From 68872cdbc094dcde840eb75dedeedbc4f145ed52 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Tue, 10 Mar 2020 17:26:53 +0100
Subject: [PATCH] theory

---
 Manuscript/EIvsW_n5.pdf |   Bin 106242 -> 106294 bytes
 Manuscript/EvsL_5.pdf   |   Bin 73872 -> 73872 bytes
 Manuscript/EvsW_n5.pdf  |   Bin 87260 -> 87289 bytes
 Manuscript/eDFT.bib     |     4 +-
 Manuscript/eDFT.tex     |   217 +-
 Notebooks/eDFT_FUEG.nb  | 10374 ++++++++------------------------------
 6 files changed, 2212 insertions(+), 8383 deletions(-)

diff --git a/Manuscript/EIvsW_n5.pdf b/Manuscript/EIvsW_n5.pdf
index f764a6059f8e503e51206c4eb5d9ef0ad8d95a4b..11956590296a5a6071106d9210201388389f9735 100644
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diff --git a/Manuscript/eDFT.bib b/Manuscript/eDFT.bib
index 9d41d3d..ae5a45e 100644
--- a/Manuscript/eDFT.bib
+++ b/Manuscript/eDFT.bib
@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-03-09 08:48:49 +0100 
+%% Created for Pierre-Francois Loos at 2020-03-10 16:50:31 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -8418,6 +8418,7 @@
 
 @article{Levy_2014,
 	Author = {Levy, Mel and Zahariev, Federico},
+	Date-Modified = {2020-03-10 16:48:46 +0100},
 	Doi = {10.1103/PhysRevLett.113.113002},
 	File = {/Users/loos/Zotero/storage/4C87D4GM/Levy and Zahariev - 2014 - Ground-State Energy as a Simple Sum of Orbital Ene.pdf},
 	Issn = {0031-9007, 1079-7114},
@@ -8425,6 +8426,7 @@
 	Language = {en},
 	Month = sep,
 	Number = {11},
+	Pages = {113002},
 	Shorttitle = {Ground-{{State Energy}} as a {{Simple Sum}} of {{Orbital Energies}} in {{Kohn}}-{{Sham Theory}}},
 	Title = {Ground-{{State Energy}} as a {{Simple Sum}} of {{Orbital Energies}} in {{Kohn}}-{{Sham Theory}}: {{A Shift}} in {{Perspective}} through a {{Shift}} in {{Potential}}},
 	Volume = {113},
diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
index f8ebe27..9698082 100644
--- a/Manuscript/eDFT.tex
+++ b/Manuscript/eDFT.tex
@@ -48,6 +48,7 @@
 \newcommand{\be}[2]{\overline{\eps}_\text{#1}^{#2}}
 \newcommand{\bv}[2]{\overline{f}_\text{#1}^{#2}}
 \newcommand{\n}[2]{n_{#1}^{#2}}
+\newcommand{\dn}[2]{\Delta n_{#1}^{#2}}
 \newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
 \newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
 
@@ -132,10 +133,10 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{abstract}
 We report a first generation of local, weight-dependent correlation density-functional approximations (DFAs) that incorporate information about both ground and excited states in the context of density-functional theory for ensembles (eDFT). 
-These density-functional approximations for ensembles (eDFAs) are specially designed for the computation of single and double excitations within \titou{Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for excited states)}, and can be seen as a natural extension of the ubiquitous local-density approximation for ensemble (eLDA).
+These density-functional approximations for ensembles (eDFAs) are specially designed for the computation of single and double excitations within Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for excited states), and can be seen as a natural extension of the ubiquitous local-density approximation for ensemble (eLDA).
 The resulting eDFAs, based on both finite and infinite uniform electron gas models, automatically incorporate the infamous derivative discontinuity contributions to the excitation energies through their explicit ensemble weight dependence. 
 Their accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
-\titou{Although the present weight-dependent functional has been specifically designed for one-dimensional systems, the methodology proposed here is directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.}
+Although the present weight-dependent functional has been specifically designed for one-dimensional systems, the methodology proposed here is directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.
 \end{abstract}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -171,7 +172,7 @@ functional of the time-dependent density $\n{}{} \equiv \n{}{}(\br,t)$ and, as
 such, it should incorporate memory effects. Standard implementations of TDDFT rely on 
 the adiabatic approximation where these effects are neglected. \cite{Dreuw_2005} In other
 words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012}
-As a result, double electronic excitations \titou{(where two electrons are simultaneously promoted by a single photon)} are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
+As a result, double electronic excitations (where two electrons are simultaneously promoted by a single photon) are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
 
 When affordable (\ie, for relatively small molecules), time-independent
 state-averaged wave function methods
@@ -222,9 +223,9 @@ extension of the LDA, will be referred to as eLDA in the remaining of this paper
 As a proof of concept, we apply this general strategy to
 ensemble correlation energies (that we combine with 
 ensemble exact exchange energies) in the particular case of
-\emph{strict} one-dimensional (1D) \trashPFL{and}
+\emph{strict} one-dimensional (1D)
 spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b} 
-In other words, the Coulomb interaction used in this work \titou{corresponds} to
+In other words, the Coulomb interaction used in this work corresponds to
 particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
 Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires. \cite{Meyer_2009, Deshpande_2010}
 %Early models of 1D atoms using this interaction have been used to study the effects of external fields upon Rydberg atoms \cite{Burnett_1993, Mayle_2007} and the dynamics of surface-state electrons in liquid helium. \cite{Nieto_2000, Patil_2001}
@@ -254,7 +255,7 @@ Atomic units are used throughout.
 In this section we give a brief review of GOK-DFT and discuss the
 extraction of individual energy levels \cite{Deur_2019,Fromager_2020} with a particular focus on exact
 individual exchange energies. 
-Let us start by introducing the GOK ensemble energy: \cite{Gross_1988a}
+Let us start by introducing the GOK ensemble energy \cite{Gross_1988a}
 \beq\label{eq:exact_GOK_ens_ener}
 	\E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)},
 \eeq
@@ -395,7 +396,7 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
 Note that, when $\bw=0$, the ensemble correlation functional reduces to the
 conventional (ground-state) correlation functional $E_{\rm c}[n]$. As a
 result, the regular KS-DFT expression is recovered from
-Eq.~(\ref{eq:exact_ener_level_dets}) for the ground-state energy:
+Eq.~(\ref{eq:exact_ener_level_dets}) for the ground-state energy
 \beq
 \E{}{(0)}=\mel*{\Det{(0)}}{\hH}{\Det{(0)}} +
 \E{c}{}[\n{\Det{(0)}}{}],
@@ -472,7 +473,7 @@ where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatia
 }
 \fi%%%%%%%%%%%%%%%%%%%%%
 then the density matrix of the   
-determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
+determinant $\Det{(K)}$ can be expressed as follows in the AO basis
 \beq
 	\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
 \eeq
@@ -668,7 +669,7 @@ following approximation:
 	}.
 \eeq
 The minimizing ensemble density matrix in Eq.~(\ref{eq:min_with_HF_ener_fun}) fulfills the following
-stationarity condition:
+stationarity condition
 \beq\label{eq:commut_F_AO}
 	\bF{\bw} \bGam{\bw} \bS = \bS \bGam{\bw} \bF{\bw},
 \eeq
@@ -812,7 +813,7 @@ Eq.~\eqref{eq:var_ener_gokdft}. This procedure is actually general, \ie,
 applicable to not-necessarily spin polarized and real (higher-dimensional) systems. 
 As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, inserting the
 ensemble density matrix into the HF interaction energy functional
-introduces unphysical \textit{ghost interaction} errors \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
+introduces unphysical \textit{ghost interaction} errors \titou{(GIE)} \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
 as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
 \beq\label{eq:WHF}
 \begin{split}
@@ -828,17 +829,15 @@ These errors are essentially removed when evaluating the individual energy
 levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.
 
 Turning to the density-functional ensemble correlation energy, the
-following ensemble local density \textit{approximation} (eLDA) will be employed:  
+following ensemble local-density \textit{approximation} (eLDA) will be employed  
 \beq\label{eq:eLDA_corr_fun}
 	\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
 \eeq
-\manu{
 where the ensemble correlation energy per particle 
 \beq\label{eq:decomp_ens_correner_per_part}
 \e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{})
 \eeq
-}
-is \textit{weight dependent}. 
+is \titou{explicitly} \textit{weight dependent}. 
 As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed
 from a finite uniform electron gas model. 
 %\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC.
@@ -853,40 +852,48 @@ reads
 %Manu, would it be useful to add this equation and the corresponding text?
 %I think it is useful for the discussion later on when we talk about the different contributions to the excitation energies.
 %This shows clearly that there is a correction due to the correlation functional itself as well as a correction due to the ensemble correlation derivative
-
-Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression within KS-eLDA: 
-\beq\label{eq:EI-eLDA}
+Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our \titou{final expression of the KS-eLDA energy level} 
+\titou{\beq\label{eq:EI-eLDA}
 \begin{split}
 	\E{{eLDA}}{(I)} 
-	& = 
-        \E{HF}{(I)}
-	\\
-%\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
-	& + \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
+	= 
+	\E{HF}{(I)} 
+	+ \Xi_\text{c}^{(I)}
+	+ \Upsilon_\text{c}^{(I)},
+\end{split}
+\eeq}
+where
+\beq\label{eq:ind_HF-like_ener}
+\E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
+\eeq   
+is the analog for ground and excited states (within an ensemble) of the HF energy, \titou{and
+\begin{gather}
+\begin{split}
+	\Xi_\text{c}^{(I)} 
+	& = \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
 	\\
 	&
 	+ \int \n{\bGam{\bw}}{}(\br{}) \qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
 	\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})} d\br{}
 	\\
-	& + \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{},
 \end{split}
-\eeq
-where
-\beq\label{eq:ind_HF-like_ener}
-\E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
-\eeq   
-is the analog for ground and excited states (within an ensemble) of the HF energy.  
+\\
+	\Upsilon_\text{c}^{(I)} 
+	= \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{}.
+\end{gather}}
+
 If, for analysis purposes, we Taylor expand the density-functional
 correlation contributions
 around the $I$th KS state density
 $\n{\bGam{(I)}}{}(\br{})$, the sum of 
-the second and third terms on the right-hand side
+the \titou{second term} on the right-hand side
 of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
-$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: 
+ \titou{$\dn{\bGam{\bw}}{(I)}(\br{}) = \n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$}: 
 \beq\label{eq:Taylor_exp_ind_corr_ener_eLDA}
-\int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
-+\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right).
+	\titou{\Xi_\text{c}^{(I)}}
+	= \int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
+	+ \titou{\order{[\dn{\bGam{\bw}}{(I)}(\br{})]^2}}.
 \eeq
 Therefore, it can be identified as 
 an individual-density-functional correlation energy where the density-functional
@@ -898,46 +905,40 @@ Let us stress that, to the best of our knowledge, eLDA is the first
 density-functional approximation that incorporates ensemble weight
 dependencies explicitly, thus allowing for the description of derivative
 discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
-comment that follows] {\it via} the last term on the right-hand side
-of Eq.~\eqref{eq:EI-eLDA}. \manu{According to the decomposition of
+comment that follows] {\it via} the \titou{third} on the right-hand side
+of Eq.~\eqref{eq:EI-eLDA}. According to the decomposition of
 the ensemble
 correlation energy per particle in Eq.
-\eqref{eq:decomp_ens_correner_per_part}, the latter can be recast as
-follows:
-\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 \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
-\Big(\be{c}{(K)}(\n{\bGam{\bw}}{}(\br{}))
--
-\be{c}{(0)}(\n{\bGam{\bw}}{}(\br{}))
-\Big)
-d\br{}
-\\
-&=\int 
-\Big(\be{c}{(I)}(\n{\bGam{\bw}}{}(\br{}))
+\eqref{eq:decomp_ens_correner_per_part}, the latter can be recast
+\begin{equation}
+\titou{\Upsilon_\text{c}^{(I)}}
+%&=
+%\int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
+%\Big(\be{c}{(K)}(\n{\bGam{\bw}}{}(\br{}))
+%-
+%\be{c}{(0)}(\n{\bGam{\bw}}{}(\br{}))
+%\Big)
+%d\br{}
+%\\
+=\int 
+\qty[\be{c}{(I)}(\n{\bGam{\bw}}{}(\br{}))
 -
 \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{}))
-\Big)\,\n{\bGam{\bw}}{}(\br{})
+] \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
+\end{equation}
 thus leading to the following Taylor expansion through first order in
-$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: 
+\titou{$\dn{\bGam{\bw}}{(I)}(\br{})$}
+ %$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: 
 \beq\label{eq:Taylor_exp_DDisc_term}
 \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{}
-\\
+\titou{\Upsilon_\text{c}^{(I)}}
+%& = \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 \be{c}{(I)}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
 \\
 &+\int \Bigg[
 \n{\bGam{(I)}}{}(\br{}) 
@@ -950,24 +951,22 @@ $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
 \\
 &+\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)
+\e{c}{\bw}(\n{\bGam{(I)}}{}(\br{}))\Bigg]
+\dn{\bGam{\bw}}{(I)}(\br{})
 d\br{}
 \\
 &
-+\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right).
++ \titou{\order{[\dn{\bGam{\bw}}{(I)}(\br{})]^2}}.
 \end{split}
 \eeq
 As readily seen from Eqs. \eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}, the 
-role of the correlation ensemble derivative [last term on the right-hand side of Eq.
-\eqref{eq:EI-eLDA}] is, through zeroth order, to substitute the expected
+role of the correlation ensemble derivative \titou{$\Upsilon_\text{c}^{(I)}$} \trashPFL{[last term on the right-hand side of Eq.
+\eqref{eq:EI-eLDA}]} is, through zeroth order, to substitute the expected
 individual correlation energy per particle for the ensemble one.
-}
 
-\manu{
 Let us finally note that, while the weighted sum of the
-individual KS-eLDA energy levels delivers a \manu{\it ghost-interaction-corrected (GIC)} version of
-the KS-eLDA ensemble energy: 
+individual KS-eLDA energy levels delivers a \textit{ghost-interaction-corrected} (GIC) version of
+the KS-eLDA ensemble energy, \ie,
 \beq\label{eq:Ew-eLDA}
 \begin{split}
 \E{GIC-eLDA}{\bw}&=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)}
@@ -982,15 +981,15 @@ expressions in Eq. \eqref{eq:EI-eLDA} simply reads
 \beq\label{eq:Om-eLDA}
 \begin{split}
 	\Ex{eLDA}{(I)} 
-	=& 
+	&= 
 	\Ex{HF}{(I)}
-	+ \int 
+\\
+	&+ \int 
 \qty[\e{c}{{\bw}}(\n{}{})+n\pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}}]	
 _{\n{}{} =
 \n{\bGam{\bw}}{}(\br{})}
-\\
-&\times\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{} 
-+ \DD{c}{(I)},
+\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{} 
+\\ & + \DD{c}{(I)},
 \end{split}
 \eeq
 where the HF-like excitation energies $\Ex{HF}{(I)} = \E{HF}{(I)} -
@@ -1001,7 +1000,6 @@ where the HF-like excitation energies $\Ex{HF}{(I)} = \E{HF}{(I)} -
 	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
 \eeq
 is the eLDA correlation ensemble derivative contribution to the $I$th excitation energy.
-}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Density-functional approximations for ensembles}
@@ -1013,7 +1011,7 @@ is the eLDA correlation ensemble derivative contribution to the $I$th excitation
 \label{sec:paradigm}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (IUEG) model (also known as jellium). \cite{ParrBook, Loos_2016}
+Most of the standard local and semi-local \titou{density-functional approximations (DFAs)} rely on the infinite uniform electron gas (IUEG) model (also known as jellium). \cite{ParrBook, Loos_2016}
 One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states are not easily accessible like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
 Moreover, because the IUEG model is a metal, it is gapless, which means that both the fundamental and optical gaps are zero.
 From this point of view, using finite UEGs (FUEGs), \cite{Loos_2011b,
@@ -1046,9 +1044,9 @@ The present weight-dependent eDFA is specifically designed for the
 calculation of excited-state energies within GOK-DFT.
 To take into account both single and double excitations simultaneously, we consider a three-state ensemble including: 
 (i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
-\titou{To ensure the GOK variational principle, \cite{Gross_1988a} the
+To ensure the GOK variational principle, \cite{Gross_1988a} the
 triensemble weights must fulfil the following conditions: \cite{Deur_2019}
-$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$, where $\ew{1}$ and $\ew{2}$ are the weights associated with the singly- and doubly-excited states, respectively.}
+$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$, where $\ew{1}$ and $\ew{2}$ are the weights associated with the singly- and doubly-excited states, respectively.
 All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$, where $R$ is the radius of the ring on which the electrons are confined.
 We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
 Generalization to a larger number of states is straightforward and is left for future work.
@@ -1209,7 +1207,7 @@ In order to test the present eLDA functional we perform various sets of calculat
 To get reference excitation energies for both the single and double excitations, we compute full configuration interaction (FCI) energies with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
 For the single excitations, we also perform time-dependent LDA (TDLDA)
 calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}]. 
-\titou{Its Tamm-Dancoff approximation version  (TDA-TDLDA) is also considered.} \cite{Dreuw_2005}
+Its Tamm-Dancoff approximation version  (TDA-TDLDA) is also considered. \cite{Dreuw_2005}
 
 Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
 (ground-state) limit where $\bw = (0,0)$ and the
@@ -1225,28 +1223,25 @@ equi-tri-ensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$
 	\includegraphics[width=\linewidth]{EvsW_n5}
 	\caption{
 	\label{fig:EvsW}
-	Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
+	Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost-interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
 	}
 \end{figure*}
 %%% %%% %%%
 
-First, we discuss the linearity of the \manu{computed} (approximate)
-ensemble \manu{energies}.
+First, we discuss the linearity of the computed (approximate)
+ensemble energies.
 To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
 The deviation from linearity of the three-state ensemble energy
 $\E{}{(\ew{1},\ew{2})}$ (\ie, the deviation from the
-\trashEF{hypothetical} \manu{linearly-interpolated} ensemble energy
-\trashEF{obtained by linear interpolation}) is represented
-in Fig.~\ref{fig:EvsW} as a function of \trashEF{both} $\ew{1}$
-\trashEF{and} \manu{or} $\ew{2}$ while
+linearly-interpolated ensemble energy) is represented
+in Fig.~\ref{fig:EvsW} as a function of $\ew{1}$ or $\ew{2}$ while
 fulfilling the restrictions on the ensemble weights to ensure the GOK
 variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
-To illustrate the magnitude of the ghost interaction error (GIE), we report the KS-eLDA ensemble energy with and without ghost interaction correction (GIC) as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
+To illustrate the magnitude of the \titou{GIE}, we report the KS-eLDA ensemble energy with and without \titou{GIC} as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
 As one can see in Fig.~\ref{fig:EvsW}, without GIC, the
 ensemble energy becomes less and less linear as $L$
-gets larger, while the GIC \trashEF{makes the ensemble energy almost
-linear} reduces \manu{the curvature of the ensemble energy
-drastically}. 
+gets larger, while the GIC reduces the curvature of the ensemble energy
+drastically. 
 %\manu{This
 %is a strong statement I am not sure about. The nature of the excitation
 %should also be invoked I guess (charge transfer or not, etc ...). If we look at the GIE:
@@ -1295,7 +1290,6 @@ ensemble KS orbitals in the presence of GIE {[see Eqs.~\eqref{eq:min_with_HF_ene
 %%% %%% %%%
 
 Figure \ref{fig:EIvsW} reports the behavior of the three KS-eLDA individual energies as functions of the weights.
-\manu{
 Unlike in the exact theory, we do not obtain
 straight horizontal lines when plotting these
 energies, which is in agreement with  
@@ -1315,7 +1309,6 @@ excitation energy is reduced as the correlation increases. On the other hand, sw
 systematically enhances the weight dependence, due to the lowering of the
 ground-state energy in this case, as $w_2$ increases.         
 The reverse is observed for the second excitation energy.
-}
 
 %%% FIG 3 %%%
 \begin{figure}
@@ -1331,7 +1324,7 @@ The reverse is observed for the second excitation energy.
 Figure \ref{fig:EvsL} reports the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
 Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
 For small $L$, the single and double excitations can be labeled as
-``pure'', \manu{as revealed by a thorough analysis of the FCI wavefunctions}.
+``pure'', as revealed by a thorough analysis of the FCI wavefunctions.
 In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
 However, when the box gets larger (\ie, $L$ increases), there is a strong mixing between the different excitation degrees.
 In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
@@ -1405,8 +1398,7 @@ the same quality as the one obtained in the linear response formalism
 excitation energy only deviates
 from the FCI value by a few tenth of percent.
 Moreover, we note that, in the strong correlation regime
-(left\manu{Manu: you mean right?} graph of
-Fig.~\ref{fig:EvsN}), the single excitation
+(\titou{right} graph of Fig.~\ref{fig:EvsN}), the single excitation
 energy obtained at the equiensemble KS-eLDA level remains in good
 agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
 This also applies to the double excitation, the discrepancy
@@ -1431,11 +1423,9 @@ electrons.
 %%% %%% %%%
 
 It is also interesting to investigate the influence of the
-\manu{correlation ensemble derivative contribution} $\DD{c}{(I)}$
-\manu{to the $I$th excitation energy} [see Eq.~\eqref{eq:DD-eLDA}].
-\manu{In
-our case, both single ($I=1$) and double ($I=2$) excitations are
-considered}.
+correlation ensemble derivative contribution $\DD{c}{(I)}$
+to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}].
+In our case, both single ($I=1$) and double ($I=2$) excitations are considered.
 To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, in the case of 3-boxium, the error percentage (with respect to FCI) as a function of the box length $L$
 on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
 %\manu{Manu: there is something I do not understand. If you want to
@@ -1474,12 +1464,13 @@ still substantial in the strongly correlated limit of the equi
 triensemble for the double
 excitation.  
 }
-\titou{Note also that, in our case, the second term in
+Note also that, in our case, the second term in
 Eq.~\eqref{eq:Om-eLDA}, which involves the weight-dependent correlation
 potential and the density difference between ground and excited states,
-has a negligible effect on the excitation energies.}\manu{Manu: Is this
-something that you checked but did not show? It feels like we can see
-this in the Figure but we cannot, right?}
+has a negligible effect on the excitation energies \titou{(results not shown)}.
+%\manu{Manu: Is this
+%something that you checked but did not show? It feels like we can see
+%this in the Figure but we cannot, right?}
 %\manu{Manu: well, we
 %would need the exact derivative value to draw such a conclusion. We can
 %only speculate. Let us first see how important the contribution in
@@ -1503,8 +1494,8 @@ undoubtedly show that, even in the strong correlation regime, the
 ensemble correlation derivative has a small impact on the double
 excitations \manu{Manu: well, the impact is larger than the one on the single
 excitation in the equiensemble} with a slight tendency of worsening the excitation energies
-in the case of equal weights, \manu{as the number of electrons
-increases. It has} a rather large influence on the single
+in the case of equal weights, as the number of electrons
+increases. It has a rather large influence on the single
 excitation energies obtained in the zero-weight limit, showing once
 again that the usage of equal weights has the benefit of significantly reducing the magnitude of the ensemble correlation derivative.
 
@@ -1530,12 +1521,12 @@ from the finite uniform electron gas eLDA is
 combined with eLDA, delivers accurate excitation energies for both
 single and double excitations, especially when an equiensemble is used.
 In the latter case, the same weights are assigned to each state belonging to the ensemble.
-\titou{The improvement on the excitation energies brought by the KS-eLDA scheme is particularly impressive in the strong correlation regime where usual methods, such as TDLDA, fail.
+The improvement on the excitation energies brought by the KS-eLDA scheme is particularly impressive in the strong correlation regime where usual methods, such as TDLDA, fail.
 We have observed that, although the ensemble correlation discontinuity has a
 non-negligible effect on the excitation energies (especially for the
 single excitations), its magnitude can be significantly reduced by
 performing equiweight calculations instead of zero-weight
-calculations.}
+calculations.
 
 Let us finally stress that the present methodology can be extended
 straightforwardly to other types of ensembles like, for example, the
@@ -1554,7 +1545,7 @@ See {\SI} for the additional details about the construction of the functionals,
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{acknowledgements}
-\titou{The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.}
+The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.
 This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.} 
 \end{acknowledgements}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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