Manu: some polishing in V.

Manuscript/eDFT.tex

@@ -1277,16 +1277,25 @@ length $L$ in the case of 5-boxium.\\
 evaluate the importance of the ensemble correlation derivatives you
 should only remove the following contribution from the $K$th KS-eLDA
 excitation energy:  
-\beq
+\beq\label{eq:DD_term_to_compute}
 \int  \n{\bGam{\bw}}{}(\br{})
       \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
 \eeq
 %rather than $E^{(I)}_{\rm HF}$
 }
-The influence of the derivative discontinuity is clearly more important in the strong correlation regime.
-Its contribution is also significantly larger in the case of the single excitation; the derivative discontinuity hardly influences the double excitation.
-Importantly, one realizes that the magnitude of the derivative discontinuity is much smaller in the case of state-averaged calculations (as compared to the zero-weight calculations).
-This could explain why equiensemble calculations are clearly more accurate as it reduces the influence of the derivative discontinuity: for a given method, state-averaged orbitals partially remove the burden of modeling properly the derivative discontinuity.
+The influence of the ensemble correlation derivative is clearly more important in the strong correlation regime.
+Its contribution is also significantly larger in the case of the single
+excitation; the ensemble correlation derivative hardly influences the double excitation.
+Importantly, one realizes that the magnitude of the ensemble correlation
+derivative is much smaller in the case of equal-weight calculations (as compared to the zero-weight calculations).
+This could explain why equiensemble calculations are clearly more
+accurate as it reduces the influence of the ensemble correlation derivative:
+for a given method, equiensemble orbitals partially remove the burden
+of modeling properly the ensemble correlation derivative.\manu{Manu: well, we
+would need the exact derivative value to draw such a conclusion. We can
+only speculate. Let us first see how important the contribution in
+Eq.~\eqref{eq:DD_term_to_compute} is. What follows should also be
+updated in the light of the new results.}
 
 %%% FIG 5 %%%
 \begin{figure}
@@ -1294,13 +1303,20 @@ This could explain why equiensemble calculations are clearly more accurate as it
 	\caption{
 	\label{fig:EvsN_HF}
 	Error with respect to FCI in single and double excitation energies for $\nEl$-boxium (with a box length of $L=8\pi$) as a function of the number of electrons $\nEl$ at the KS-eLDA (solid lines) and eHF (dashed lines) levels. 
-	Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, black and red lines) and state-averaged (\ie, $\ew{1} = \ew{2} = 1/3$, blue and green lines) calculations are reported. 
+	Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, black and red lines) and
+equal-weight (\ie, $\ew{1} = \ew{2} = 1/3$, blue and green lines) calculations are reported. 
 	}
 \end{figure}
 %%% %%% %%%
 
 Finally, in Fig.~\ref{fig:EvsN_HF}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime). 
-The difference between the eHF and KS-eLDA excitation energies undoubtedly show that, even in the strong correlation regime, the derivative discontinuity has a small impact on the double excitations with a slight tendency of worsening the excitation energies in the case of state-averaged weights, and a rather large influence on the single excitation energies obtained in the zero-weight limit, showing once again that the usage of state-averaged weights has the benefit of significantly reducing the magnitude of the derivative discontinuity.
+The difference between the eHF and KS-eLDA excitation energies
+undoubtedly show that, even in the strong correlation regime, the
+ensemble correlation derivative has a small impact on the double
+excitations with a slight tendency of worsening the excitation energies
+in the case of equal weights, and a rather large influence on the single
+excitation energies obtained in the zero-weight limit, showing once
+again that the usage of equal weights has the benefit of significantly reducing the magnitude of the ensemble correlation derivative.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Concluding remarks}