Manu: saving work in the discussion

Manuscript/eDFT.tex

@@ -1295,18 +1295,21 @@ energies, which is in agreement with
 the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the
 individual energies do not vary in the same way depending on the state
 considered and the value of the weights. 
-\titou{We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
-the ground and first excited-state increase with respect to the first-excited-state weight $\ew{1}$, thus showing that we
-``deteriorate'' these states by optimizing the orbitals. The reverse actually occurs for the ground state in the triensemble
-as $\ew{2}$ increases.} The variations in the ensemble
+We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
+the ground and first excited-state increase with respect to the
+first-excited-state weight $\ew{1}$, thus showing that, \manu{in this
+case}, we
+``deteriorate'' these states by optimizing the orbitals \manu{for the
+ensemble, rather than for each state individually}. The reverse actually occurs for the ground state in the triensemble
+as $\ew{2}$ increases. The variations in the ensemble
 weights are essentially linear or quadratic. They are induced by the
 eLDA functional, as readily seen from
 Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
 \eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
-\titou{excited-state energy} is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
+excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
 systematically enhances the weight dependence, due to the lowering of the
-ground-state energy in this case, as $\ew{2}$ increases.         
-The reverse is observed for the second \titou{excited state}.
+ground-state energy, as $\ew{2}$ increases.         
+The reverse is observed for the second excited state.
 
 %%% FIG 3 %%%
 \begin{figure}
@@ -1325,7 +1328,7 @@ For small $L$, the single and double excitations can be labeled as
 ``pure'', as revealed by a thorough analysis of the FCI wavefunctions.
 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 \titou{disputable}. \cite{Loos_2019}
+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
@@ -1353,8 +1356,8 @@ When the box gets larger, they start to deviate.
 For the single excitation, TDLDA is extremely accurate up to $L = 2\pi$, but yields more significant errors at larger $L$ by underestimating the excitation energies.
 TDA-TDLDA slightly corrects this trend thanks to error compensation. 
 Concerning the eLDA functional, our results clearly evidence that the equiweight [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
-This is especially true for the single excitation
-which is significantly improved by using equal weights, \titou{especially in the strong correlation regime}.
+This is especially true\manu{, in the strong correlation regime,} for the single excitation
+which is significantly improved by using equal weights.
 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.
@@ -1424,7 +1427,7 @@ It is also interesting to investigate the influence of the
 correlation ensemble derivative contribution $\DD{c}{(I)}$
 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}, \titou{for $\nEl = 3$, $5$, and $7$}, the error percentage (with respect to FCI) as a function of the box length $L$
+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 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
@@ -1436,21 +1439,37 @@ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}
 %\eeq
 %%rather than $E^{(I)}_{\rm HF}$
 %}
-\titou{We first stress that although for $\nEl=3$ both single and double excitation energies are 
+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
 taking into account  
-the correlation ensemble derivative, this is not systematically the case for larger number of electrons.}
-\trashPFL{This statement holds in both zero-weight and equal-weight limits.}
+the correlation ensemble derivative, this is not
+\trashEF{systematically} \manu{always} the case for larger numbers of electrons.
 The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime.
-\titou{For 3-boxium, in the zero-weight limit, its contribution is also significantly larger in the case of the single
-excitation; the reverse is observed in the equal-weight triensemble
+For 3-boxium, in the zero-weight limit, its contribution is
+\trashEF{also} significantly larger \manu{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
 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 the weight-dependent functional.
-Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of the eLDA functional is based on a two-electron finite uniform electron gas.
-Therefore, it might be more appropriate to model the derivative discontinuity in few-electron systems.}
+This non-systematic behavior in terms of the number of electrons might
+be a consequence of how we constructed \trashEF{the weight-dependent
+functional} \manu{eLDA}.
+Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of
+the eLDA functional is based on a \manu{\it two-electron} finite uniform electron gas.
+\manu{Incorporating an $N$-dependence in the functional through the
+curvature of the Fermi hole, in the spirit of Ref. \cite{Loos_2017a}, would be
+valuable in this respect. This is left for future work.}
+\trashEF{Therefore, it might be more appropriate to model the derivative
+discontinuity in few-electron systems.}\\
+\\
+\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).
@@ -1461,16 +1480,33 @@ 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
-%do not like this statement. As I wrote above, the ensemble derivative is
-%still substantial in the strongly correlated limit of the equi
-%triensemble for the double
-%excitation.  
-%}
-Note also that, in our case, the second term in
+}\\
+\manu{Manu: I propose to rephrase this part as follows:}\\
+\\
+\manu{
+Interestingly, for the single excitation in the 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,
+even in the strongly correlated regime. A possible interpretation is
+that, at least for the single excitation, equiensemble orbitals partially remove the burden
+of modelling properly the correlation ensemble derivative. 
+This conclusion does not hold for larger
+$N=5$ or $N=7$ numbers of
+electrons, possibly because eLDA extracts density-functional correlation ensemble
+derivatives from a two-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 
+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
 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 \titou{(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?}
@@ -1494,8 +1530,8 @@ has a negligible effect on the excitation energies \titou{(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 \titou{a rather significant impact} on the double
-excitations \titou{(around $10\%$)} with a slight tendency of worsening the excitation energies
+ensemble correlation derivative has a rather significant impact on the double
+excitations (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
 excitation energies obtained in the zero-weight limit, showing once