{12}) 
+	+ \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) 
+	+ \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
 	\\
-	& + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
-\end{split}
-	\\
-\begin{split}
-	\eps{2}{\ew{},\eLDA} & = \eHc{2} +  (1-\ew{})(2\eJ{12} - \eK{12}) + \ew{}(2\eJ{22} - \eK{22}) 
-	\\
-	& + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
-\end{split}
+	\eps{2}{\ew{},\eLDA} 
+	& = \eHc{2} +  (1-\ew{})(2\eJ{12} - \eK{12}) + \ew{}(2\eJ{22} - \eK{22}) 
+	+ \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) 
+	+ \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
 \end{align}
 \end{subequations}
 respectively.
 
 The derivative discontinuity is modelled by the last term of the RHS of Eq.~\eqref{eq:dEdw}.
 Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}].
+\end{widetext}