revision in progress

Response_Letter/Response_Letter.tex

@@ -1,5 +1,5 @@
 \documentclass[10pt]{letter}
-\usepackage{UPS_letterhead,xcolor,mhchem,mathpazo,ragged2e,hyperref}
+\usepackage{UPS_letterhead,xcolor,mhchem,mathpazo,ragged2e,hyperref,physics,amsmath}
 \newcommand{\alert}[1]{\textcolor{red}{#1}}
 \definecolor{darkgreen}{HTML}{009900}
 
@@ -38,35 +38,36 @@ We look forward to hearing from you.
 	The approach is demonstrated with a clever use of one-dimensional boxes that allow investigation of its behavior in different correlation-strength regimes. 
 
 	The paper is significant, well written, and thoughtful. 
-	I recommend it's publication, though I do have some suggestions for improving its accessibility to a wider audience. 
+	I recommend its publication, though I do have some suggestions for improving its accessibility to a wider audience. 
 	I also have some questions, answers to which might improve the work's interest and utility for formal DFT investigators, if the answers are readily available. }
 	\\
-	\alert{bla bla bla}
+	\alert{We thank the reviewer for his/her kind comments.}
 
 	\item 
 	{Page 4: barred correlation per particle is introduced implicitly in Eqn 41. 
 	Some pieces of the later discussion of this quantity might be helpful here, since this quantity appears in discussion of eqns 48 and 49, }
 	\\
-	\alert{bla bla bla}
+	\alert{This part is for you Manu.}
 
 	\item 
 	{Page 4: It would be helpful to note clearly near eqn 40 that the authors' focus on LDA correlation differs from what many readers will be used to in ground-state LDA, namely using LDA exchange as well, instead of combining it with HF Hartree-exchange as is done here. 
 	The combination of exchange and correlation leads to some well-known behavior that readers should not rely upon when reading this work, so it would be a helpful gesture to remind them. 
 }
 	\\
-	\alert{bla bla bla}
+	\alert{The reviewer is right. 
+	We have added a note to clarify this point just below Eq.~(40).}
 
 	\item 
 	{Page 5: The authors' expansion of the correlation energy around the $I$th state and its resulting neglect of correlation effects between states more remote from one another might affect evaluation and analysis of the approximation and/or the embedding scheme. 
 	Could the authors note around eqn 47 somewhere how they decided this expansion was valid and useful, as well as how they anticipate this approximation might affect their later evaluation and analysis of the approximation, and similar for the embedding scheme? 
 	This is mentioned elsewhere in the text, but treating it here would bolster the authors' narrative and support their choices more. }
 	\\
-	\alert{bla bla bla}
+	\alert{This part is for you Manu.}
 
 	\item 
 	{Page 5: Though the ringium model is developed elsewhere in the literature in great detail, a diagram for readers not as familiar with it would be a kindness. }
 	\\
-	\alert{bla bla bla}
+	\alert{Following the reviewer's suggestion, we have added a figure to illustrate the ringium model in the supporting information.}
 
 	\item 
 	{Page 6: How well does equation 53 work in the low-density limit, or in the regime approaching the low-density limit? 
@@ -74,7 +75,23 @@ We look forward to hearing from you.
 	Is it important here? 
 	To mimic parametrizations used in ground-state LDA, both pieces of correlation behavior would be captured, so readers may question this as I do. }
 	\\
-	\alert{bla bla bla}
+	
+	\alert{By construction, Eq.~(53) is exact in the high-density limit as the coefficient $a_1^{(I)}$ is set to do so.
+	This point is mentioned in the original manuscript.
+	In the low-density (i.e., large $R$) regime, Eq.~(53) behaves as 
+	\begin{equation}
+		\varepsilon_c^{(I)} = \frac{a_1^{(I)}}{a_3^{(I)} \pi} \frac{1}{R} - \frac{a_1^{(I)} a_2^{(I)}}{(a_2^{(I)})^2 \pi^{3/2}} \frac{1}{R^{3/2}} + \order{R^{-2}}
+	\end{equation}
+	which is the correct behavior in terms of $R$, the first term (proportional to $R^{-1}$) representing the classical Coulomb repulsion between electrons and the second term (proportional to $R^{-3/2}$) representing the zero-point vibrational energy of the electrons.
+	This is the usual Wigner crystal representation as mentionned in our previous works.
+	The asymptotic behavior is then correct in the large-$R$ limit.
+	However, the coefficients do not exactly match the exact ones.
+	From a numerical and practical point of view, we have not found that enforcing the exact values does not improve the results.
+	Actually, it usually worsen them as enforcing the correct coefficients in the large-$R$ limit usually deteriorates the results for intermediate $R$ values.
+	Finally, let us mention that the logarithmic behavior does not occur for the correlation energy.
+	It only occurs in the thermodynamic limit (where the number of electrons gets very large) in the Hartree-exchange term.
+	In any case, for small, finite number of electrons, there is no logarithmic behavior.
+	We have added a small paragraph in the supporting information to further explain the behavior of Eq.~(53) in the low-density limit.}
 
 	\item 
 	{Page 6: Does the embedding scheme mean that the weight-dependence of the two-electron finite uniform system would ideally capture the weight-dependence for the infinite system? 
@@ -82,12 +99,14 @@ We look forward to hearing from you.
 	Or is that not the correct way to think about this way of embedding? 
 	I wonder if an explanatory diagram for the embedding scheme might clarify my thinking here, particularly if my thinking is wrong! }
 	\\
-	\alert{bla bla bla}
+	\alert{As mentioned in the original manuscript, the impurity carries the weight dependence of the functional.
+	Performing a simple gedanken experiment, one can imagine that, in the infinite system, the excitation will occur locally on the impurity.}
 
 	\item 
 	{Page 7: Another diagram suggestion: unfamiliar readers might be helped by a pedagogical diagram of your box systems, to help folks see how the different box lengths correspond to different correlation-strength regimes. }
 	\\
-	\alert{bla bla bla}
+	\alert{Very good suggestion. Accordingly, we have added a figure showing the electron density for a very small box and a very large box.
+	This illustrates how electrons localize when the density gets smaller.}
 
 	\item 
 	{Page 7: Does strong correlation always result in non-linear ghosts uncorrected by the GIC-eLDA, or is it particularly difficult or changed by the embedding scheme somehow? 
@@ -109,7 +128,7 @@ We look forward to hearing from you.
 	\item 
 	{Fig. 2: Why does the crossover point for the 1st excitation curves disappear for $L=8\pi$? }
 	\\
-	\alert{bla bla bla}
+	\alert{We do not know.}
 
 	\item 
 	{Page 8: If the authors have evidence of behavior between $w=(0,0)$ and the equiensemble, instead of just these endpoints, that would be interesting to mention for the eDFT crowd. }
@@ -126,29 +145,37 @@ We look forward to hearing from you.
 	{Figure 3 discussion: Will eLDA always overestimate double excitations? 
 	It's hard to see differences for small $L$. }
 	\\
-	\alert{bla bla bla}
+	\alert{The corresponding numerical data are reported in Supporting information.}
 
 	\item 
 	{Clear statement that $w=(0,0)$ is the conventional ground-state HF X+LDA C KS orbital energy difference result should come earlier in the manuscript, to guide less familiar readers. 
 	It currently occurs on page 9. }
 	\\
-	\alert{bla bla bla}
+	\alert{This is now mentioned earlier in the manuscript (see Computational Details section).}
 
 	\item 
 	{Page 9: Is the negligible effect of the second term of Eqn 51 on the excitation energies due to the focus on uniform systems or verified via FCI on boxium? }
 	\\
-	\alert{bla bla bla}
+	\alert{As mentioned in the original manuscript (see Results and Discussion section), we believe that it might be a consequence of how we constructed the eLDA functional, as the weight dependence of the eLDA functional is based on a two-electron uniform electron gas.
+	We do not think this is due to the uniformity of its density, though.
+	Incorporating a $N$-dependence in the functional through the curvature of the Fermi hole might be valuable in this respect. 
+	This is left for future work.}
 
 	\item 
 	{Figure 6: Do the ground-state and equiensemble results for doubles converge as $N$ goes to infinity?}
 	\\
-	\alert{bla bla bla}
+	\alert{This is an excellent question. Currently, we do not have the data to check this fact as FCI calculations cannot be easily performed for larger number of electrons.
+	Moreover, GOK-DFT calculations for larger number of electrons would require a larger one-electron basis set to be converged.
+	Currently, our one-electron basis set is fixed to 30 basis functions.
+	We hope to be able to answer this question in the near future.}
 
 	\item 
 	{Figure 6: Why is $N=7$ the number of particles where the DeltaC influence vanishes? 
 	Does it remain negligible for higher particle numbers? }
 	\\
-	\alert{bla bla bla}
+	\alert{Currently, we do not have any explanation on why for $N=7$ the DeltaC influence vanishes.
+	Our guess is that it might change sign but we could not compute the FCI energies for more than 7 electrons.
+	Therefore, we cannot fully answer this question here.}
 	
 \end{itemize}
 

Revised_Manuscript/eDFT.tex

@@ -197,7 +197,6 @@ The weight dependence of the xc functional plays a crucial role in the
 calculation of excitation energies.
 \cite{Gross_1988b,Yang_2014,Deur_2017,Deur_2019,Senjean_2018,Senjean_2020} 
 It actually accounts for the derivative discontinuity contribution to energy gaps. \cite{Levy_1995, Perdew_1983}
-%\titou{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
 
 Even though GOK-DFT is in principle able to
 describe near-degenerate situations and multiple-electron excitation
@@ -234,7 +233,6 @@ spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
 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}
 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}
 
@@ -463,22 +461,6 @@ ensemble KS orbitals (from which the determinants are constructed) in an atomic
 \beq
 	\MO{p}{}(\br{}) =  \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
 \eeq
-\iffalse%%%%%%%%%%%%%%%%%%%%%%%%
-\titou{\beq
-	\SO{p}{}(\bx{}) = s(\omega) \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
-\eeq
-where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatial degrees of freedom, and
-\beq
-	s(\omega)
-	= 
-	\begin{cases}
-		\alpha(\omega),	&	\text{for spin-up electrons,}	\\
-		\text{or}	\\
-		\beta(\omega),	&	\text{for spin-down electrons,}	
-	\end{cases}
-\eeq
-}
-\fi%%%%%%%%%%%%%%%%%%%%%
 then the density matrix of the   
 determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
 \beq
@@ -490,26 +472,6 @@ as follows:
 \beq
 	\n{\bGam{(K)}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{(K)} \AO{\nu}(\br{}),
 \eeq 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Manu's derivation %%%
-\iffalse%%
-\blue{
-\beq
-n_{\bmg^{(K)}}(\br{})&=&\sum_\sigma\left\langle\hat{\Psi}^\dagger(\br{}\sigma)\hat{\Psi}(\br{}\sigma)\right\rangle^{(K)}
-\nonumber\\
-&=&\sum_\sigma\sum_{pq}\varphi^\sigma_p(\br{})\varphi^\sigma_q(\br{})\left\langle\hat{a}_{p^\sigma,\sigma}^\dagger\hat{a}_{q^\sigma,\sigma}\right\rangle^{(K)}
-\nonumber\\
-&=&\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}\left(\varphi^\sigma_p(\br{})\right)^2
-\nonumber\\
-&=&\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}\sum_{\mu\nu}c^\sigma_{{\mu
-p}}c^\sigma_{{\nu p}}\AO{\mu}(\br{})\AO{\nu}(\br{})
-\nonumber\\
-&=&\sum_{\mu\nu}\AO{\mu}(\br{})\AO{\nu}(\br{})\sum_\sigma\sum_{\varphi^\sigma_p\in(K)}c^\sigma_{{\mu
-p}}c^\sigma_{{\nu p}}
-\eeq
-}
-\fi%%%
-%%%% end Manu
 while the ensemble density matrix
 and the ensemble density read
 \beq
@@ -560,103 +522,11 @@ with
 \beq
 	\ERI{\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}.
 \eeq
-%Note that, in Sec.~\ref{sec:results}, the theory is applied to (1D) spin
-%polarized systems in which $\eGam{\mu\nu}{(K)\beta}=0$ and
-%$G_{\mu\nu\lambda\omega}^{\alpha\alpha}\equiv G_{\mu\nu\lambda\omega}=({\mu}{\nu}\vert{\lambda}{\omega})
-%-(\mu\omega\vert\lambda\nu)$. 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%% Hx energy ...
-%%% Manu's derivation
-\iffalse%%%%
-\blue{
-\beq
-&&\dfrac{1}{2}\sum_{PQRS}\langle PQ\vert\vert
-RS\rangle\eGam{PR}^{(K)}\eGam{QS}^{(L)}
-\nonumber\\
-&&
-=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}RS}
-\nonumber\\
-&&\Big(\langle p^\sigma\sigma q^\tau\tau\vert RS\rangle -\langle
-p^\sigma\sigma q^\tau\tau
-\vert SR\rangle
-\Big)\Gamma^{(K)}_{p^\sigma\sigma,R}\Gamma^{(L)}_{q^\tau\tau, S}
-\nonumber\\
-&&
-=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
-\nonumber\\
-&&\Big(\sum_{r^\sigma s^\tau}\langle p^\sigma q^\tau\vert r^\sigma s^\tau\rangle
-\Gamma^{(K)\sigma}_{p^\sigma r^\sigma}\Gamma^{(L)\tau}_{q^\tau s^\tau}
-\nonumber\\
-&& -\sum_{s^\sigma r^\tau}\langle
-p^\sigma q^\tau
-\vert s^\sigma r^\tau\rangle
-\delta_{\sigma\tau}\Gamma^{(K)\sigma}_{p^\sigma
-r^\sigma}\Gamma^{(L)\sigma}_{q^\sigma s^\sigma}\Big)
-\nonumber\\
-&&=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
-\nonumber\\
-&&\left(\langle p^\sigma q^\tau\vert p^\sigma q^\tau\rangle
-n_{p^\sigma}^{(K)\sigma}n_{q^\tau}^{(L)\tau}
--\delta_{\sigma\tau}\langle p^\sigma q^\sigma\vert q^\sigma p^\sigma \rangle
-n_{p^\sigma}^{(K)\sigma}n_{q^\sigma}^{(L)\sigma}\right) 
-\nonumber\\
-&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
-\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
--\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\sigma}
-\Big)
-\nonumber\\
-&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
--\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle
-\Big)
-\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
-\nonumber\\
-&&=\dfrac{1}{2}\sum_{\mu\nu\lambda\omega}\sum_{\sigma,\tau}\Big[({\mu}{\nu}\vert{\lambda}{\omega})
--\delta_{\sigma\tau}(\mu\omega\vert\lambda\nu)
-\Big]
-\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
-\eeq
-}
-\fi%%%%%%%
-%%%%
-%%%%%%%%%%%%%%%%%%%%%
-\iffalse%%%% Manu's derivation ...
-\blue{
-\beq
-n^{\bw}({\br{}})&=&\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
-w}_Kn^{(K)}({\bfx})
-\nonumber\\
-&=&
-\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
-w}_K\sum_{pq}\varphi_p({\bfx})\varphi_q({\bfx})\Gamma_{pq}^{(K)}
-\nonumber\\
-&=&
-\sum_{\sigma=\alpha,\beta}
-\sum_{K\geq 0}
-{\tt
-w}_K\sum_{p\in (K)}\varphi^2_p({\bfx})
-\nonumber\\
-&=&
-\sum_{\sigma=\alpha,\beta}
-\sum_{K\geq 0}
-{\tt
-w}_K
-\sum_{\mu\nu}
-\sum_{p\in (K)}c_{\mu p}c_{\nu p}\AO{\mu}({\bfx})\AO{\nu}({\bfx})
-\nonumber\\
-&=&\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\bfx})\AO{\nu}({\bfx}){\Gamma}^{\bw}_{\mu\nu}
-\eeq
-}
-\fi%%%%%%%% end
-%%%%%%%%%%%%%%%
-%\subsection{Hybrid GOK-DFT}
-%%%%%%%%%%%%%%%
-
 
 %%%%%%%%%%%%%%%
 \subsection{Approximations}\label{subsec:approx}
 %%%%%%%%%%%%%%%
 
-
 In the following, GOK-DFT will be applied
 to 1D
 spin-polarized systems where 
@@ -691,128 +561,7 @@ with
 	\eh{\mu\nu}{\bw} 
 	= \eh{\mu\nu}{} + \int \AO{\mu}(\br{}) \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} \AO{\nu}(\br{}) d\br{}.
 \eeq
-%%%%%%%%%%%%%%%
-\iffalse%%%%%%
-% Manu's derivation %%%%
-\color{blue}
-I am teaching myself ...\\
-Stationarity condition
-\beq
-&&0=\sum_{K\geq 0}w_K\sum_{t^\sigma}\Big(f_{p^\sigma\sigma,t^\sigma\sigma}\Gamma^{(K)\sigma}_{t^\sigma
-q^\sigma}-\Gamma^{(K)\sigma}_{p^\sigma
-t^\sigma}f_{t^\sigma\sigma,q^\sigma\sigma}\Big)
-\nonumber\\
-&&=\sum_{K\geq 0}w_K
-\Big(f_{p^\sigma\sigma,q^\sigma\sigma}n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}f_{p^\sigma\sigma,q^\sigma\sigma}\Big)
-\nonumber\\
-&&
-=\sum_{\mu\nu}\sum_{K\geq 0}w_KF_{\mu\nu}^\sigma c^\sigma_{\mu
-p}c^\sigma_{\nu q}\left(n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}\right)
-\eeq
-thus leading to
-\beq
-&&0=\sum_{p^\sigma q^\sigma}c^\sigma_{\lambda
-p}c^\sigma_{\omega q}\left(\sum_{\mu\nu}\sum_{K\geq 0}w_KF_{\mu\nu}^\sigma c^\sigma_{\mu
-p}c^\sigma_{\nu q}\left(n^{(K)\sigma}_{q^\sigma}-n^{(K)\sigma}_{p^\sigma}\right)\right)
-\nonumber\\
-&&=\sum_{\mu\nu}\sum_{K\geq 0}w_K
-F_{\mu\nu}^\sigma\left(\Gamma^{(K)\sigma}_{\nu\omega}\sum_{p^\sigma}c^\sigma_{\lambda
-p}c^\sigma_{\mu
-p}-\Gamma^{(K)\sigma}_{\mu\lambda}\sum_{q^\sigma}c^\sigma_{\omega q}c^\sigma_{\nu q}\right)
-\nonumber\\
-\eeq
-If we denote $M^\sigma_{\lambda\mu}=\sum_{p^\sigma}c^\sigma_{\lambda
-p}c^\sigma_{\mu
-p}$ it comes 
-\beq
-S_{\mu\nu}=\sum_{\lambda\omega}S_{\mu\lambda}M^\sigma_{\lambda\omega}S_{\omega\nu}
-\eeq
-which simply means that
-\beq
-{\bm S}={\bm S}{\bm M}{\bm S}
-\eeq
-or, equivalently,
-\beq
-{\bm M}={\bm S}^{-1}.
-\eeq
-The stationarity condition simply reads
-\beq
-\sum_{\mu\nu}F_{\mu\nu}^\sigma\left(\Gamma^{\bw\sigma}_{\nu\omega}
-\left[{\bm S}^{-1}\right]_{\lambda\mu}
--\Gamma^{\bw\sigma}_{\mu\lambda}\left[{\bm S}^{-1}\right]_{\omega\nu}\right)
-=0
-\eeq
-thus leading to
-\beq
-{\bm S}^{-1}{{\bm F}^\sigma}{\bm \Gamma}^{\bw\sigma}={\bm \Gamma}^{\bw\sigma}{{\bm F}^\sigma}{\bm S}^{-1}
-\eeq
-or, equivalently,
-\beq
-{{\bm F}^\sigma}{\bm \Gamma}^{\bw\sigma}{\bm S}={\bm S}{\bm
-\Gamma}^{\bw\sigma}{{\bm F}^\sigma}.
-\eeq
-%%%%%
 
-Fock operator:\\
-\beq
-&&f_{p^\sigma\sigma,q^\sigma\sigma}-\langle\varphi_p^\sigma\vert\hat{h}\vert\varphi_q^\sigma\rangle
-\nonumber\\
-&&=\sum_{L\geq 0}w_L\sum_{\tau}\sum_{r^\tau s^\tau}
-\nonumber\\
-&&
-\Big(\langle p^\sigma r^\tau\vert
-q^\sigma s^\tau\rangle
--\delta_{\sigma\tau}\langle p^\sigma r^\sigma\vert
-s^\sigma q^\sigma\rangle
-\Big)
-\Gamma^{(L)\tau}_{r^\tau
-s^\tau}
-\nonumber\\
-&&
-=\sum_{L\geq 0}w_L\sum_{\tau}\sum_{r^\tau}\Big(\langle p^\sigma r^\tau\vert
-q^\sigma r^\tau\rangle
--\delta_{\sigma\tau}\langle p^\sigma r^\tau\vert
-r^\tau q^\sigma\rangle
-\Big)
-n^{(L)\tau}_{r^\tau}
-\nonumber\\
-&&=\sum_{L\geq 0}w_L
-\sum_{\lambda\omega}\sum_{\tau}\Big[\langle
-p^\sigma\lambda\vert q^\sigma\omega\rangle
--\delta_{\sigma\tau}
-\langle
-p^\sigma\lambda\vert \omega q^\sigma\rangle\Big]
-\Gamma^{(L)\tau}_{\lambda\omega}
-\nonumber\\
-&&=
-\sum_{\lambda\omega}\sum_{\tau}\Big[\langle
-p^\sigma\lambda\vert q^\sigma\omega\rangle
--\delta_{\sigma\tau}
-\langle
-p^\sigma\lambda\vert \omega q^\sigma\rangle\Big]
-\Gamma^{\bw\tau}_{\lambda\omega}
-\nonumber\\
-&&=\sum_{\mu\nu\lambda\omega}\sum_{\tau}
-\Big(\langle{\mu}{\lambda}\vert{\nu}{\omega}\rangle
--\delta_{\sigma\tau}\langle\mu\lambda\vert\omega\nu\rangle
-\Big)\Gamma^{\bw\tau}_{\lambda\omega}c^\sigma_{\mu p}c^\sigma_{\nu q}
-\nonumber\\
-\eeq
-or, equivalently,
-\beq
-f_{p^\sigma\sigma,q^\sigma\sigma}=\sum_{\mu\nu}F_{\mu\nu}^\sigma c^\sigma_{\mu p}c^\sigma_{\nu q}
-\eeq
-where
-\beq
-F_{\mu\nu}^\sigma=h_{\mu\nu}+\sum_{\lambda\omega}\sum_\tau
-G_{\mu\nu\lambda\omega}^{\sigma\tau}\Gamma^{\bw\tau}_{\lambda\omega}
-\eeq
-and
-\color{black}
-\\
-\fi%%%%%%%%%%%
-%%%%% end Manu 
-%%%%%%%%%%%%%%%%%%%%
 Note that, within the approximation of Eq.~\eqref{eq:min_with_HF_ener_fun}, the ensemble density matrix is
 optimized with a non-local exchange potential rather than a
 density-functional local one, as expected from
@@ -847,8 +596,8 @@ where the ensemble correlation energy per particle
 is 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.
-%What do you think?}
+\titou{Note that, here, only the correlation part is treated at the KS level while we rely on exact HF exchange.
+This is different from the usual context where both exchange and correlation are treated at the LDA level which gives compensation of errors.}
 
 The resulting KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun}
 reads 
@@ -856,9 +605,6 @@ reads
 \E{eLDA}{\bw}=\Tr[\bGam{\bw}\bh] + \WHF[\bGam{\bw}] +\int
 \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{\bw}}{}(\br{}) d\br{}.
 \eeq
-%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 expression of the
 KS-eLDA energy levels 
@@ -922,30 +668,18 @@ correlation energy per particle in Eq.
 \eqref{eq:decomp_ens_correner_per_part}, the latter can be recast
 \begin{equation}
 \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{}))
 ] \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{equation}
 thus leading to the following Taylor expansion through first order in
 $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: 
 \beq\label{eq:Taylor_exp_DDisc_term}
 \begin{split}
 \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 \qty[ \be{c}{(I)}(\n{\bGam{(I)}}{}(\br{})) - \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) ] \n{\bGam{(I)}}{}(\br{}) d\br{}
 \\
@@ -1059,27 +793,7 @@ $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$, where $\ew{1}$
 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.
-%The constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed
-%to consider an equi-bi-ensemble
-%for which $\ew{1}=1/2$. This possibility is excluded with your
-%inequalities. The correct constraints are given in Ref.~\cite{Deur_2019}
-%and are the ones you also mentioned, \ie, $0 \le \ew{2} \le 1/3$ and
-%$\ew{2} \le \ew{1} \le (1-\ew{2})/2$.} 
-%\manu{
-%Just in case, starting from
-%\beq
-%\begin{split}
-%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{1}-\ew{2})
-%\\
-%\end{split}
-%\eeq
-%we obtain
-%\beq
-%0\leq \ew{2}\leq \ew{1}\leq (1-\ew{2})/2 
-%\eeq
-%which implies $\ew{2}\leq(1-\ew{2})/2$ or, equivalently, $\ew{2}\leq
-%1/3$.
-%}
+
 %%% TABLE 1 %%%
 \begin{table*}
 	\caption{
@@ -1124,7 +838,7 @@ Combining these, one can build the following three-state weight-dependent correl
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 One of the main driving force behind the popularity of DFT is its ``universal'' nature, as xc density functionals can be applied to any electronic system.
 Obviously, the two-electron-based density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the finite uniform electron gas.
-However, one can partially cure this dependency by applying a simple embedding scheme in which the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath).
+However, one can partially cure this dependency by applying a simple \titou{``embedding''} scheme in which the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath).
 The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in Eq.~\eqref{eq:ecw}], while the remaining correlation effects are provided by the bath (\ie, the usual LDA correlation functional).
 Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} as follows:
 \begin{equation}
@@ -1194,6 +908,18 @@ Our testing playground for the validation of the eLDA functional is the ubiquito
 In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \nEl \le 7$. 
 These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
 For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime. \cite{Rogers_2017,Rogers_2016}
+\titou{The one-electron density in these two regimes of correlation is represented in Fig.~\ref{fig:rho}.}
+
+%%% FIG 1 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{rho}
+	\caption{
+	\titou{Ground-state one-electron density $\n{}{}(x)$ of 4-boxium (\ie, $N = 4$) for $L = \pi/32$ (left) and $L = 32\pi$ (right).
+	In the weak correlation regime (small box length), the one-electron density is much more delocalized than in the strong correlation regime (large box length).}
+	\label{fig:rho}
+	}
+\end{figure}
+%%% %%% %%%
 
 We use as basis functions the (orthonormal) orbitals of the one-electron system, \ie,
 \begin{equation}
@@ -1208,7 +934,6 @@ with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations.
 The convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw}
 \bS - \bS \bGam{\bw} \bF{\bw}}}$ [see Eq.~\eqref{eq:commut_F_AO}] of the KS-DFT self-consistent calculation is set
 to $10^{-5}$. 
-%For comparison, regular HF and KS-DFT calculations are performed with the same threshold.
 In order to compute the various density-functional
 integrals that cannot be performed in closed form, 
 a 51-point Gauss-Legendre quadrature is employed.
@@ -1222,6 +947,8 @@ Its Tamm-Dancoff approximation version  (TDA-TDLDA) is also considered. \cite{Dr
 Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
 (ground-state) limit where $\bw = (0,0)$ and the
 equi-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$.
+\titou{Note that a zero-weight calculation does correspond to a conventional ground-state KS calculation with exact exchange and LDA correlation.}
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results and discussion}
@@ -1252,24 +979,6 @@ 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 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:
-%\beq
-%\WHF[
-%\bGam{\bw}]-\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}]
-%\eeq
-%For a bi-ensemble ($w_1=w$) it can be written as
-%\beq
-%\dfrac{1}{2}\left[(w^2-1)W_0+w(w-2)W_1\right]+w(1-w)W_{01} 
-%\eeq
-%If, for some reason, $W_0\approx W_1\approx W_{01}=W$, then the error
-%reduces to $-W/2$, which is weight-independent (it fits for example with
-%what you see in the weakly correlated regime). Such an assumption depends on the nature of the
-%excitation, not only on the correlation strength, right? Nevertheless,
-%when looking at your curves, this assumption cannot be made when the
-%correlation is strong. It is not clear to me which integral ($W_{01}?$)
-%drives the all thing.\\} 
 It is important to note that, even though the GIC removes the explicit
 quadratic Hx terms from the ensemble energy, a non-negligible curvature
 remains in the GIC-eLDA ensemble energy when the electron
@@ -1280,21 +989,6 @@ energies [see Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
 \eqref{eq:Taylor_exp_DDisc_term}], and (ii) the optimization of the
 ensemble KS orbitals in the presence of ghost-interaction errors [see
 Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:WHF}].
-%However, this orbital-driven error is small (in our case at
-%least) \trashEF{as the correlation part of the ensemble KS potential $\delta
-%\E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared
-%to the Hx contribution}.\manu{Manu: well, I guess that the problem arises
-%from the density matrices (or orbitals) that are used to compute
-%individual Coulomb-exchange energies (I would not expect the DFT
-%correlation part to have such an impact, as you say). The best way to check is to plot the
-%ensemble energy without the correlation functional.}\\
-%\\
-%\manu{Manu: another idea. As far as I can see we do
-%not show any individual energies (excitation energies are plotted in the
-%following). Plotting individual energies (to be compared with the FCI
-%ones) would immediately show if there is some curvature (in the ensemble
-%energy). The latter would
-%be induced by any deviation from the expected horizontal straight lines.}
 
 %%% FIG 2 %%%
 \begin{figure*}
@@ -1347,27 +1041,7 @@ In other words, each excitation is dominated by a sole, well-defined reference S
 However, when the box gets larger (\ie, as $L$ 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 disputable. \cite{Loos_2019}
 This can be clearly evidenced by the weights of the different
-configurations in the FCI wave function.\\
-% TITOU: shall we keep the paragraph below?
-%Therefore, it is paramount to construct a two-weight correlation functional
-%(\ie, a triensemble functional, as we have done here) which
-%allows the mixing of singly- and doubly-excited configurations.
-%Using a single-weight (\ie, a biensemble) functional where only the ground state and the lowest singly-excited states are taken into account, one would observe a neat deterioration of the excitation energies (as compared to FCI) when the box gets larger.
-%\titou{Titou might add results for the biensemble to illustrate this.}
-%\manu{Well, neglecting the second excited state is not the same as
-%considering the $w_2=0$ limit. I thought you were referring to an
-%approximation where the triensemble calculation is performed with
-%the biensemble functional. This is not the same as taking $w_2=0$
-%because, in this limit, you may still have a derivative discontinuity
-%correction. The latter is absent if you truly neglect the second excited
-%state in your ensemble functional. This should be clarified.}\\
-%\manu{Are the results in the supp mat? We could just add "[not
-%shown]" if not. This is fine as long as you checked that, indeed, the
-%results deteriorate ;-)}
-%\manu{Should we add that, in the bi-ensemble case, the ensemble
-%correlation derivative $\partial \epsilon^\bw_{\rm c}(n)/\partial w_2$
-%is neglected (if this is really what you mean (?)). I guess that this is the reason why
-%the second excitation energy would not be well described (?)}
+configurations in the FCI wave function.
 
 As shown in Fig.~\ref{fig:EvsL}, all methods provide accurate estimates of the excitation energies in the weak correlation regime (\ie, small $L$).
 When the box gets larger, they start to deviate.
@@ -1380,17 +1054,7 @@ The effect on the double excitation is less pronounced.
 Overall, one clearly sees that, with 
 equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
 This conclusion is verified for smaller and larger numbers of electrons
-(see {\SI}).\\
-%\\
-%\manu{Manu: now comes the question that is, I believe, central in this
-%work. How important are the
-%ensemble correlation derivatives $\partial \epsilon^\bw_{\rm
-%c}(n)/\partial w_I$  that, unlike any functional
-%in the literature, the eLDA functional contains. We have to discuss this
-%point... I now see, after reading what follows that this question is
-%addressed later on. We should say something here and then refer to the
-%end of the section, or something like that ...}
-
+(see {\SI}).
 
 %%% FIG 4 %%%
 \begin{figure*}
@@ -1447,16 +1111,6 @@ 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}, for $\nEl = 3$, $5$, and $7$, the error percentage (with respect to FCI) as a function of the box length $L$
 on the excitation energies obtained at the KS-eLDA level 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
-%evaluate the importance of the ensemble correlation derivatives you
-%should only remove the following contribution from the $K$th KS-eLDA
-%excitation energy:  
-%\beq\label{eq:DD_term_to_compute}
-%\int  \n{\bGam{\bw}}{}(\br{})
-%      \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
-%\eeq
-%%rather than $E^{(I)}_{\rm HF}$
-%}
 We first stress that although for $\nEl=3$ both single and double excitation energies are 
 systematically improved (as the strength of electron correlation
 increases) when
@@ -1473,30 +1127,9 @@ This non-systematic behavior in terms of the number of electrons might
 be a consequence of how we constructed eLDA.
 Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of
 the eLDA functional is based on a \textit{two-electron} finite uniform electron gas.
-Incorporating an $\nEl$-dependence in the functional through the
+Incorporating a $\nEl$-dependence in the functional through the
 curvature of the Fermi hole, in the spirit of Ref.~\onlinecite{Loos_2017a}, would be
 valuable in this respect. This is left for future work.
-%\\
-%\manu{Manu: I am sorry to insist but I have a real problem with what follows. If
-%we look at the N=3 results, one has the impression that, indeed, for the
-%single excitation, a zero-weight calculation with the ensemble derivative
-%is almost equivalent to an equal-weight calculation without the
-%derivative. This is not the case for $N=5$ or 7, maybe because our
-%derivative is based on two electrons. }\\
-%{\it
-%Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble 
-%derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
-%compared to the zero-weight calculations).
-%%\manu{Manu: well, this is not
-%%really the case for the double excitation, right? I would remove this
-%%sentence or mention the single excitation explicitly.}
-%This could explain why equiensemble calculations are clearly more
-%accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative:
-%for a given method, equiensemble orbitals partially remove the burden
-%of modelling properly the ensemble correlation derivative.
-%}\\
-%\manu{Manu: I propose to rephrase this part as follows:}\\
-%\\
 Interestingly, for the single excitation in 3-boxium, the magnitude of the correlation ensemble 
 derivative is substantially reduced when switching from a zero-weight to
 an equal-weight calculation, while giving similar excitation energies,
@@ -1516,15 +1149,7 @@ of
 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 (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
-%Eq.~\eqref{eq:DD_term_to_compute} is. What follows should also be
-%updated in the light of the new results.}
+shown).
 
 %%% FIG 6 %%%
 \begin{figure}
@@ -1592,6 +1217,11 @@ dependence. We hope to report on this in the near future.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 See {\SI} for the additional details about the construction of the functionals, raw data and additional graphs.
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Data availability statement}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+The data that supports the findings of this study are available within the article [and its supplementary material].
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{acknowledgements}
 The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.