Merge branch 'master' of https://git.irsamc.ups-tlse.fr/loos/FarDFT

Manuscript/FarDFT.bib

@@ -1,13 +1,58 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-04-07 20:33:37 +0200 
+%% Created for Pierre-Francois Loos at 2020-04-08 14:13:22 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Fromager_2020,
+	Archiveprefix = {arXiv},
+	Author = {Emmanuel Fromager},
+	Date-Added = {2020-04-08 14:13:18 +0200},
+	Date-Modified = {2020-04-08 14:13:18 +0200},
+	Eprint = {2001.08605},
+	Primaryclass = {physics.chem-ph},
+	Title = {Individual correlations in ensemble density-functional theory: State-driven/density-driven decomposition without additional Kohn-Sham systems},
+	Year = {2020}}
+
+@article{Bottcher_1974,
+	Author = {C. Bottcher and K. Docken},
+	Date-Added = {2020-04-08 13:03:40 +0200},
+	Date-Modified = {2020-04-08 13:05:55 +0200},
+	Doi = {10.1088/0022-3700/7/1/002},
+	Journal = {J. Phys. B: At. Mol. Phys.},
+	Pages = {L5},
+	Title = {Autoionizing States of the Hydrogen Molecule.},
+	Volume = {7},
+	Year = {1974}}
+
+@article{Mielke_2005,
+	Author = {S. L. Mielke and D. W. Schwenke and K. A. Peterson},
+	Date-Added = {2020-04-08 12:47:49 +0200},
+	Date-Modified = {2020-04-08 12:49:45 +0200},
+	Doi = {10.1063/1.1917838},
+	Journal = {J. Chem. Phys.},
+	Pages = {224313},
+	Title = {Benchmark calculations of the complete configuration-interaction limit of Born-Oppenheimer diagonal corrections to the saddle points of isotopomers of the {{H+H2}} reaction.},
+	Volume = {122},
+	Year = {2005},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.1917838}}
+
+@article{Sun_2016,
+	Author = {J. Sun and J. P. Perdew and Z. Yang and H. Peng},
+	Date-Added = {2020-04-08 10:56:23 +0200},
+	Date-Modified = {2020-04-08 10:56:47 +0200},
+	Doi = {10.1063/1.4950845},
+	Journal = {J. Chem. Phys.},
+	Pages = {191101},
+	Title = {Near-locality of exchange and correlation density functionals for 1- and 2-electron systems},
+	Volume = {144},
+	Year = {2016},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.4950845}}
+
 @article{Slater_1951,
 	Author = {J. C. Slater},
 	Date-Added = {2020-04-07 19:53:52 +0200},
@@ -17,7 +62,8 @@
 	Pages = {385},
 	Title = {A Simplification of the Hartree-Fock Method},
 	Volume = {81},
-	Year = {1981}}
+	Year = {1981},
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.81.385}}
 
 @book{Slater_1974,
 	Date-Added = {2020-04-07 19:48:23 +0200},

Manuscript/FarDFT.tex

@@ -67,8 +67,11 @@
 \newcommand{\Ec}{E_\text{c}}
 \newcommand{\HF}{\text{HF}}
 \newcommand{\LDA}{\text{LDA}}
-\newcommand{\eLDA}{\text{eLDA}}
-\newcommand{\CID}{\text{CID}}
+\newcommand{\SD}{\text{S}}
+\newcommand{\VWN}{\text{VWN5}}
+\newcommand{\SVWN}{\text{SVWN5}}
+\newcommand{\LIM}{\text{LIM}}
+\newcommand{\MOM}{\text{MOM}}
 \newcommand{\Hxc}{\text{Hxc}}
 \newcommand{\Ha}{\text{H}}
 \newcommand{\ex}{\text{x}}
@@ -173,13 +176,12 @@ We believe that it is partly due to the lack of accurate approximations for GOK-
 In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation for ensemble (eDFA) has never been developed for atoms and molecules.
 The present contribution is a small step towards this goal.
 
-\titou{When one talks about constructing functionals, the local-density approximation (LDA) is never far away.
+When one talks about constructing functionals, the local-density approximation (LDA) is never far away.
 The LDA, as we know it, is based on the uniform electron gas (UEG) also known as jellium, an hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively-charged jelly. \cite{Loos_2016}
 Although the Hohenberg--Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behaviour in a real system. \cite{Kohn_1965}
 However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct LDA functionals. \cite{Loos_2014a,Loos_2014b,Loos_2017a}
 Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b}
-Here, we combine these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA functional for ensembles (eLDA).
-The present eLDA functional is specifically designed to compute double excitations within GOK-DFT, and it automatically incorporates the infamous derivative discontinuity contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}}
+In particular, we combine these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT, and automatically incorporates the infamous derivative discontinuity contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}
 
 %The paper is organised as follows.
 %In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented.
@@ -250,10 +252,13 @@ Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes t
 
 From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019}
 \begin{equation}
+\begin{split}
 	\label{eq:dEdw}
 	\pdv{\E{}{\bw}}{\ew{I}} 
-	= \E{}{(I)} - \E{}{(0)} 
-	= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}(\br{})},
+	& = \E{}{(I)} - \E{}{(0)} 
+	\\
+	& = \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}(\br{})},
+\end{split}
 \end{equation}
 where 
 \begin{align}
@@ -272,19 +277,23 @@ is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orb
 The latters are determined by solving the ensemble KS equation
 \begin{equation}
 \label{eq:eKS}
-	\qty( \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
+	\qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
 \end{equation}
 where $\hHc(\br{}) = -\nabla^2/2  + \vne(\br{})$, and
 \begin{equation}
-\begin{split}
 	\fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})} 
-	& = \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
-	\\
-	& = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}' 
-	+ \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{}))
-\end{split}
+	= \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
 \end{equation}
-is the Hxc potential.
+is the Hxc potential, with
+\begin{subequations}
+\begin{align}
+	\fdv{\E{\Ha}{}[\n{}{}]}{\n{}{}(\br{})} 
+	& = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}',
+	\\
+	 \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} 
+	 & = \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})).
+\end{align}
+\end{subequations}
 Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view.
 Note that the individual densities $\n{\Det{I}{\bw}}{}(\br{})$ defined in Eq.~\eqref{eq:nI} do not match the \textit{exact} individual-state densities as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density. 
 Nevertheless, 
@@ -310,24 +319,30 @@ Numerical quadratures are performed with the \texttt{numgrid} library using 194
 This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
 Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered. 
 Although we should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint.
-Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
+Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
 Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Results}
-\label{sec:res}
+\section{Hydrogen molecule}
+\label{sec:H2}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-First, we compute the ensemble energy of the \ce{H2} molecule using the aug-cc-pVTZ basis set and the weight-independent Slater local exchange functional, \cite{Dirac_1930, Slater_1951} \bruno{notée S51 sur les figures?} which is explicitly given by 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Weight-independent exchange functional}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+First, we compute the ensemble energy of the \ce{H2} molecule (at equilibrium bond length, \ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent Slater-Dirac local exchange functional, \cite{Dirac_1930, Slater_1951} which is explicitly given by 
 \begin{align}
 	\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
 	&
 	\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
 \end{align}
-The ensemble energy $\E{}{w}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of the weight $0 \le \ew{} \le 1$.
+In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state and the lowest doubly-excited state of configuration $1\sigma_u^2$, which has an autoionising resonance nature. \cite{Bottcher_1974}
+The ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of the weight $0 \le \ew{} \le 1$.
 Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
-As anticipated, $\E{}{w}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies significantly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}).
-Note that the exact xc correlation ensemble functional would yield a perfectly linear energy and the same value of the excitation energy independently of $\ew{}$.
+As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies significantly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}).
+Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/2$.
+Note that the exact xc correlation ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\ew{}$.
 
 \begin{figure}
 	\includegraphics[width=\linewidth]{Ew_H2}
@@ -345,29 +360,37 @@ Note that the exact xc correlation ensemble functional would yield a perfectly l
 	}
 \end{figure}
 
-Second, in order to remove this spurious curvature of the ensemble energy (which is partly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
-Doing so, we have found that the present weight-dependent exchange functional (denoted as MSFL in the following), represented in Fig.~\ref{fig:Cx_H2},
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Weight-dependent exchange functional}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Second, in order to remove this spurious curvature of the ensemble energy (which is mostly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
+Doing so, we have found that the present weight-dependent exchange functional (denoted as GIC-S in the following as its main purpose is to correct for the ghost-interaction error), represented in Fig.~\ref{fig:Cx_H2},
 \begin{equation}
-	\e{\ex}{\ew{},\text{MSFL}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
+	\e{\ex}{\ew{},\text{GIC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
 \end{equation}
 with 
 \begin{equation}
 \label{eq:Cxw}
-	\Cx{\ew{}} = \Cx{} \qty{ 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ]}
+	\frac{\Cx{\ew{}}}{\Cx{}} = 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ]
 \end{equation}
 and 
+\begin{subequations}
 \begin{align}
 	\alpha & = + 0.575\,178,
-	&
+	\\
 	\beta & = - 0.021\,108,
-	&
+	\\
 	\gamma & = - 0.367\,189,
 \end{align}
-makes the ensemble much more linear (see Fig.~\ref{fig:Ew_H2})\bruno{C'est celle notée ``GIC'' sur la figure ? Pourquoi pas MSFL ? A clarifier pour le lecteur}, and the excitation energy much more stable and closer to the full configuration interaction (FCI) reference of $28.75$ eV \cite{Barca_2018a} (see Fig.~\ref{fig:Om_H2})
+\end{subequations}
+makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}).
 As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$.
-Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider $1/2 < \ew{} \le 1$.
-We ensure that the weight-dependent functional does not affect the two ghost-interaction-free limit at $\ew{} = 0$ and $1$.
- 
+Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is strictly forbidden by the GOK variational principle. \cite{Gross_1988a} 
+However, it is important to ensure that the weight-dependent functional does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $\ew{} = 1$.
+Therefore, by construction, the weight-dependent correction vanishes for these two limiting weight values (see Fig.~\ref{fig:Cx_H2}).
+Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear.
 \begin{figure}
 	\includegraphics[width=0.8\linewidth]{Cx_H2}
 	\caption{
@@ -376,9 +399,14 @@ We ensure that the weight-dependent functional does not affect the two ghost-int
 	}
 \end{figure}
 
-In a third time, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Weight-independent correlation functional}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Third, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980}
 For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
-The combination of the Slater and VWN5 functionals (SVWN5) yield a highly non-linear ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the MSFL and VWN5 functionals exhibit a small curvature and improved excitation energies, especially at small weights.
+The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the GIC-S and VWN5 functionals (GIC-SVWN5) exhibit a small curvature (the ensemble energy is now slightly concave) and improved excitation energies, especially at small weights, where the SVWN5 excitation energy is almost spot on. 
 
 %%%%%%%%%%%%%%%%%%
 %%% FUNCTIONAL %%%
@@ -391,12 +419,17 @@ The combination of the Slater and VWN5 functionals (SVWN5) yield a highly non-li
 %The construction of these two functionals is described below.
 %Extension to spin-polarised systems will be reported in future work.
 
-Fourth, in the spirit of our recent work \cite{Loos_2020}, we have designed a weight-dependent correlation functional. 
-To build this weight-dependent correlation functional, we consider the singlet ground state and the first singlet doubly-excited state of a two-electron finite UEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Weight-dependent correlation functional}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+Fourth, in the spirit of our recent work, \cite{Loos_2020} we have designed a weight-dependent correlation functional. 
+To build this weight-dependent correlation functional, we consider the singlet ground state and the first singlet doubly-excited state of a two-electron FUEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
 Notably, these two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
 Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalised hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}  
 As mentioned above, we confine our attention to paramagnetic (or unpolarised) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron, thus yielding an electron density that is uniform on the 3-sphere.
-Note that the present paradigm is equivalent to the IUEG model in the thermodynamic limit. \cite{Loos_2011b}
+Note that the present paradigm is equivalent to the conventional IUEG model in the thermodynamic limit. \cite{Loos_2011b}
 We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -408,8 +441,10 @@ The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states
 \begin{subequations}
 \begin{align}
 	\e{\HF}{(0)}(\n{}{}) & = \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
+	\label{eq:eHF_0}
 	\\
 	\e{\HF}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3} + \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}.
+	\label{eq:eHF_1}
 \end{align}
 \end{subequations}
 %These two energies can be conveniently decomposed as 
@@ -455,14 +490,7 @@ The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states
 %	\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}.
 %\end{equation}
 %Conveniently, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient; an expected feature from a theoretical point of view, yet a nice property from a more practical aspect.
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%\subsection{Weight-dependent correlation functional}
-%\label{sec:Ec}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-Based on highly-accurate calculations, \cite{Loos_2009a,Loos_2009c,Loos_2010e} one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
+Thanks to highly-accurate calculations and the expressions of the HF energies provided by Eqs.~\eqref{eq:eHF_0} and \eqref{eq:eHF_1}, \cite{Loos_2009a,Loos_2009c,Loos_2010e} one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant \cite{Sun_2016,Loos_2020}
 \begin{equation}
 \label{eq:ec}
 	\e{\co}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
@@ -479,7 +507,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
 
 %%% FIG 1 %%%
 \begin{figure}
-	\includegraphics[width=\linewidth]{fig/fig1}
+	\includegraphics[width=0.8\linewidth]{fig1}
 	\caption{
 	Reduced (i.e., per electron) correlation energy $\e{\co}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi^2 \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarised) two-electron FUEG.
 	The data gathered in Table \ref{tab:Ref} are also reported. 
@@ -538,114 +566,117 @@ Combining these, we build a two-state weight-dependent correlation functional:
 %\subsection{LDA-centered functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-Our intent is 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).
-Hence, we employ a simple embedding scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
-The weight-dependence of the xc functional is then carried exclusively by the impurity [\ie, the functionals defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA xc functional).
+Because our intent is 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), we employ a simple embedding scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
+As explained further in Ref.~\onlinecite{Loos_2020}, this embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles originally derived by Franck and Fromager. \cite{Franck_2014} 
+The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA correlation functional).
  
-Consistently with such a strategy, Eqs.~\eqref{eq:exw} and \eqref{eq:ecw} are ``centred'' on their corresponding jellium reference
-\bruno{you commented the exchange part, why ?}
+Consistently with such a strategy, Eq.~\eqref{eq:ecw} is ``centred'' on its corresponding weight-independent VWN5 LDA reference
 \begin{equation}
 \label{eq:becw}
-	\be{\xc}{\ew{}}(\n{}{}) =  (1-\ew{}) \be{\xc}{(0)}(\n{}{}) + \ew{} \be{\xc}{(1)}(\n{}{})
+	\be{\co}{\ew{}}(\n{}{}) =  (1-\ew{}) \be{\co}{(0)}(\n{}{}) + \ew{} \be{\co}{(1)}(\n{}{})
 \end{equation}
 via the following shift:
 \begin{equation}
-	\be{\xc}{(I)}(\n{}{}) = \e{\xc}{(I)}(\n{}{}) + \e{\xc}{\LDA}(\n{}{}) - \e{\xc}{(0)}(\n{}{}).
+	\be{\co}{(I)}(\n{}{}) = \e{\co}{(I)}(\n{}{}) + \e{\co}{\VWN}(\n{}{}) - \e{\co}{(0)}(\n{}{}).
 \end{equation}
-The LDA xc functional is similarly decomposed as
-\begin{equation}
-	\e{\xc}{\LDA}(\n{}{}) = \e{\ex}{\LDA}(\n{}{}) + \e{\co}{\LDA}(\n{}{}),
-\end{equation}
-where we consider here the Dirac exchange functional \cite{Dirac_1930} 
-\begin{equation}
-	\e{\ex}{\LDA}(\n{}{}) = \Cx{\LDA} \n{}{1/3},
-\end{equation}
-with 
-\begin{equation}
-	\Cx{\LDA} = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3},
-\end{equation}
-and the VWN5 correlation functional \cite{Vosko_1980}
-\begin{equation}
-	\e{\co}{\LDA}(\n{}{}) \equiv \e{\co}{\text{VWN5}}(\n{}{}).
-\end{equation}
-For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
 
 Equation \eqref{eq:becw} can be recast
 \begin{equation}
 \label{eq:eLDA}
 \begin{split}
-	\be{\xc}{\ew{}}(\n{}{}) 
-	& =  \e{\xc}{\LDA}(\n{}{}) + \ew{} \qty[\e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})]
+	\be{\co}{\ew{}}(\n{}{}) 
+	& =  \e{\co}{\VWN}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})]
 	\\
-	& =  \e{\xc}{\LDA}(\n{}{}) + \ew{} \pdv{\e{\xc}{\ew{}}(\n{}{})}{\ew{}},
+	& =  \e{\co}{\VWN}(\n{}{}) + \ew{} \pdv{\e{\co}{\ew{}}(\n{}{})}{\ew{}},
 \end{split}
 \end{equation}
-which nicely highlights the centrality of the LDA in the present eDFA. 
-In particular, $\be{\xc}{(0)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{})$.
-Consequently, in the following, we name this weight-dependent xc functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
+which nicely highlights the centrality of the LDA in the present weight-dependent density-functional approximation for ensembles.
+In particular, $\be{\co}{(0)}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$.
+Consequently, in the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the VWN5 local correlation functional for ensembles.
 Also, we note that, by construction,
 \begin{equation}
 \label{eq:dexcdw}
-	\pdv{\be{\xc}{\ew{}}(\n{}{})}{\ew{}}
-	= \be{\xc}{(1)}(n(\br)) - \be{\xc}{(0)}(n(\br)).
+	\pdv{\be{\co}{\ew{}}(\n{}{})}{\ew{}}
+	= \be{\co}{(1)}(n) - \be{\co}{(0)}(n),
+\end{equation}
+which shows that the weight correction is purely linear in eVWN5.
+
+As shown in Fig.~\ref{fig:Ew_H2}, the SGIC-eVWN5 is slightly less concave than its SGIC-VWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).
+
+%This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
+%\begin{equation}
+%\label{eq:GACE}
+%	\E{\xc}{\bw}[\n{}{}] 
+%	= \E{\xc}{}[\n{}{}] 
+%	+ \sum_{I=1}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi,
+%\end{equation}
+%(where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014} 
+%Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional. 
+%In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir 
+%$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?}
+
+For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
+In particular, we report the excitation energies obtained with GOK-DFT in the zero-weight limit (\ie, $\ew{} = 0$) and for the equi-ensemble (\ie, $\ew{} = 1/2$).
+For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016} which are defined as
+\begin{equation}
+	\Ex{\LIM}{(1)} = 2 (\E{}{\ew{}=1/2} - \E{}{\ew{}=0}),
+\end{equation}
+as well as the MOM excitation energies. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
+We point out that the MOM excitation energy is obtained by the difference in energy between the doubly-excited state at $\ew{} = 1$ and the ground state at $\ew{} = 0$.
+MOM excitation energies can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
+\begin{equation}
+	\Ex{\MOM}{(1)} = \E{}{\ew{}=1} - \E{}{\ew{}=0}.
 \end{equation}
 
-This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
-\begin{equation}
-\label{eq:GACE}
-	\E{\xc}{\bw}[\n{}{}] 
-	= \E{\xc}{}[\n{}{}] 
-	+ \sum_{I=1}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi,
-\end{equation}
-(where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014} 
-Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional. 
-In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir 
-$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?}
+The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the GIC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to GIC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
+The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less satisfactory, but still remains in good agreement with FCI, with again a small improvement as compared to GIC-SVWN5.
 
 %%% TABLE I %%%
 \begin{table*}
 \caption{
-Excitation energies (in eV) of \ce{H2} with $\RHH = 1.4$ bohr for various methods and basis sets. \bruno{Why you don't report results from the eLDA functional which is not system-specific like MSFL ? What is MSFL for the correlation part ? Is it what you referred to eLDA in the text ?}
-\label{tab:Energies}
+Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 1.4$ bohr for various methods, combinations of xc functionals, and basis sets.
+\label{tab:BigTab_H2}
 }
 \begin{ruledtabular}
 \begin{tabular}{llccccc}
-	\mc{2}{c}{xc functional} 	\\
-	\cline{1-2}
-	exchange	&	correlation		&	Basis			&	GOK($\ew{} = 0$)	&	GOK($\ew{} = 1/2$)	&	LIM		&	MOM		\\
+	\mc{2}{c}{xc functional} 		&					&	\mc{2}{c}{GOK}	\\
+	\cline{1-2}												\cline{4-5}
+	exchange	&	correlation		&	Basis			&	 $\ew{} = 0$	&	$\ew{} = 1/2$	&	LIM		&	MOM		\\
 	\hline
-	HF			&					&	aug-cc-pVDZ		&	38.52				&	30.86				&	34.55	&	28.65	\\
-				&					&	aug-cc-pVTZ		&	38.58				&	35.82				&	35.68	&	28.65	\\
-				&					&	aug-cc-pVQZ		&	39.12				&	35.94				&	35.64	&	28.65	\\
+	HF			&					&	aug-cc-pVDZ		&	38.52			&	30.86			&	34.55	&	28.65	\\
+				&					&	aug-cc-pVTZ		&	38.58			&	35.82			&	35.68	&	28.65	\\
+				&					&	aug-cc-pVQZ		&	39.12			&	35.94			&	35.64	&	28.65	\\
 	\\
-	S			&					&	aug-cc-pVDZ		&	38.40				&	27.35				&	23.54	&	26.60	\\
-				&					&	aug-cc-pVTZ		&	39.21				&	27.42				&	23.62	&	26.67	\\
-				&					&	aug-cc-pVQZ		&	39.78				&	27.42				&	23.62	&	26.67	\\
+	S			&					&	aug-cc-pVDZ		&	38.40			&	27.35			&	23.54	&	26.60	\\
+				&					&	aug-cc-pVTZ		&	39.21			&	27.42			&	23.62	&	26.67	\\
+				&					&	aug-cc-pVQZ		&	39.78			&	27.42			&	23.62	&	26.67	\\
 	\\
-	S			&	VWN5			&	aug-cc-pVDZ		&	38.10				&	27.76				&	24.40	&	27.10	\\
-				&					&	aug-cc-pVTZ		&	38.54				&	27.81				&	24.46	&	27.17	\\
-				&					&	aug-cc-pVQZ		&	38.81				&	27.81				&	24.46	&	27.17	\\
+	S			&	VWN5			&	aug-cc-pVDZ		&	38.10			&	27.76			&	24.40	&	27.10	\\
+				&					&	aug-cc-pVTZ		&	38.54			&	27.81			&	24.46	&	27.17	\\
+				&					&	aug-cc-pVQZ		&	38.81			&	27.81			&	24.46	&	27.17	\\
 	\\
-	MSFL		&					&	aug-cc-pVDZ		&	26.83				&	26.51				&	26.53	&	26.60	\\
-				&					&	aug-cc-pVTZ		&	26.88				&	26.59				&	26.61	&	26.67	\\
-				&					&	aug-cc-pVQZ		&	26.82				&	26.60				&	26.62	&	26.67	\\
+	GIC-S		&					&	aug-cc-pVDZ		&	26.83			&	26.51			&	26.53	&	26.60	\\
+				&					&	aug-cc-pVTZ		&	26.88			&	26.59			&	26.61	&	26.67	\\
+				&					&	aug-cc-pVQZ		&	26.82			&	26.60			&	26.62	&	26.67	\\
 	\\
-	MSFL		&	VWN5			&	aug-cc-pVDZ		&	28.54				&	26.94				&	27.48	&	27.10	\\
-				&					&	aug-cc-pVTZ		&	28.66				&	27.00				&	27.56	&	27.17	\\
-				&					&	aug-cc-pVQZ		&	28.64				&	27.00				&	27.56	&	27.17	\\
+	GIC-S		&	VWN5			&	aug-cc-pVDZ		&	28.54			&	26.94			&	27.48	&	27.10	\\
+				&					&	aug-cc-pVTZ		&	28.66			&	27.00			&	27.56	&	27.17	\\
+				&					&	aug-cc-pVQZ		&	28.64			&	27.00			&	27.56	&	27.17	\\
 	\\
-	MSFL		&	MSFL			&	aug-cc-pVDZ		&	28.78				&	27.10				&	27.56	&	27.27	\\
-				&					&	aug-cc-pVTZ		&	28.90				&	27.16				&	27.64	&	27.34	\\
-				&					&	aug-cc-pVQZ		&	28.89				&	27.16				&	27.65	&	27.34	\\
+	GIC-S		&	eVWN5			&	aug-cc-pVDZ		&	28.78			&	27.10			&	27.56	&	27.27	\\
+				&					&	aug-cc-pVTZ		&	28.90			&	27.16			&	27.64	&	27.34	\\
+				&					&	aug-cc-pVQZ		&	28.89			&	27.16			&	27.65	&	27.34	\\
 	\\
-	B88			&	LYP				&	aug-mcc-pV8Z	&					&					&		&		28.42\fnm[1]	\\
-	B3			&	LYP				&	aug-mcc-pV8Z	&					&					&		&		27.77\fnm[1]	\\
-	HF			&	LYP				&	aug-mcc-pV8Z	&					&					&		&		29.18\fnm[1]	\\
-	HF			&					&	aug-mcc-pV8Z	&					&					&		&		28.65\fnm[1]	\\
-	HF			&	FCI				&	aug-mcc-pV8Z	&					&					&		&		28.75\fnm[1]	\\
+	B88			&	LYP				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		28.42\fnm[2]	\\
+	B3			&	LYP				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		27.77\fnm[2]	\\
+	HF			&	LYP				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		29.18\fnm[2]	\\
+	HF			&					&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		28.65\fnm[2]	\\
+	\\
+	HF			&	FCI				&	aug-mcc-pV8Z\fnm[1]	&					&					&		&		28.75\fnm[2]	\\
 \end{tabular}
 \end{ruledtabular}
-\fnt[1]{Reference \onlinecite{Barca_2018a}.}
+\fnt[1]{Reference \onlinecite{Mielke_2005}.}
+\fnt[2]{Reference \onlinecite{Barca_2018a}.}
 \end{table*}
 %%% %%% %%% %%%