Manu: done with the discussion

Manuscript/eDFT.tex

@@ -1349,7 +1349,8 @@ For small $L$, the single and double excitations can be labeled as
 In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
 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.
+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
@@ -1382,7 +1383,7 @@ 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}).
+(see {\SI}).\\
 %\\
 %\manu{Manu: now comes the question that is, I believe, central in this
 %work. How important are the
@@ -1465,11 +1466,11 @@ increases) when
 taking into account  
 the correlation ensemble derivative, this is not
 always the case for larger numbers of electrons.
-For 3-boxium, in the zero-weight limit, the ensemble derivative is
+For 3-boxium, in the zero-weight limit, the correlation ensemble derivative is
 significantly larger for the single
 excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
 case.
-However, for 5- and 7-boxium, the correlation ensemble derivative hardly
+However, for 5- and 7-boxium, it hardly
 influences the double excitation (except when the correlation is strong), and slightly deteriorates the single excitation in the intermediate and strong correlation regimes.
 This non-systematic behavior in terms of the number of electrons might
 be a consequence of how we constructed eLDA.
@@ -1499,7 +1500,6 @@ valuable in this respect. This is left for future work.
 %}\\
 %\manu{Manu: I propose to rephrase this part as follows:}\\
 %\\
-\titou{
 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,
@@ -1511,16 +1511,15 @@ numbers of electrons ($N=5$ or $7$), possibly because eLDA extracts density-func
 derivatives from a two-electron uniform electron gas, as mentioned previously.
 For the double excitation, the ensemble derivative remains important, even in
 the equiensemble case. 
-To summarize, in all cases, the equiensemble calculation 
+To summarize, the equiensemble calculation 
 is always more accurate than a zero-weight
 (\ie, a conventional ground-state DFT) one, with or without including the ensemble 
-derivative correction.    
-}
-\\
-Note that the second term in
+derivative correction. Note that the second term on the right-hand side
+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).
+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?}
@@ -1543,13 +1542,17 @@ has a negligible effect on the excitation energies (results not shown).
 
 Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime). 
 The difference between the solid and dashed curves
-undoubtedly show that, even in the strong correlation regime, the
-ensemble correlation derivative has a rather significant impact on the double
-excitations (around $10\%$) with a slight tendency of worsening the excitation energies
+undoubtedly show that \trashEF{, even in the strong correlation regime,}
+\manu{Manu: in the light of what we already discussed, we expect the
+derivative to be important in the strongly correlated regime so the
+sentence "even in ..." is useless (that's why I would remove it)} the
+correlation ensemble derivative has a rather significant impact on the double
+excitation (around $10\%$) with a slight tendency of worsening the excitation energies
 in the case of equal weights, as the number of electrons
-increases. It has a rather large influence on the single
+increases. It has a rather large influence \manu{(which decreases with the
+number of electrons)} on the single
 excitation energies obtained in the zero-weight limit, showing once
-again that the usage of equal weights has the benefit of significantly reducing the magnitude of the ensemble correlation derivative.
+again that the usage of equal weights has the benefit of significantly reducing the magnitude of the correlation ensemble derivative.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Concluding remarks}