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Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and equiweight (\ie, $\ew{1} = \ew{2} = 1/3$, blue lines) calculations are reported. } \end{figure} %%% %%% %%% It is also interesting to investigate the influence of the ensemble correlation derivative $\DD{c}{I}$ [defined in Eq.~\eqref{eq:DD-eLDA}] on both the single and double excitations. -To do so, we have reported in Fig.~\ref{fig:EvsLHF}, in the case of 3-boxium, the error percentage (with respect to FCI) as a function of the box length $L$ +To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, in the case of 3-boxium, the error percentage (with respect to FCI) as a function of the box length $L$ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{I}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}]. %\manu{Manu: there is something I do not understand. If you want to %evaluate the importance of the ensemble correlation derivatives you @@ -1344,18 +1344,17 @@ of modeling properly the ensemble correlation derivative. %%% FIG 6 %%% \begin{figure} - \includegraphics[width=\linewidth]{EvsN_HF} + \includegraphics[width=\linewidth]{EvsN_DD} \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 -equiweight (\ie, $\ew{1} = \ew{2} = 1/3$, blue and green lines) calculations are reported. + \label{fig:EvsN_DD} + 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 level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$. + Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and equiweight (\ie, $\ew{1} = \ew{2} = 1/3$, blue 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 +Finally, in Fig.~\ref{fig:EvsN_DD}, 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 solid and dashed lines 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