Added more discussion on the results about LIM (or MOM).

Manuscript/FarDFT.tex

@@ -901,13 +901,16 @@ Eqs.~\eqref{eq:MOM1} and \eqref{eq:MOM2}.
 
 The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
 The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI.
+Interestingly, the CC-S functional
+leads to a substantial improvement of the LIM
+excitation energy, getting close to the reference value
+(with an error of up to 0.24 eV) when no correlation
+functional is used. When correlation functionals are
+added (\ie VWN5 or eVWN5), LIM tends to overestimate
+the excitation energy by about 1 eV but still performs
+better than when no correction of the curvature is considered.
 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}
@@ -1003,6 +1006,9 @@ For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the exci
 %work. I guess the latter option is what you did. We need to explain more
 %what we do!!!} 
 As expected from the linearity of the ensemble energy, the CC-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.
+As a direct consequence of this linearity, LIM and MOM
+do not provide any noticeable improvement on the excitation
+energy.
 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. 
@@ -1074,7 +1080,13 @@ The CC-S exchange functional attenuates significantly this dependence, and when
 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.
+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. Here again, the LIM excitation energy
+when the CC-S functional is used is very accurate with
+only 22 millihartree error compared to the reference value,
+while adding the correlation contribution to the functional
+tends to overestimate the excitation energy.
+Hence, in the light of the results obtained in this paper, it seems that the weight-dependent curvature correction to the exchange functional makes the biggest impact in providing
+accurate excitation energies.
 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.
 
 %%% TABLE V %%%