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0.4721 1.0633 1.9492 3.1633 4.7359


0.5606 1.1764 2.0855 3.3217 4.9156


0.7990 1.4749 2.4547 3.7572 5.4146


0.0885 0.1131 0.1363 0.1583 0.1797


0.3269 0.4116 0.5055 0.5938 0.6787


0.0257 0.0386 0.0506 0.0612 0.0702


0.0008 0.0008 0.0033 0.0078 0.0131


0.0209 0.0149 0.0196 0.0271 0.0360


0.0266 0.0394 0.0540 0.0690 0.0834


0.0467 0.0534 0.0703 0.0884 0.1063


0.0000 0.0000 0.0000 0.0000 0.0000


0.0072 0.0080 0.0080 0.0075 0.0065


0.0050 0.0028 0.0005 0.0046 0.0095


0.0022 0.0015 0.0049 0.0067 0.0079


0.0004 0.0044 0.0095 0.0160 0.0209


0.0300 0.0300 0.0325 0.0355 0.0389


0.0026 0.0029 0.0047 0.0093 0.0130


0.0323 0.0315 0.0373 0.0423 0.0469


0.0045 0.0040 0.0029 0.0013 0.0007


0.0032 0.0044 0.0056 0.0066 0.0076


0.0013 0.0004 0.0027 0.0053 0.0083

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Manuscript/EvsN_DD.pdf
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Manuscript/EvsN_DD.pdf
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@ 1307,15 +1307,15 @@ electrons.


\begin{figure}


\includegraphics[width=\linewidth]{EvsL_3_DD}


\caption{


\label{fig:EvsLHF}


Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 3boxium at the KSeLDA level.


\label{fig:EvsL_DD}


Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 3boxium at the KSeLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.


Zeroweight (\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:DDeLDA}] on both the single and double excitations.


To do so, we have reported in Fig.~\ref{fig:EvsLHF}, in the case of 3boxium, 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 3boxium, the error percentage (with respect to FCI) as a function of the box length $L$


on the excitation energies obtained at the KSeLDA with and without $\DD{c}{I}$ [\ie, the last term in Eq.~\eqref{eq:OmeLDA}].


%\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 KSeLDA (solid lines) and eHF (dashed lines) levels.


Zeroweight (\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 KSeLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$.


Zeroweight (\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 KSeLDA 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 KSeLDA 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



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