res H2 1st draft

Manuscript/FarDFT.tex

@@ -387,9 +387,10 @@ and
 \end{subequations}
 makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}).
 As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$.
-Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider $1/2 < \ew{} \le 1$.
-We ensure that the weight-dependent functional does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $1$.
-It is interesting to note that, around $\ew{} = 0$, the behavior of Eq.~\eqref{eq:Cxw} is linear.
+Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is strictly forbidden by the GOK variational principle. \cite{Gross_1988a} 
+However, it is important to ensure that the weight-dependent functional does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $\ew{} = 1$.
+Therefore, by construction, the weight-dependent correction vanishes for these two limiting weight values (see Fig.~\ref{fig:Cx_H2}).
+Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear.
 \begin{figure}
 	\includegraphics[width=0.8\linewidth]{Cx_H2}
 	\caption{