From e59eea073678661c04bf9f4a7f1a6d899c519906 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Mon, 25 Nov 2019 22:21:10 +0100
Subject: [PATCH] expand Ew

---
 Manuscript/FarDFT.bib | 13 ++++++-------
 Manuscript/FarDFT.tex | 20 +++++++++++++-------
 2 files changed, 19 insertions(+), 14 deletions(-)

diff --git a/Manuscript/FarDFT.bib b/Manuscript/FarDFT.bib
index bbb3834..9d58519 100644
--- a/Manuscript/FarDFT.bib
+++ b/Manuscript/FarDFT.bib
@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2019-11-23 21:56:54 +0100 
+%% Created for Pierre-Francois Loos at 2019-11-25 22:20:38 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -17,7 +17,8 @@
 	Pages = {2406--2412},
 	Title = {Selected applications of hyperspherical harmonics in quantum theory},
 	Volume = {97},
-	Year = {1993}}
+	Year = {1993},
+	Bdsk-Url-1 = {https://doi.org/10.1021/j100112a048}}
 
 @book{AveryBook,
 	Address = {Dordrecht},
@@ -31,14 +32,12 @@
 @article{Loos_2019,
 	Author = {Loos, Pierre-Fran{\c c}ois and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
 	Date-Added = {2019-11-21 21:56:17 +0100},
-	Date-Modified = {2019-11-21 21:56:23 +0100},
+	Date-Modified = {2019-11-25 22:14:42 +0100},
 	Doi = {10.1021/acs.jctc.8b01205},
-	Eprint = {https://doi.org/10.1021/acs.jctc.8b01205},
 	Journal = {J. Chem. Theory Comput.},
 	Number = {3},
 	Pages = {1939--1956},
 	Title = {Reference Energies for Double Excitations},
-	Url = {https://doi.org/10.1021/acs.jctc.8b01205},
 	Volume = {15},
 	Year = {2019},
 	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}}
@@ -1935,10 +1934,10 @@
 @article{Boggio-Pasqua_2007,
 	Author = {{Boggio-Pasqua}, Martial and Bearpark, Michael J. and Robb, Michael A.},
 	Date-Added = {2018-10-24 22:38:52 +0200},
-	Date-Modified = {2018-10-24 22:38:52 +0200},
+	Date-Modified = {2019-11-25 22:15:16 +0100},
 	Doi = {10.1021/jo070452v},
 	Issn = {0022-3263, 1520-6904},
-	Journal = {The Journal of Organic Chemistry},
+	Journal = {J. Org. Chem.},
 	Language = {en},
 	Month = jun,
 	Number = {12},
diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex
index d5f1d6f..b79e508 100644
--- a/Manuscript/FarDFT.tex
+++ b/Manuscript/FarDFT.tex
@@ -600,8 +600,7 @@ For HF, we have
 \end{split}
 \end{equation}
 which is clearly quadratic with respect to $\ew{}$ due to the ghost interaction error in the Hartree term.
-
-For LDA, we have
+In the case of the LDA, it reads
 \begin{equation}
 \begin{split}
 	\bE{\LDA}{\ew{}} 
@@ -613,12 +612,11 @@ For LDA, we have
 	+ 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
 	\\
 	& + (1-\ew{}) \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{}
-	+ \ew{} \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{}
+	+ \ew{} \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{},
 \end{split}
 \end{equation}
 which is also clearly quadratic with respect to $\ew{}$ because the (weight-independent) LDA functional cannot compensate the ``quadraticity'' of the Hartree term.
-
-For eLDA, we have
+For eLDA, the ensemble energy can be decomposed as
 \begin{equation}
 \begin{split}
 	\bE{\eLDA}{\ew{}} 
@@ -634,9 +632,17 @@ For eLDA, we have
 	\\
 	& + (1-\ew{})\ew{} \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{}
 	+ \ew{}(1-\ew{}) \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{}
+	\\
+	& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} 
+	+ (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{} ]
+	+ \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{} ]
+	\\
+	& + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12} 
+	+ \frac{1}{2} \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{}
+	+ \frac{1}{2} \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{} ],
 \end{split}
 \end{equation}
-which \textit{could} be linear with respect to the weight if the weight-dependent xc functional is very well constructed.
+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}
 
@@ -645,7 +651,7 @@ This would be, for example, the case with the exact xc functional.
 %%%%%%%%%%%%%%%%%%
 \section{Conclusion}
 \label{sec:ccl}
-As concluding remarks, we would like to say that, what we have done is awesome.
+As concluding remarks, we would like to say that what we have done is awesome.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %%% ACKNOWLEDGEMENTS %%%