clean up by T2
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@ -705,7 +705,7 @@ However, these results also clearly evidence that special care has to be taken f
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\subsection{Doubly-Excited States of the Carbon Dimer}
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\label{sec:C2}
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%=======================
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In order to have a miscellaneous test set of excitations, in a fourth time, we propose to study some doubly-excited states of the carbon dimer \ce{C2}, a prototype system for strongly correlated and multireference systems. \cite{AbrShe-JCP-04, AbrShe-CPL-05, Var-JCP-08, PurZhaKra-JCP-09, AngCimPas-MP-12, BooCleThoAla-JCP-11, Sha-JCP-15, SokCha-JCP-16, HolUmrSha-JCP-17, VarRoc-PTRSMPES-18}
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In order to have a miscellaneous test set of excitations, in a third time, we propose to study some doubly-excited states of the carbon dimer \ce{C2}, a prototype system for strongly correlated and multireference systems. \cite{AbrShe-JCP-04, AbrShe-CPL-05, Var-JCP-08, PurZhaKra-JCP-09, AngCimPas-MP-12, BooCleThoAla-JCP-11, Sha-JCP-15, SokCha-JCP-16, HolUmrSha-JCP-17, VarRoc-PTRSMPES-18}
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These two valence excitations --- $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ and $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ --- are both of $(\pi,\pi) \ra (\si,\si)$ character.
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They have been recently studied with state-of-the-art methods, and have been shown to be ``pure'' doubly-excited states as they do not involve single excitations. \cite{LooBogSceCafJac-JCTC-19}
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The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}.
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@ -723,48 +723,29 @@ In other words, the UEG on-top density used in the LDA and PBE functionals (see
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\end{figure}
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%%% %%% %%%
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It is interesting to study the behavior of \manu{the key quantities involved in the basis set correction} for different states as the basis set incompleteness error is obviously state specific.
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It is interesting to study the behavior of the key quantities involved in the basis set correction for different states as the basis set incompleteness error is obviously state specific.
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%\manu{To do so, we report the value of the range separation parameter in real space $\rsmu{}{\Bas}(\br{})$, the value of the energetic correction $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{}) $ and the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ computed with different basis sets for the ground state and second excited state of the carbon dimer which are both of $\Sigma_g^+$ symmetry, in Figures . }
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We report $\rsmu{}{\Bas}(z)$, along the nuclear axis ($z$) for these two electronic states computed with the AVDZ, AVTZ and AVQZ basis sets.
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We report $\rsmu{}{\Bas}(z)$, along the nuclear axis ($z$) for the two $^1 \Sigma_g^+$ electronic states of \ce{C2} computed with the AVDZ, AVTZ and AVQZ basis sets.
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\manu{These figures illustrate several important things:
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i) the maximal values of $\rsmu{}{\Bas}(\br{})$ are systematically close to the nuclei, a signature of the atom-centered basis set,
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ii) the overall values of $\rsmu{}{\Bas}(\br{})$ increase with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set,
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iii) the value of $\rsmu{}{\Bas}(\br{})$ are slightly larger near the oxygen atom, which traduces the fact that the inter-electronic distance is higher than close to the carbon atom due to a higher nuclear charge. }
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%%% FIG 4 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{C2_mu}
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\caption{ C$_2$: $\rsmu{}{\Bas}(z)$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
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\begin{figure*}
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\includegraphics[height=0.45\linewidth]{C2_mu}
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\hspace{0.5cm}
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\includegraphics[height=0.45\linewidth]{C2_PBEot}
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\includegraphics[height=0.45\linewidth]{C2_PBE.pdf}
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\hspace{0.5cm}
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\includegraphics[height=0.45\linewidth]{C2_n2.pdf}
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\caption{$\rsmu{}{\Bas}$ (top left), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBEot}$ (top right), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBE}$ (bottom left) and $\n{2}{\Bas}$ (bottom right) along the molecular axis ($z$) for the ground state (black curve) and second doubly-excited state (red curve) of \ce{C2} for various basis sets $\Bas$.
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The two electronic states are both of $\Sigma_g^+$ symmetry.
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The carbon nuclei are located at $z= \pm 1.180$ bohr and represented by the thin black lines.}
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\label{fig:C2_mu}
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\end{figure}
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\end{figure*}
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%%% %%% %%%
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%%% FIG 4 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{C2_PBEot.pdf}
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\caption{ C$_2$: $\pbeotint$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
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\label{fig:C2_PBEot}
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\end{figure}
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%%% %%% %%%
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%%% FIG 4 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{C2_PBE.pdf}
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\caption{ C$_2$: $\pbeint$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
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\label{fig:C2_PBE}
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\end{figure}
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%%% %%% %%%
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%%% FIG 4 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{C2_n2.pdf}
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\caption{ C$_2$: $\n{2}{\Bas}(\br{})$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
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\label{fig:C2_n2}
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\end{figure}
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%%% %%% %%%
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%=======================
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\subsection{Ethylene}
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\label{sec:C2H4}
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@ -200,13 +200,13 @@ C 0.000000 0.000000 -0.624021
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\end{verbatim}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Carbon monoxyde}
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%\subsection{Carbon monoxyde}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{verbatim}
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C 0.000000 0.000000 -1.249421
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0 0.000000 0.000000 0.892667
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\end{verbatim}
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%\begin{verbatim}
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%C 0.000000 0.000000 -1.249421
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%0 0.000000 0.000000 0.892667
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%\end{verbatim}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Ethylene}
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@ -449,17 +449,17 @@ Here, we report the absolute energetic corrections for each state of each molecu
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& -0.049\,208 & -0.021\,292 & -0.01\,0257
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\\
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\\
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Carbon monoxyde & $1\,^{1}\Sigma^+$
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& -0.074\,328 & -0.031\,117 & -0.015\,510
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& -0.084\,655 & -0.035\,318 & -0.017\,142
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& -0.076\,668 & -0.077\,437 & -0.018\,768
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\\
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& $1\,^{1}\Pi$
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& -0.075\,790 & -0.031\,456 & -0.016\,083
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& -0.085\,494 & -0.035\,255 & -0.017\,182
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& -0.036\,301 & -0.036\,359 & -0.018\,855
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\\
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\\
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% Carbon monoxyde & $1\,^{1}\Sigma^+$
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% & -0.074\,328 & -0.031\,117 & -0.015\,510
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% & -0.084\,655 & -0.035\,318 & -0.017\,142
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% & -0.076\,668 & -0.077\,437 & -0.018\,768
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% \\
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% & $1\,^{1}\Pi$
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% & -0.075\,790 & -0.031\,456 & -0.016\,083
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% & -0.085\,494 & -0.035\,255 & -0.017\,182
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% & -0.036\,301 & -0.036\,359 & -0.018\,855
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% \\
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% \\
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Water & $1\,^{1}A_1$
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& -0.058\,765 & -0.024\,014 & -0.011\,990
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& -0.066\,603 & -0.027\,236 & -0.013\,127
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