Added corrections and comments.

Manuscript/FarDFT.tex

@@ -524,7 +524,7 @@ via the following global, state-independent shift:
 	\be{\co}{(I)}(\n{}{}) = \e{\co}{(I)}(\n{}{}) + \e{\co}{\VWN}(\n{}{}) - \e{\co}{(0)}(\n{}{}).
 \end{equation}
 In the following, we name this weight-dependent correlation functional ``eVWN5'' as it is a natural extension of the VWN5 local correlation functional for ensembles.
-Also, Eq.~\eqref{eq:becw} can be recast
+Also, Eq.~\eqref{eq:becw} can be recast as
 \begin{equation}
 \label{eq:eLDA}
 	\e{\co}{\ew{},\eVWN}(\n{}{}) =  \e{\co}{\VWN}(\n{}{}) + \ew{} \qty[\e{\co}{(1)}(\n{}{}) - \e{\co}{(0)}(\n{}{})]
@@ -539,7 +539,7 @@ 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 SGIC-eVWN5 is slightly less concave than its SGIC-VWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).
+As shown in Fig.~\ref{fig:Ew_H2}, the GIC-SeVWN5 is slightly less concave than its GIC-SVWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).
 
 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 the equi-ensemble (\ie, $\ew{} = 1/2$).
@@ -559,7 +559,12 @@ They can then be obtained via GOK-DFT ensemble calculations by performing a line
 The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the GIC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to GIC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
 The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less satisfactory, but still remains in good agreement with FCI, with again a small improvement as compared to GIC-SVWN5.
 It is also important to mention that the GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
-Finally, note that, 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 $\ew{} = 0$ and $1$.
+Note that by construction,
+for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper),
+LIM and MOM can be reduced to a single calculation
+at $w = 1/4$ and $w=1/2$, respectively, instead of performing an interpolation between two different calculations.
+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 $\ew{} = 0$ and $1$.
+
 
 %%% TABLE III %%%
 \begin{table}
@@ -634,13 +639,16 @@ The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (gree
 
 One clearly sees that the correction brought by GIC-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 ghost-interaction ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
+Note that this linearity at $\RHH = 3.7$ bohr was
+also observed using weight-independent xc-functionals in Ref.~\cite{Senjean_2015}.
 Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers basis set converged results), the same set of calculations as in Table \ref{tab:BigTab_H2}.
 As a reference value, we computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015}
-For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the best match being reached with HF exchange. 
-The GIC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits.
+For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the best match being reached with HF exchange \bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}. 
+As expected from the linearity of the ensemble energy, the GIC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits.
 Nonetheless, the excitation energy is still off by 3 eV.
 The fundamental theoretical reason of such a poor agreement is not clear.
 The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error.
+\bruno{I'm a bit surprise that the ensemble correction to the correlation functional does not improve things at all... Is the derivative discontinuity, computed with this functional, almost 0 here ?}
 
 %%% TABLE IV %%%
 \begin{table}
@@ -692,8 +700,11 @@ The parameters of the GIC-S weight-dependent exchange functional (computed with
 In other words, the ghost-interaction hole is deeper.
 The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
 
-The GIC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the GIC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) 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.
+The GIC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the GIC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value.\bruno{But also with GIC-SVWN5, as in the rest of this article, so one could wonder about the usefulness of the eVWN5 functional...}
+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.~\cite{Loos_2020} thus strengthening the importance of
+developing weight-dependent functionals that gives linear ensemble
+energies, \ie, 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 GIC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.