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\begin{letter}%
{To the Members of the Faraday Discussions Scientific Committee,}

\opening{Dear Members of the Faraday Discussions Scientific Committee,}

\justifying
Please find attached a revised version of the manuscript entitled 
\begin{quote}
\textit{``Weight Dependence of Local Exchange-Correlation Functionals in Ensemble Density-Functional Theory: Double Excitations in Two-Electron Systems''}.
\end{quote}
We thank the reviewer for his/her constructive comments.
Our detailed responses to his/her comments can be found below.

We look forward to hearing from you.

\closing{Sincerely, the authors.}

%%% REVIEWER 1 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#1}

	{The authors describe the ensemble formulation of DFT or the Gross-Oliveira-Kohn DFT (GOK-DFT) in its Kohn-Sham formulation as a viable method for excited state calculations. They provide a very clear summary of the theory, followed by the main work of the paper which is the investigation of weight-dependent LDA-type xc functionals for eDFT calculations. The provide important insights on small systems with 2 electrons and functionals that are tailored for double excitations in these systems. The manuscript makes an important contribution to the field of DFT and should be accepted for publication. However, I would be grateful if the authors modify the paper slightly to address the following minor points and corrections:}
	\\
	\alert{We thank the reviewer for his/her support. 
	His/her comments are addressed below.}


\begin{enumerate}

	\item 
	{They should comment about what is needed (or even if it is possible) to develop a weight-dependent universal xc functional for eDFT calculations instead of application-specific functionals as presented in this paper.}
	\\
	\alert{This is a good point. For clarity, we complemented the
theory section as follows.\\
{\it 
``As shown in Sec. IV A 4, the weight dependence of the correlation
energy can be extracted from a FUEG model. In order to make the
resulting weight-dependent correlation functional truly universal, i.e.
independent on the number of electrons in the FUEG, one could use the
curvature of the Fermi hole [88] as an additional variable in the
density-functional approximation. The development of such a generalized
correlation eLDA is left for future work. Even though a similar
strategy could be applied to the weight-dependent exchange part, we
explore in the present work a different path where the
(system-dependent) exchange functional parameterization relies on the
ensemble energy linearity constraint (see Sec. IV A 2). Finally, let us
stress that, in order to further improve the description of the ensemble
correlation energy, a post-treatment of the recently revealed
density-driven correlations [62,92-94] (which, by construction, are absent from FUEGs) might be necessary. An orbital-dependent correction derived in Ref. 92 might be used for that purpose. Work is currently in progress in this direction.
''}}

	\item 
	{In the captions of Figures 1 and 2 replace "functional's" with "functionals'"}
	\\
	\alert{This has been fixed.}

	\item 
	{Not clear what density is used in equation 21: From the discussion that follows this appears to not be the ensemble density (which is weight dependent and I would expect it to be used here) but the density only for the ground state Slater determinant. The authors should explain why this is so. }
	\\
	\alert{The density $n$ used in Eqs.~(21), (9), (10) can be any density, and, mathematically, there is no need to specify its origin. 
	In the case of Eq.~(21), we simply define the well-known Dirac exchange functional. 
	By definition of a density functional, we do not need to specify in its formulation to which density it is applied but only that it is a mathematical object applying to any density $n(r)$.
	Of course, when using this functional (or any other ones) in our work we surely apply it to the {\it minimizing} ensemble density $n^w(r)$ [see Eqs. (5) and (11)] and the notation is carefully modified accordingly.}
	
	\item 
	{Change "Third, we add up correlation effects" to "Third, we include correlation effects"}	
	\\
	\alert{This has been fixed.}	

	
	\item 
	{Change "studied in excruciated details" to "studied extensively"}	
	\\
	\alert{This has been fixed.}	
	
		
	\item 
	{They need to be a bit more consistent with their notation as in equation 9 and elsewhere "$n(r)$" should be the ensemble (weight-dependent) density "$n^w(r)$". 
	I don't believe they defined "$n(r)$" in the paper so I don?t know which density it represents. }	
	\\
	\alert{See our response to 3.}	
		
	\item 
	{Even if it sounds trivial, they should explain why the exact xc functional should have linear dependence in the excitation energies as a function of the weight value.}	
	\\
	\alert{As clearly stated in the original manuscript, GOK variational principle states that the expectation value of the ensemble energy admits a lower bond which is linear with respect to each of the ensemble weight $w_I$ and is the exact ensemble energy of the studied system [Eq.~(1)]. 
	Moreover, by construction, one can easily see that the slope of the exact ensemble energy with respect to a specific weight $w_I$ corresponds to the excitation energy of the system defined between the ground state and the $I$th excited state associated to this specific weight [Eq.~(4)]. 
	It is important that the reader keeps in mind that the exact excitation energies are based on pure-state energies and, therefore, do not depend on the weights of the ensemble.
	In practice, the ensemble energy is rarely linear in $w$ because of the approximate nature of the xc functionals. 
	Indeed, by inserting the ensemble density in the Hartree interaction functional [Eq.~(9)], one introduces, in the ensemble energy, spurious quadratic curvature with respect to the weight. 
	Some of those terms are responsible of the unphysical phenomenon called ghost interaction, as explained in the manuscript. 
	Therefore, the ensemble Khon-Sham gap is, somehow, "weight contaminated" and does not possess the correct weight dependence [see first two terms of the right-hand side of Eq.~(16)].
	By taking its derivative with respect to the weight, the weight-dependent xc functional is expected to compensate those spurious quadratic terms in order to retrieve a linear behavior of the exact ensemble energy: only a weight-dependent xc functional can achieve such a feat.
	In other words, the xc functional does not have to be linear with respect to $w$. 
	We have clarified this point in the revised manuscript. 
	We emphasize that the exact ensemble xc functional has the ideal weight dependency, and would make the corresponding ensemble energy perfectly linear, hence leading to weight-independent excitation energies.
	As shown in the present manuscript, the use of an approximate weight-dependent xc functional reduces the ensemble energy curvature as well as the variation of the excitation energies with respect to $w$.
	This illustrates why the construction of reliable weight-dependent xc functionals is a challenging task in eDFT.
	As a final comment, we insist after Eq.~(28) on the fact that, in the exact theory, the xc ensemble density functional has no reason to vary (for a fixed density $n$) linearly with the ensemble weights. 
	We refer to Ref.~92 where exact expressions for the individual xc energies are derived.}	
\end{enumerate}

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