Theory section done

Manuscript/eDFT.bib

@@ -1,13 +1,148 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2019-09-05 12:13:36 +0200 
+%% Created for Pierre-Francois Loos at 2019-09-09 09:45:30 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Boblest_2014,
+	Author = {S. Boblest and C. Schimeczek and G. Wunner},
+	Date-Added = {2019-09-09 09:42:01 +0200},
+	Date-Modified = {2019-09-09 09:43:17 +0200},
+	Journal = {Phys. Rev. A},
+	Pages = {012505},
+	Volume = {89},
+	Year = {2014}}
+
+@article{Stopkowicz_2015,
+	Author = {S. Stopkowicz and J. Gauss and K. K. Lange and E. I. Tellgren and T. Helgaker},
+	Date-Added = {2019-09-09 09:41:46 +0200},
+	Date-Modified = {2019-09-09 09:43:56 +0200},
+	Journal = {J. Chem. Phys.},
+	Pages = {074110},
+	Volume = {143},
+	Year = {2015}}
+
+@article{Tellgren_2008,
+	Author = {E. I. Tellgren and A. Soncini and T. Helgaker},
+	Date-Added = {2019-09-09 09:41:39 +0200},
+	Date-Modified = {2019-09-09 09:44:14 +0200},
+	Journal = {J. Chem. Phys.},
+	Pages = {154114},
+	Volume = {129},
+	Year = {2008}}
+
+@article{Tellgren_2009,
+	Author = {E. I. Tellgren and T. Helgaker and A. Soncini},
+	Date-Added = {2019-09-09 09:41:39 +0200},
+	Date-Modified = {2019-09-09 09:44:18 +0200},
+	Journal = {Phys. Chem. Chem. Phys..},
+	Pages = {5489},
+	Volume = {11},
+	Year = {2009}}
+
+@article{Schmelcher_1997,
+	Author = {P. Schmelcher and L. S. Cederbaum},
+	Date-Added = {2019-09-09 09:41:33 +0200},
+	Date-Modified = {2019-09-09 09:45:27 +0200},
+	Journal = {Int. J. Quantum Chem.},
+	Pages = {501},
+	Volume = {64},
+	Year = {1997}}
+
+@article{Schmelcher_1990,
+	Author = {P. Schmelcher and L. S. Cederbaum},
+	Date-Added = {2019-09-09 09:41:22 +0200},
+	Date-Modified = {2019-09-09 09:45:20 +0200},
+	Journal = {Phys. Rev. A},
+	Pages = {4936},
+	Volume = {41},
+	Year = {1990}}
+
+@article{Lange_2012,
+	Author = {K. K. Lange and E. I. Tellgren and M. R. Hoffmann and T. Helgaker},
+	Date-Added = {2019-09-09 09:41:15 +0200},
+	Date-Modified = {2019-09-09 09:44:10 +0200},
+	Journal = {Science},
+	Pages = {327},
+	Volume = {337},
+	Year = {2012}}
+
+@article{Schmelcher_2012,
+	Author = {P. Schmelcher},
+	Date-Added = {2019-09-09 09:41:08 +0200},
+	Date-Modified = {2019-09-09 09:45:15 +0200},
+	Journal = {Science},
+	Pages = {302},
+	Volume = {337},
+	Year = {2012}}
+
+@article{Patil_2001,
+	Author = {S. H. Patil},
+	Date-Added = {2019-09-09 09:40:48 +0200},
+	Date-Modified = {2019-09-09 09:45:03 +0200},
+	Journal = {Phys. Rev. A},
+	Pages = {064902},
+	Title = {Electron near a helium liquid surface},
+	Volume = {64},
+	Year = {2001}}
+
+@article{Nieto_2000,
+	Author = {Michael Martin Nieto},
+	Date-Added = {2019-09-09 09:40:42 +0200},
+	Date-Modified = {2019-09-09 09:44:54 +0200},
+	Journal = {Phys. Rev. A},
+	Pages = {034901},
+	Title = {Electrons above a helium surface and the one-dimensional Rydberg atom},
+	Volume = {61},
+	Year = {2000}}
+
+@article{Mayle_2007,
+	Author = {Michael Mayle and Bernd Hezel and Igor Lesanovsky and Peter Schmelcher},
+	Date-Added = {2019-09-09 09:40:30 +0200},
+	Date-Modified = {2019-09-09 09:44:39 +0200},
+	Journal = {Phys. Rev. Lett.},
+	Pages = {113004},
+	Title = {One-Dimensional Rydberg Gas in a Magnetoelectric Trap},
+	Volume = {99},
+	Year = {2007}}
+
+@article{Burnett_1993,
+	Author = {K. Burnett and V. C. Reed and P. L. Knight},
+	Date-Added = {2019-09-09 09:40:14 +0200},
+	Date-Modified = {2019-09-09 09:43:47 +0200},
+	Journal = {J. Phys. B: At. Mol. Opt. Phys.},
+	Pages = {561},
+	Title = {Atoms in ultra-intense laser fields},
+	Volume = {26},
+	Year = {1993}}
+
+@article{Astrakharchik_2011,
+	Author = {G. E. Astrakharchik and M. D. Girardeau},
+	Date-Added = {2019-09-09 09:24:25 +0200},
+	Date-Modified = {2019-09-09 09:24:51 +0200},
+	Journal = {Phys. Rev. B},
+	Pages = {153303},
+	Title = {Exact ground-state properties of a one-dimensional Coulomb gas},
+	Volume = {83},
+	Year = {2011}}
+
+@article{Lee_2011a,
+	Author = {Lee, R. M. and Drummond, N. D.},
+	Date-Added = {2019-09-09 09:22:34 +0200},
+	Date-Modified = {2019-09-09 09:23:15 +0200},
+	Issue = {24},
+	Journal = {Phys. Rev. B},
+	Numpages = {12},
+	Pages = {245114},
+	Publisher = {American Physical Society},
+	Title = {Ground-state properties of the one-dimensional electron liquid},
+	Volume = {83},
+	Year = {2011}}
+
 @article{Perdew_1983,
 	Author = {J. P. Perdew and M. Levy},
 	Date-Added = {2019-09-05 12:04:19 +0200},
@@ -4475,10 +4610,10 @@
 	Volume = {136},
 	Year = {2012}}
 
-@article{Lee_2011,
+@article{Lee_2011b,
 	Author = {Lee, R. M. and Conduit, G. J. and Nemec, N. and L{\'o}pez R{\'\i}os, P. and Drummond, N. D.},
 	Date-Added = {2018-10-24 22:38:52 +0200},
-	Date-Modified = {2018-10-24 22:38:52 +0200},
+	Date-Modified = {2019-09-09 09:22:45 +0200},
 	Doi = {10.1103/physreve.83.066706},
 	Issn = {1550-2376},
 	Journal = {Phys. Rev. E},

Manuscript/eDFT.tex

@@ -27,7 +27,7 @@
 % operators
 \newcommand{\hHc}{\Hat{h}}
 \newcommand{\hT}{\Hat{T}}
-\newcommand{\vne}{\Hat{v}_\text{ne}}
+\newcommand{\hVext}{\Hat{V}_\text{ext}}
 \newcommand{\hWee}{\Hat{W}_\text{ee}}
 
 % functionals, potentials, densities, etc
@@ -37,7 +37,7 @@
 \newcommand{\bE}[2]{\bar{E}_\text{#1}^{#2}}
 \newcommand{\be}[2]{\bar{\eps}_\text{#1}^{#2}}
 \newcommand{\bv}[2]{\bar{f}_\text{#1}^{#2}}
-\newcommand{\n}[1]{n^{#1}}
+\newcommand{\n}[2]{n_{#1}^{#2}}
 \newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
 \newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
 
@@ -124,7 +124,7 @@ Their accuracy is illustrated by computing single and double excitations in one-
 \label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964, Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials. \cite{ParrBook}
-The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}(\br)$, the latter being a much simpler quantity than the many-electron wave function. 
+The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}{}(\br)$, the latter being a much simpler quantity than the many-electron wave function. 
 The complexity of the many-body problem is then transferred to the xc functional. 
 Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} (KS-DFT) suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003, Tozer_1999, Dreuw_2003, Sobolewski_2003, Dreuw_2004, Tozer_1998, Tozer_2000, Casida_1998, Casida_2000, Tapavicza_2008, Levine_2006}
 The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge. 
@@ -152,6 +152,11 @@ In both cases, the key problem, namely the design of weight-dependent DFAs for e
 A first step towards this goal is presented in this article with the ambition to turn, in the near future, eDFT into a practical computational method for modeling excited states in molecules and extended systems.
 \alert{Mention WIDFA?}
 In the following, the present methodology is illustrated on \emph{strict} one-dimensional (1D), spin-polarized electronic systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
+In other words, the Coulomb interaction used in this work describes 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}
+This description of 1D systems also has interesting connections with the exotic chemistry of ultra-high magnetic fields (such as those in white dwarf stars), where the electronic cloud is dramatically compressed perpendicular to the magnetic field. \cite{Schmelcher_1990, Lange_2012, Schmelcher_2012}
+In these extreme conditions, where magnetic effects compete with Coulombic forces, entirely new bonding paradigms emerge. \cite{Schmelcher_1990, Schmelcher_1997, Tellgren_2008, Tellgren_2009, Lange_2012, Schmelcher_2012, Boblest_2014, Stopkowicz_2015}
 Atomic units are used throughout.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -164,7 +169,21 @@ Atomic units are used throughout.
 \label{sec:geKS}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\alert{Manu, you might want to add here details about the general KS-eDFT procedure.}
+\alert{Manu, you might want to add general details about the eDFT here.}
+
+%In eDFT, the ensemble energy
+%\begin{equation}
+%	\E{}{\bw} = \qty(1-\sum_{I>0}\ew{I})\E{}{(0)}+\sum_{I>0} \ew{I} \E{}{(I)}
+%\end{equation}
+%is obtained variationally as follows,
+%\begin{equation}
+%	\E{}{\bw} = \qty(1-\sum_{I>0}\ew{I})\E{}{(0)}+\sum_{I>0} \ew{I} \E{}{(I)}
+%\end{equation}
+%
+%In analogy with ground-state generalized KS-DFT, we consider the following partitioning of the ensemble Levy-Lieb functional 
+%\begin{equation}
+%	F^{\bw}[n]=\underset{\hat{\Gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\Gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
+%\end{equation}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{KS-eDFT for excited states}
@@ -181,7 +200,7 @@ By definition, the ensemble energy is
 The $\E{}{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the single and double excitation, respectively.
 To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions: 
 $0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$.
-Note that, in order to extract individual energies from a single KS-eDFT calculation (see below), the weights must remain independent.
+Note that, in order to extract individual energies from a single KS-eDFT calculation [see Subsec.~\ref{sec:E_I}], the weights must remain independent.
 By construction, the excitation energies are
 \begin{equation}
 \label{eq:Ex}
@@ -208,7 +227,7 @@ The coefficients $\cMO{\mu p}{\bw}$ used to construct the density matrix $\bGamm
 	\eF{\mu\nu}{\bw} 
 	= \eHc{\mu\nu} + \sum_{\la\si} \eGamma{\la\si}{\bw} \eG{\mu\nu\la\si} 
 	\\
-	+ \int \left. \fdv{\bE{Hxc}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)}  \AO{\mu}(\br) \AO{\nu}(\br) d\br,
+	+ \int \left. \fdv{\bE{Hxc}{\bw}[\n{}{}]}{\n{}{}(\br)} \right|_{\n{}{} = \n{}{\bw}(\br)}  \AO{\mu}(\br) \AO{\nu}(\br) d\br,
 \end{multline}
 which itself depends on $\bGamma{\bw}$.
 In Eq.~\eqref{eq:F}, $\hHc$ is the core Hamiltonian (including kinetic and electron-nucleus attraction terms), $\eG{\mu\nu\la\si} = (\mu\nu|\la\si) - (\mu\si|\la\nu)$,
@@ -216,62 +235,120 @@ In Eq.~\eqref{eq:F}, $\hHc$ is the core Hamiltonian (including kinetic and elect
 	(\mu\nu|\la\si) = \iint  \frac{\AO\mu(\br_1) \AO\nu(\br_1) \AO\la(\br_2) \AO\si(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
 \end{equation}
 are two-electron repulsion integrals,
-$\bE{Hxc}{\bw}[\n{}(\br)] = \n{}(\br) \be{Hxc}{\bw}[\n{}(\br)]$ and $\be{Hxc}{\bw}[\n{}(\br)]$ is the weight-dependent correlation functional to be built in the present study.
+$\bE{Hxc}{\bw}[\n{}{}(\br)] = \n{}{}(\br) \be{Hxc}{\bw}[\n{}{}(\br)]$ and $\be{Hxc}{\bw}[\n{}{}(\br)]$ is the weight-dependent correlation functional to be built in the present study.
 The one-electron ensemble density is
 \begin{equation}
-	\n{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br),
+	\n{}{\bw}(\br) = \sum_{\mu\nu} \AO{\mu}(\br) \, \eGamma{\mu\nu}{\bw} \, \AO{\nu}(\br),
 \end{equation}
-with a similar expression for $\n{(I)}(\br)$, while the ensemble energy reads 
+with a similar expression for $\n{}{(I)}(\br)$, while the ensemble energy reads 
 \begin{equation}
 \label{eq:Ew}
 	\E{}{\bw} 
 	= \Tr(\bGamma{\bw} \, \bHc) 
 	+ \frac{1}{2} \Tr(\bGamma{\bw} \, \bG \, \bGamma{\bw}) 
 %	\\
-%	+ \int \e{c}{\bw}[\n{\bw}(\br)] \n{\bw}(\br) d\br.
+%	+ \int \e{c}{\bw}[\n{}{\bw}(\br)] \n{}{\bw}(\br) d\br.
-	+ \int \bE{Hxc}{\bw}[\n{\bw}(\br)] d\br.
+	+ \int \bE{Hxc}{\bw}[\n{}{\bw}(\br)] d\br.
 \end{equation}
 The self-consistent process described above is carried on until $\max \abs{\bF{\bw} \, \bGamma{\bw} \, \bS - \bS \, \bGamma{\bw} \, \bF{\bw}} < \tau$, where $\tau$ is a user-defined threshold and $\eS{\mu\nu} = \braket{\AO{\mu}}{\AO{\nu}}$ are elements of the overlap matrix $\bS$. 
 Note that because the second term of the RHS of Eq.~\eqref{eq:Ew} is quadratic in $\bGamma{\bw}$, the weight-dependent energy contains the so-called ghost interaction which makes the ensemble energy non linear. \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} 
-Below, we propose a ghost-interaction correction in order to minimize this error.
+Below, we propose a ghost-interaction correction (GIC) in order to minimize this error.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Extracting individual energies}
-\label{sec:ind_energy}
+\label{sec:E_I}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Following Deur and Fromager, \cite{Deur_2018b} it is possible to extract individual energies, $\E{}{(I)}$, from the ensemble energy [see Eq.~\eqref{eq:Ew}] as follows:
 \begin{multline}
 	\E{}{(I)} = \Tr(\bGamma{(I)} \, \bHc)  + \frac{1}{2} \Tr(\bGamma{(I)} \, \bG \, \bGamma{(I)})
 			\\
-			+ \int \left. \fdv{\bE{Hxc}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{(I)}(\br) d\br
+			+ \int \left. \fdv{\bE{Hxc}{\bw}[\n{}{}]}{\n{}{}(\br)} \right|_{\n{}{} = \n{}{\bw}(\br)} \n{}{(I)}(\br) d\br
 			+ \LZ{Hxc}{} + \DD{Hxc}{(I)}.
 \end{multline}
 Note that a \emph{single} KS-eDFT calculation is required to extract the three individual energies.
 The correlation part of the (state-independent) Levy-Zahariev shift and the so-called derivative discontinuity are given by
 \begin{align}
-	\LZ{Hxc}{} 	& =  - \int \left. \fdv{\be{Hxc}{\bw}[\n{}]}{\n{}(\br)} \right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br)^2 d\br,
+	\LZ{Hxc}{} 	& =  - \int \left. \fdv{\be{Hxc}{\bw}[\n{}{}]}{\n{}{}(\br)} \right|_{\n{}{} = \n{}{\bw}(\br)} \n{}{\bw}(\br)^2 d\br,
 	\\
-	\DD{Hxc}{(I)} & = \sum_{J>0} (\delta_{IJ} - \ew{J}) \int  \left. \pdv{\be{Hxc}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} \n{\bw}(\br) d\br.
+	\DD{Hxc}{(I)} & = \sum_{J>0} (\delta_{IJ} - \ew{J}) \int  \left. \pdv{\be{Hxc}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br)} \n{}{\bw}(\br) d\br.
 \end{align}
 Because the Levy-Zahariev shift is state independent, it does not contribute to excitation energies [see Eq.~\eqref{eq:Ex}].
-The only remaining piece of information to define at this stage is the weight-dependent correlation functional $\be{Hxc}{\bw}(\n{})$.
+The only remaining piece of information to define at this stage is the weight-dependent correlation functional $\be{Hxc}{\bw}(\n{}{})$.
 \alert{Mention LIM?}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Uniform electron gases}
+\section{Density-functional approximations for ensembles}
-\label{sec:UEG}
+\label{sec:eDFA}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas model (also known as jellium). \cite{ParrBook, Loos_2016}
+We decompose the weight-dependent functional as
+\begin{equation}
+	\be{Hxc}{\bw}(\n{}{}) = \be{Hx}{\bw}(\n{}{}) + \be{c}{\bw}(\n{}{}),
+\end{equation}
+where $\be{Hx}{\bw}(\n{}{})$ is a weight-dependent Hartree-exchange functional designed to correct the ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} [see Subsec.~\ref{sec:GIC}] and $\be{c}{\bw}(\n{}{})$ is a weight-dependent correlation functional [see Subsec.~\ref{sec:Ec}].
+The construction of these two functionals is described below.
+Note that, because we consider strict 1D systems, one cannot decompose further the Hartree-exchange contribution as each component diverges independently but their sum is finite. \cite{Astrakharchik_2011, Lee_2011a, Loos_2012, Loos_2013, Loos_2013a}
+
+Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (UEG) 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 cannot be easily identified like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
-From this point of view, using finite uniform electron gases \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs is much more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
+Moreover, because the infinite UEG 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 \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs can be seen as more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
 Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems. 
-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}
 
 As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e., a circle). \cite{Loos_2012, Loos_2013a, Loos_2014b}
 The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform. 
 As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary. 
 This is a necessary condition for being able to model derivative discontinuities. 
 
+The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
+As mentioned previously, we consider a three-state ensemble including the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
+All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
+We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Ghost-interaction correction}
+\label{sec:GIC}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+The GIC weight-dependent Hartree-exchange functional is defined as  
+\begin{multline}
+	\be{Hx}{\bw}(\n{}{\bw}) = (1-\sum_{I>0} \ew{I}) \be{Hx}{}(\n{}{(0)}) + \sum_{I>0} \ew{I} \be{Hx}{}(\n{}{(I)}) 
+	\\
+	- \be{Hx}{(I)}(\n{}{\bw}),
+\end{multline}
+where
+\begin{equation}
+	\be{Hx}{}(\n{}{}) = \iint \frac{\n{}{}(\br_1) \n{}{}(\br_2) - \n{}{}(\br_1,\br_2)^2}{r_{12}} d\br_1 d\br_2,
+\end{equation}
+and 
+\begin{equation}
+	\n{}{(I)}(\omega) = (\pi R)^{-1} \cos[(I+1) \omega/2]
+\end{equation}
+is the first-order density matrix with $\omega$ the interelectronic angle.
+It yields
+\begin{equation}
+	\be{Hx}{}(\n{}{}) = \n{}{} \qty[ a_1 \ew{1} (\ew{1} - 1) +  a_2 \ew{1} \ew{2} + a_3 \ew{2} (\ew{2} - 1)],
+\end{equation}
+with $a_1 = 2 \ln 2 - 1/3$, $a_2 = 8/3$ and $a_3 = 32/15$.
+ 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Weight-dependent correlation functional}
+\label{sec:Ec}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
+\begin{equation}
+\label{eq:ec}
+	\e{c}{(I)}(\n{}{}) = \frac{c_1^{(I)}\,\n{}{}}{\n{}{} + c_2^{(I)} \sqrt{\n{}{}} + c_3^{(I)}},
+\end{equation}
+where the $c_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
+The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
+Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
+Combining these, one can build a three-state weight-dependent correlation eDFA:
+\begin{equation}
+\label{eq:ecw}
+	\e{c}{\bw}(\n{}{}) =  (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
+\end{equation}
+
 %%% TABLE 1 %%%
 \begin{table*}
 	\caption{
@@ -280,7 +357,7 @@ This is a necessary condition for being able to model derivative discontinuities
 %	\begin{ruledtabular}
 		\begin{tabular}{lcddd}
 			\hline\hline
-			State					&	$I$		&	\tabc{$a^{(I)}$}	&	\tabc{$b^{(I)}$}	&	\tabc{$c^{(I)}$}	\\
+			State					&	$I$		&	\tabc{$c_1^{(I)}$}	&	\tabc{$c_2^{(I)}$}	&	\tabc{$c_3^{(I)}$}	\\
 			\hline
 			Ground state			&	$0$		&	-0.0137078			&	0.0538982			&	0.0751740			\\
 			Singly-excited state	&	$1$		&	-0.0238184			&	0.00413142			&	0.0568648			\\
@@ -291,108 +368,53 @@ This is a necessary condition for being able to model derivative discontinuities
 \end{table*}
 %%% %%% %%% %%%
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Density-functional approximations for ensembles}
-\label{sec:eDFA}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-We decompose the weight-dependent functional $\be{Hxc}{\bw}(\n{})$ as
-\begin{equation}
-	\be{Hxc}{\bw}(\n{}) = \be{Hx}{\bw}(\n{}) + \be{c}{\bw}(\n{}),
-\end{equation}
-where $\be{Hx}{\bw}(\n{})$ is an Hartree-exchange functional designed to correct the ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} and $\be{c}{\bw}(\n{})$ is a weight-dependent correlation functional.
-The construction of these two functionals is described below.
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Ghost-interaction correction}
-\label{sec:GIC}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Weight-dependent correlation functional}
-\label{sec:Ec}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
-As mentioned previously, we consider a three-state ensemble including the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
-All these states have the same (uniform) density $\n{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
-We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
-
-Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
-\begin{equation}
-\label{eq:ec}
-	\e{c}{(I)}(\n{}) = \frac{a^{(I)}\,\n{}}{\n{} + b^{(I)} \sqrt{\n{}} + c^{(I)}},
-\end{equation}
-where $b^{(I)}$ and $c^{(I)}$ are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
-The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
-Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
-Combining these, one can build a three-state weight-dependent correlation eDFA:
-\begin{equation}
-\label{eq:ecw}
-	\e{c}{\bw}(\n{}) =  (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}) + \ew{1} \e{c}{(1)}(\n{}) + \ew{2} \e{c}{(2)}(\n{}).
-\end{equation}
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{LDA-centered functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
 \begin{equation}
 \label{eq:becw}
-	\be{c}{\bw}(\n{}) =  (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}) + \ew{1} \be{c}{(1)}(\n{}) + \ew{2} \be{c}{(2)}(\n{}),
+	\be{c}{\bw}(\n{}{}) =  (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
 \end{equation}
 where
 \begin{equation}
-	\be{c}{(I)}(\n{}) = \e{c}{(I)}(\n{}) + \e{c}{\text{LDA}}(\n{}) - \e{c}{(0)}(\n{}).
+	\be{c}{(I)}(\n{}{}) = \e{c}{(I)}(\n{}{}) + \e{c}{\text{LDA}}(\n{}{}) - \e{c}{(0)}(\n{}{}).
 \end{equation}
 The local-density approximation (LDA) correlation functional,
 \begin{equation}
-	\e{c}{\text{LDA}}(\n{}) = a^\text{LDA} \, F\qty[1,\frac{3}{2},c^\text{LDA}, \frac{a^\text{LDA}(1-c^\text{LDA})}{b^\text{LDA}} {\n{}}^{-1}],
+	\e{c}{\text{LDA}}(\n{}{}) = c_1^\text{LDA} \, F\qty[1,\frac{3}{2},c_3^\text{LDA}, \frac{c_1^\text{LDA}(1-c_3^\text{LDA})}{c_2^\text{LDA}} {\n{}{}}^{-1}],
 \end{equation}
 specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
 \begin{align}
-	a^\text{LDA} & = - \frac{\pi^2}{360},
+	c_1^\text{LDA} & = - \frac{\pi^2}{360},
 	&
-	b^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
+	c_2^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
 	& 
-	c^\text{LDA} & = 2.408779.
+	c_3^\text{LDA} & = 2.408779.
 \end{align}
 Equation \eqref{eq:becw} can be recast
 \begin{equation}
 \label{eq:eLDA}
 \begin{split}
-	\be{c}{\bw}(\n{}) 
+	\be{c}{\bw}(\n{}{}) 
-	& =  \e{c}{\text{LDA}}(\n{}) 
+	& =  \e{c}{\text{LDA}}(\n{}{}) 
 	\\
-	& + \ew{1} \qty[\e{c}{(1)}(\n{})-\e{c}{(0)}(\n{})] + \ew{2} \qty[\e{c}{(2)}(\n{})-\e{c}{(0)}(\n{})],
+	& + \ew{1} \qty[\e{c}{(1)}(\n{}{})-\e{c}{(0)}(\n{}{})] + \ew{2} \qty[\e{c}{(2)}(\n{}{})-\e{c}{(0)}(\n{}{})],
 \end{split}
 \end{equation}
 which nicely highlights the centrality of the LDA in the present eDFA. 
-In particular, $\be{c}{(0,0)}(\n{}) = \e{c}{\text{LDA}}(\n{})$.
+In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$.
 Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
+
+This procedure can be theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) which was originally derived by Franck and Fromager. \cite{Franck_2014} 
+Within this in-principle-exact formalism, the (weight-dependent) correlation energy of the ensemble is constructed from the (weight-independent) ground-state functional (such as the LDA), yielding Eq.~\eqref{eq:eLDA}.
+This is a crucial point as we intend to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons).
+
 Finally, we note that, by construction,
 \begin{equation}
-	\left. \pdv{\be{c}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} = \be{c}{(J)}[\n{\bw}(\br)] - \be{c}{(0)}[\n{\bw}(\br)].
+	\left. \pdv{\be{c}{\bw}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\bw}(\br)} = \be{c}{(J)}[\n{}{\bw}(\br)] - \be{c}{(0)}[\n{}{\bw}(\br)].
 \end{equation}
 \alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
-%%% FIG 1 %%%
-\begin{figure}
-	\includegraphics[width=\linewidth]{EvsL_5}
-	\caption{
-	\label{fig:EvsL}
-	Error with respect to FCI in single and double excitation energies for 5-boxium for various methods and box length $L$.
-	Graphs for additional values of $\Nel$ can be found as {\SI}.
-	}
-\end{figure}
-%%% %%% %%%
-
-%%% FIG 2 %%%
-\begin{figure}
-	\includegraphics[width=\linewidth]{EvsN_1}
-	\caption{
-	\label{fig:EvsN}
-	Error with respect to FCI in single and double excitation energies for $\Nel$-boxium for various methods and number of electrons $\Nel$ at $L=\pi$.
-	Graphs for additional values of $L$ can be found as {\SI}.
-	}
-\end{figure}
-%%% %%% %%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}
@@ -441,12 +463,34 @@ Overall, one clearly sees that, with state-averaged weights, the eLDA functional
 This conclusion is verified for smaller and larger number of electrons (see {\SI}).
 \alert{Shall I test the one-electron system for self-interaction?}
 
+%%% FIG 1 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{EvsL_5}
+	\caption{
+	\label{fig:EvsL}
+	Error with respect to FCI in single and double excitation energies for 5-boxium for various methods and box length $L$.
+	Graphs for additional values of $\Nel$ can be found as {\SI}.
+	}
+\end{figure}
+%%% %%% %%%
+
 Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies, for the same methods, as a function of $\Nel$ and fixed $L$ (in this case $L=\pi$).
 The graphs associated with other $L$ values are reported as {\SI}.
 Again, the graph for $L=\pi$ is quite typical and we draw similar conclusions as in the previous paragraph: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations. 
 As a rule of thumb, we see that eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDHF or TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$, an error of the same order as CIS or TDA-TDLDA.
 Even for larger boxes, the discrepancy between FCI and eLDA for double excitations is only a few percent.
 
+%%% FIG 2 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{EvsN_1}
+	\caption{
+	\label{fig:EvsN}
+	Error with respect to FCI in single and double excitation energies for $\Nel$-boxium for various methods and number of electrons $\Nel$ at $L=\pi$.
+	Graphs for additional values of $L$ can be found as {\SI}.
+	}
+\end{figure}
+%%% %%% %%%
+
 \alert{Need further discussion on DD and LZ shift. Linearity of energy wrt weights?}
 
 \alert{For small $L$, the single and double excitations are ``pure''. In other words, the excitation is dominated by a single reference Slater determinant.