revsision under progress

Response_Letter/Response_Letter.tex

@@ -100,7 +100,10 @@ We look forward to hearing from you.
 	I wonder if an explanatory diagram for the embedding scheme might clarify my thinking here, particularly if my thinking is wrong! }
 	\\
 	\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.}
+	Performing a simple gedanken experiment, one can imagine that, in the infinite system, the excitation will occur locally, i.e., on the impurity.
+	Therefore, we can assume, as a first approximation, that the weight dependence will originate mainly from this impurity, most of the bath being unaffected by this local excitation. This is, roughly speaking, the philosophy that we have followed.
+	We believe this is also the reviewer's way of thinking.
+	We have added a figure to illustrate this in the revised version of the manuscript.}
 
 	\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. }
@@ -112,7 +115,7 @@ We look forward to hearing from you.
 	{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? 
 	Is this result influenced by the state mixing shown by FCI in Figure 3's discussion? }
 	\\
-	\alert{bla bla bla}
+	\alert{According to our observation, yes.}
 
 	\item 
 	{Page 7: In the penultimate paragraph on this page, the discussion of Eqns 47 and 49 and the variation in the ensemble weights touches on one of the more subtle results of the GIC-eLDA, in my opinion, so it would be best to more explicitly describe this and its tie to the aforementioned equations.}
@@ -133,19 +136,21 @@ We look forward to hearing from you.
 	\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. }
 	\\
-	\alert{bla bla bla}
+	\alert{Nothing to mention here.}
 
 	\item 
 	{Are there similar issues with combining HF exchange with LDA C as seen in the ground-state? 
 	If not, why not? }
 	\\
-	\alert{bla bla bla}
+	\alert{Yes, similar issues appear for excited states.}
 
 	\item 
 	{Figure 3 discussion: Will eLDA always overestimate double excitations? 
 	It's hard to see differences for small $L$. }
 	\\
-	\alert{The corresponding numerical data are reported in Supporting information.}
+	\alert{Except for the two-electron system where we observed cases of underestimation, eLDA usually overestimates double excitations.
+	The corresponding numerical data are reported in Supporting information.
+	We have mentioned this interesting observation in the revised manuscript.}
 
 	\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. 

Revised_Manuscript/eDFT.tex

@@ -829,16 +829,27 @@ Equation \eqref{eq:ec} provides three state-specific correlation density-functio
 Combining these, one can build the following three-state weight-dependent correlation density-functional approximation:
 \begin{equation}
 \label{eq:ecw}
-	%\e{c}{\bw}(\n{}{})
- \Tilde{\epsilon}_{\rm c}^\bw(n)=  (1-\ew{1}-\ew{2}) \e{c}{(0)}(\n{}{}) + \ew{1} \e{c}{(1)}(\n{}{}) + \ew{2} \e{c}{(2)}(\n{}{}).
+ \Tilde{\epsilon}_{\rm 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}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%% FIG 1 %%%
+\begin{figure}
+	\includegraphics[width=0.7\linewidth]{embedding}
+	\caption{
+	\label{fig:embedding}
+	\titou{Schematic view of the ``embedding'' scheme: the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath).
+	The electronic excitation occurs locally, \ie, on the impurity.}
+	}
+\end{figure}
+%%% %%% %%%
+
 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 \titou{``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 \titou{(illustrated in Fig.~\ref{fig:embedding})} 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}
@@ -1055,6 +1066,7 @@ 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}).
+\titou{Except for the two-electron system where we observe cases of underestimation, eLDA usually overestimates double excitations, as evidenced by the numerical data gathered in the {\SI}).}
 
 %%% FIG 4 %%%
 \begin{figure*}