Done with 1st draft

Manuscript/FarDFT.tex

@@ -491,7 +491,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
 	\end{ruledtabular}
 \end{table}
 
-%%% TABLE 1 %%%
+%%% TABLE II %%%
 \begin{table}
 	\caption{
 	\label{tab:OG_func}
@@ -541,18 +541,6 @@ showing that the weight correction is purely linear in eVWN5 and entirely depend
 
 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}).
 
-%This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
-%\begin{equation}
-%\label{eq:GACE}
-%	\E{\xc}{\bw}[\n{}{}] 
-%	= \E{\xc}{}[\n{}{}] 
-%	+ \sum_{I=1}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi,
-%\end{equation}
-%(where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014} 
-%Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional. 
-%In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir 
-%$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?}
-
 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, $\ew{} = 1/2$).
 For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016} 
@@ -573,7 +561,7 @@ The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less
 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 $1$.
 
-%%% TABLE I %%%
+%%% TABLE III %%%
 \begin{table}
 \caption{
 Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 1.4$ bohr for various methods, combinations of xc functionals, and basis sets.
@@ -654,7 +642,7 @@ 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.
 
-%%% TABLE I %%%
+%%% TABLE IV %%%
 \begin{table}
 \caption{
 Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} at $\RHH = 3.7$ bohr obtained with the aug-cc-pVTZ basis set for various methods and combinations of xc functionals.
@@ -709,7 +697,7 @@ As in the case of \ce{H2}, the excitation energies obtained at zero-weight are m
 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.
 
 
-%%% TABLE I %%%
+%%% TABLE V %%%
 \begin{table}
 \caption{
 Excitation energies (in hartree) associated with the lowest double excitation of \ce{He} obtained with the d-aug-cc-pVQZ basis set for various methods and combinations of xc functionals.
@@ -741,7 +729,6 @@ Excitation energies (in hartree) associated with the lowest double excitation of
 \fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.}
 \end{table}
 
-
 %%% TABLE I %%%
 %\begin{table}
 %\caption{