Manu: done with my revisions

Manuscript/FarDFT.tex

@@ -609,7 +609,7 @@ Second, in order to remove some of this spurious curvature of the ensemble
 energy (which is mostly due to the ghost-interaction error \cite{Gidopoulos_2002}, but not only \cite{Loos_2020}), 
 one can easily reverse-engineer (for this particular system, geometry, basis set, and excitation) a local exchange functional to make $\E{}{(0,\ew{2})}$ as linear as possible for $0 \le \ew{2} \le 1$ assuming a perfect linearity between the pure-state limits $ \ew{1} = \ew{2} = 0$ (ground state) and $\ew{1} = 0 \land \ew{2} = 1$ (doubly-excited state).
 Doing so, we have found that the following weight-dependent exchange functional (denoted as CC-S for ``curvature-corrected'' Slater functional)
-\begin{equation}
+\begin{equation}\label{eq:ensemble_Slater_func}
 	\e{\ex}{\ew{},\text{CC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
 \end{equation}
 with 
@@ -690,9 +690,9 @@ We shall come back to this point later on.
 \subsubsection{Weight-independent correlation functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-Third, we add up correlation effects via the VWN5 local correlation functional. \cite{Vosko_1980}
+Third, we add up correlation effects via the \manu{conventional} VWN5 local correlation functional. \cite{Vosko_1980}
 For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
-The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (green curve in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies (red curve in Figs.~\ref{fig:Ew_H2} and \ref{fig:Om_H2}), especially at small weights, where the CC-SVWN5 excitation energy is almost spot on. 
+The combination of the \manu{(weight-independent)} Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (green curve in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies (red curve in Figs.~\ref{fig:Ew_H2} and \ref{fig:Om_H2}), especially at small weights, where the CC-SVWN5 excitation energy is almost spot on. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsubsection{Weight-dependent correlation functional}
@@ -729,9 +729,10 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 Combining these, we build a three-state weight-dependent correlation functional:
 \begin{equation}
 \label{eq:ecw}
-	\e{\co}{\bw}(\n{}{}) =  (1-\ew{1}-\ew{2}) \e{\co}{(0)}(\n{}{}) + \ew{1} \e{\co}{(1)}(\n{}{}) + \ew{2} \e{\co}{(2)}(\n{}{}).
+	\e{\co}{\bw}(\n{}{}) =  (1-\ew{1}-\ew{2}) \e{\co}{(0)}(\n{}{}) + \ew{1} \e{\co}{(1)}(\n{}{}) + \ew{2} \e{\co}{(2)}(\n{}{}),
 \end{equation}
-
+\manu{where, unlike in the exact theory~\cite{Fromager_2020}, the individual components are
+weight-{\it independent}.}
 %%% FIG 4 %%%
 \begin{figure}
 	\includegraphics[width=0.8\linewidth]{fig1}
@@ -792,7 +793,10 @@ Combining these, we build a three-state weight-dependent correlation functional:
 
 Because our intent is to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons), we employ a simple ``embedding'' scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
 As explained further in Ref.~\onlinecite{Loos_2020}, this embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles originally derived by Franck and Fromager. \cite{Franck_2014} 
-The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA correlation functional).
+The weight-dependence of the correlation functional is then carried
+exclusively by the impurity [\ie, the functional defined in
+\eqref{eq:ecw}], while the remaining effects are produced by the bath
+(\ie, the usual \manu{ground-state} LDA correlation functional).
  
 Consistently with such a strategy, Eq.~\eqref{eq:ecw} is ``centred'' on its corresponding weight-independent VWN5 LDA reference
 \begin{equation}
@@ -825,19 +829,33 @@ We note also that, by construction, we have
 \end{equation}
 showing that the weight correction is purely linear in eVWN5 and entirely dependent on the FUEG model.
 
-As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 is very slightly less concave than its CC-SVWN5 counterpart and it also improves (not by much) the excitation energy (see purple curve in Fig.~\ref{fig:Om_H2}).
+As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 is very slightly less
+concave than its CC-SVWN5 counterpart and it also improves (not by much)
+the excitation energy (see purple curve in Fig.~\ref{fig:Om_H2}).\\
+\manuf{Again, which value of $\ew{1}$ has been used for generating the
+results in this Figures (see my previous comments)? $\ew{1}=0$? If so, we should not claim that we follow GOK
+theory because, for H$_2$ at equilibrium, the single excitation is
+missing in the ensemble.}
 For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
 In particular, we report the excitation energies obtained with GOK-DFT
 in the zero-weight limit (\ie, $\ew{} = 0$) and for equi-weights (\ie, $\ew{} = 1/3$).
-These excitation energies are computed using Eq.~\eqref{eq:dEdw}.
+These excitation energies are computed using
+Eq.~\eqref{eq:dEdw}.\manuf{OK but, again, how do you compute the exchange ensemble
+derivative for both excited states when it seems like the functional in
+Eqs.~(\ref{eq:ensemble_Slater_func}) and (\ref{eq:Cxw})
+only depends on $\ew{}$ rather than $\ew{1}$ AND $\ew{2}$.}
 
-For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016} 
-a pragmatic way of getting weight-independent excitation energies, defined as
+For comparison \trashEF{purposes}, we also report \manu{results obtain
+with }the linear interpolation method (LIM) \trashEF{excitation
+energies.} \cite{Senjean_2015,Senjean_2016} \manu{The latter simply
+consists in extracting the excitation energies (which are
+weight-independent, by construction) from the equiensemble energies, as
+follows:} 
 \begin{subequations}
 \begin{align}
 	\Ex{\LIM}{(1)} & = 2 \qty[\E{}{\bw{}=(1/2,0)} - \E{}{\bw{}=(0,0)}], \label{eq:LIM1}
 	\\
-	\Ex{\LIM}{(2)} & = 3 \qty[\E{}{\bw{}=(1/3,1/3)} - \E{}{\bw{}=(1/2,0)}] + \frac{1}{2} \Ex{\LIM}{(1)}, \label{eq:LIM2}
+	\Ex{\LIM}{(2)} & = 3 \qty[\E{}{\bw{}=(1/3,1/3)} - \E{}{\bw{}=(1/2,0)}] + \frac{1}{2} \Ex{\LIM}{(1)}. \label{eq:LIM2}
 \end{align}
 \end{subequations}
 %\manu{
@@ -850,15 +868,16 @@ a pragmatic way of getting weight-independent excitation energies, defined as
 %$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
 %\E{}{\bw{}=(1/2,0)}]+\frac{1}{2} \Ex{\LIM}{(1)}=E_2-E_0$
 %}\\
-which require three independent calculations, as well as the MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
+\manu{As readily seen, it requires three successive calculations.} MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
 \begin{subequations}
 \begin{align}
 	\Ex{\MOM}{(1)} & = \E{}{\bw{}=(1,0)} - \E{}{\bw{}=(0,0)}, \label{eq:MOM1}
 	\\
-	\Ex{\MOM}{(2)} & = \E{}{\bw{}=(0,1)} - \E{}{\bw{}=(0,0)}. \label{eq:MOM2}
+	\Ex{\MOM}{(2)} & = \E{}{\bw{}=(0,1)} - \E{}{\bw{}=(0,0)}, \label{eq:MOM2}
 \end{align}
 \end{subequations}
-which also require three separate calculations at a different set of ensemble weights.
+which also require three separate calculations at a different set of
+ensemble weights, have been computed, for comparison.
 For a general expression with multiple (and possibly degenerate) states, we refer the reader to Eq.~(106) of Ref.~\onlinecite{Senjean_2015}, where LIM is shown to interpolate linearly the ensemble energy between equi-ensembles.
 Note that two calculations are needed to get the first LIM excitation energy, but only one is required for each higher excitation. 
 
@@ -884,6 +903,11 @@ The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitati
 The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI.
 It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes in this limit (\textit{vide supra}).
 Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between the ghost-interaction-free pure-state limits.
+\manuf{There is a quite detailed discussion about LIM but nothing about
+its performance. One should say something. LIM is a ``poor man (or
+woman)'' approach when weight-dependent functionals are not available. I
+would at least compare regular LDA LIM with results obtained (by
+differentiations) with the more advanced CC-S+eVWN5 approach.}
 
 %%% TABLE III %%%
 \begin{table}
@@ -957,7 +981,11 @@ Although we could safely restrict ourselves to a biensemble composed by the grou
 Nonetheless, one should just be careful when reading the equations reported above, as they correspond to the case where the singly-excited state is lower in energy than the doubly-excited state.
 We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a CC-S functional for this system at $\RHH = 3.7$ bohr.
 It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}].
-The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve). 
+The weight dependence of $\Cx{\ew{}}$ is illustrated in
+Fig.~\ref{fig:Cxw} (green curve).\manuf{Again, it would be nice to say
+explicitly if you construct a functional, function of $\ew{1}$ and
+$\ew{2}$ (how then) or just $\ew{}$ (how to compute the separate
+derivatives then?)} 
 
 One clearly sees that the correction brought by CC-S is much more gentle than at $\RHH = 1.4$ bohr, which means that the ensemble energy obtained with the LDA exchange functional is much more linear at $\RHH = 3.7$ bohr.
 %In other words, the curvature ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
@@ -1043,7 +1071,9 @@ The parameters of the CC-S weight-dependent exchange functional (computed with t
 The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
 
 The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy at $\ew{} = 0$ is only $18$ millihartree off the reference value.
-As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}. 
+As in the case of \ce{H2}, the excitation energies obtained at
+zero-weight are more accurate than at equi-weight, while the opposite
+conclusion was made in Ref.~\onlinecite{Loos_2020}. 
 This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy.
 As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the CC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.
 
@@ -1092,7 +1122,10 @@ Although the weight-dependent correlation functional developed in this paper (eV
 To better understand the reasons behind this, it would be particularly interesting to investigate the influence of the self-consistent procedure, 
 \ie, the variation in excitation energy when the \textit{exact} ensemble density (built with the exact individual densities) is used instead
 of the self-consistent one. 
-Density- and state-driven errors \cite{Gould_2019,Gould_2019_insights,Fromager_2020} can also be calculated to provide additional insights about the present results. 
+\manu{Exploring the impact of both density- and state-driven
+correlations}
+\cite{Gould_2019,Gould_2019_insights,Fromager_2020} \trashEF{can also be
+calculated} \manu{may provide} additional insights about the present results. 
 This is left for future work.
 
 In the light of the results obtained in this study on double excitations computed within the GOK-DFT framework, we believe that the development of more universal weight-dependent exchange and correlation functionals has a bright future, and we hope to be able to report further on this in the near future.