results
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\kcal}{kcal.mol$^{-1}$}
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% methods
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% methods
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@ -659,6 +660,40 @@ Therefore, we propose the following valence-only approximations for the compleme
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\section{Results}
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\section{Results}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{table}
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\caption{
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Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_AE}.
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Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference data.
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\label{tab:stats}}
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\begin{ruledtabular}
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\begin{tabular}{lddd}
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Method & \tabc{MAD} & \tabc{RMSD} & \tabc{MAX} \\
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\hline
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CCSD(T)/cc-pVDZ & 22.81 & 25.82 & 72.08 \\
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CCSD(T)/cc-pVTZ & 7.95 & 8.99 & 25.99 \\
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CCSD(T)/cc-pVQZ & 3.24 & 3.67 & 11.66 \\
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CCSD(T)/cc-pV5Z & 1.39 & 1.54 & 3.46 \\
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\\
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CCSD(T)+LDA/cc-pVDZ & 11.75 & 13.99 & 54.88 \\
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CCSD(T)+LDA/cc-pVTZ & 3.11 & 3.94 & 16.77 \\
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CCSD(T)+LDA/cc-pVQZ & 0.87 & 1.36 & 6.22 \\
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\\
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CCSD(T)+PBE/cc-pVDZ & 8.68 & 10.92 & 45.81 \\
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CCSD(T)+PBE/cc-pVTZ & 2.66 & 3.52 & 14.73 \\
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CCSD(T)+PBE/cc-pVQZ & 0.88 & 1.36 & 5.96 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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\begin{figure*}
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\includegraphics[width=\linewidth]{VDZ}
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\includegraphics[width=\linewidth]{VTZ}
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\includegraphics[width=\linewidth]{VQZ}
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\caption{
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Deviation (in \kcal) from CCSD(T)/CBS reference atomization energies obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
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\label{fig:G2_AE}}
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\end{figure*}
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\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
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\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
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We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$ and F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ in the case of N$_2$ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). N$_2$, O$_2$ and F$_2$ belong to the G2 set and can be considered as weakly correlated, whereas C$_2$ contains already a non negligible non dynamic correlation component.
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We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$ and F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ in the case of N$_2$ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). N$_2$, O$_2$ and F$_2$ belong to the G2 set and can be considered as weakly correlated, whereas C$_2$ contains already a non negligible non dynamic correlation component.
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@ -678,36 +713,23 @@ Also, the values obtained with the largest basis sets tends to converge toward a
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\begin{table*}
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\begin{table*}
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\caption{
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\caption{
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\label{tab:diatomics}
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\label{tab:diatomics}
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Dissociation energy ($\De$) in kcal/mol of the \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} molecules computed with various methods and basis sets.
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Dissociation energy ($\De$) in {\kcal} of the \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} molecules computed with various methods and basis sets.
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The deviation with respect to the CBS values are reported in square brackets.
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}
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}
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\begin{ruledtabular}
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\begin{ruledtabular}
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\begin{tabular}{llddddd}
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\begin{tabular}{llddddd}
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& & \mc{4}{c}{Dunning's basis set}
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& & \mc{4}{c}{Dunning's basis set}
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\\
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\\
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\\
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\cline{3-6}
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\cline{3-6}
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Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$D_e^{Q5Z}$}
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Molecule & Method & \tabc{X=D} & \tabc{X=T} & \tabc{X=Q} & \tabc{X=5} & \tabc{CBS}
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\\
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\\
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\\
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%\hline
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%\ce{C2} & (FC)FCIQMC & 130.0(1) & 139.9(3) & 143.3(2) & & 144.9 \\
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% & (FC)FCIQMC+F12 & 142.3 & 145.3 & & & \\
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\hline
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\hline
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\ce{C2} & ex (FC)FCI & 132.0 & 140.3 & 143.6 & 144.7 & 145.7 \\
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\ce{C2} & exFCI\fnm[1] & 132.0 & 140.3 & 143.6 & 144.7 & 145.7 \\
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\hline
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(cc-pVXZ) & exFCI+LDA\fnm[1] & 141.3 & 145.1 & 146.4 & 146.3 & \\
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& ex (FC)FCI+LDA-val & 141.3 & 145.1 & 146.4 & 146.3 & \\
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& exFCI+PBE\fnm[1] & 145.7 & 145.7 & 146.3 & 146.2 & \\
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& ex (FC)FCI+PBE-val & 145.7 & 145.7 & 146.3 & 146.2 & \\
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& CCSD(T)\fnm[1] & 129.2 & 139.1 & 143.0 & 144.2 & 145.4 \\
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\hline
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& CCSD(T)+LDA\fnm[1] & 139.1 & 143.7 & 145.9 & 145.9 & \\
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\hline
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& CCSD(T)+PBE\fnm[1] & 142.8 & 144.2 & 145.9 & 145.8 & \\
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& (FC)CCSD(T) & 129.2 & 139.1 & 143.0 & 144.2 & 145.4 \\
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\hline
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& (FC)CCSD(T)+LDA-val & 139.1 & 143.7 & 145.9 & 145.9 & \\
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& (FC)CCSD(T)+PBE-val & 142.8 & 144.2 & 145.9 & 145.8 & \\
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\hline
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\\
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\cline{3-6}
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& & \tabc{cc-pCVDZ} & \tabc{cc-pCVTZ} & \tabc{cc-pCVQZ} & \tabc{cc-pCV5Z} & \tabc{ $D_e^{C(Q5)Z}$ }
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\\
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\\
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\\
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% \hline
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% \hline
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%%%%%%%% & ex (FC)FC-FCI & 130.5 & 140.5 & 143.8 & 144.9 & 147.1 \\
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%%%%%%%% & ex (FC)FC-FCI & 130.5 & 140.5 & 143.8 & 144.9 & 147.1 \\
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@ -718,45 +740,25 @@ Also, the values obtained with the largest basis sets tends to converge toward a
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%%%%%%%% & ex (FC)FCI+PBE & 144.5 & 145.9 & 146.4 & 146.3 & \\
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%%%%%%%% & ex (FC)FCI+PBE & 144.5 & 145.9 & 146.4 & 146.3 & \\
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%%%%%%%% & ex (FC)FCI+PBE -val & 145.2 & 145.9 & 146.4 & 146.3 & \\
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%%%%%%%% & ex (FC)FCI+PBE -val & 145.2 & 145.9 & 146.4 & 146.3 & \\
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\hline
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\ce{C2} & exFCI\fnm[2] & 131.0 & 141.5 & 145.1 & 146.1 & 147.1 \\
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& ex FCI & 131.0 & 141.5 & 145.1 & 146.1 & 147.1 \\
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(cc-pCVXZ) & exFCI+LDA\fnm[2] & 141.4 & 146.7 & 147.8 & 147.6 & \\
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\hline
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& ex FCI+LDA & 141.4 & 146.7 & 147.8 & 147.6 & \\
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%%%%%%%% & ex FCI+LDA-val & 141.8 & 146.6 & 147.7 & 147.6 & \\
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%%%%%%%% & ex FCI+LDA-val & 141.8 & 146.6 & 147.7 & 147.6 & \\
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\hline
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& exFCI+PBE\fnm[2] & 145.1 & 147.0 & 147.7 & 147.5 & \\
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& ex FCI+PBE & 145.1 & 147.0 & 147.7 & 147.5 & \\
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%%%%%%%% & ex FCI+PBE-val & 145.7 & 147.0 & 147.6 & 147.5 & \\
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%%%%%%%% & ex FCI+PBE-val & 145.7 & 147.0 & 147.6 & 147.5 & \\
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\\
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\\
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\\
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\ce{N2} & exFCI\fnm[1] & 201.1 & 217.1 & 223.5 & 225.7 & 227.8 \\
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& & \mc{4}{c}{Dunning's basis set}
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(cc-pVXZ) & exFCI+LDA\fnm[1] & 217.9 & 225.9 & 228.0 & 228.6 & \\
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\\
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& exFCI+PBE\fnm[1] & 227.7 & 227.8 & 228.3 & 228.5 & \\
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\cline{3-6}
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Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$D_e^{Q5Z}$}
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\\
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\\
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\ce{N2} & ex (FC)FCI & 201.1 & 217.1 & 223.5 & 225.7 & 227.8 \\
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\hline
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& ex (FC)FCI+LDA-val & 217.9 & 225.9 & 228.0 & 228.6 & \\
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& ex (FC)FCI+PBE-val & 227.7 & 227.8 & 228.3 & 228.5 & \\
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% & exFCI+PBE-on-top-val & 224.8 & 226.7 & 228.3 & 228.3 & \\
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% & exFCI+PBE-on-top-val & 224.8 & 226.7 & 228.3 & 228.3 & \\
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\hline
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& CCSD(T)\fnm[1] & 199.9 & 216.3 & 222.8 & 225.0 & 227.2 \\
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\hline
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& (FC)CCSD(T) & 199.9 & 216.3 & 222.8 & 225.0 & 227.2 \\
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\hline
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%%%%%%%%& ex (FC)CCSD(T)+LDA & 214.7 & 221.9 & ----- & ----- & \\
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%%%%%%%%& ex (FC)CCSD(T)+LDA & 214.7 & 221.9 & ----- & ----- & \\
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%%%%%%%%& ex (FC)CCSD(T)+PBE & 223.4 & 224.3 & ----- & ----- & \\
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%%%%%%%%& ex (FC)CCSD(T)+PBE & 223.4 & 224.3 & ----- & ----- & \\
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& (FC)CCSD(T)+LDA-val & 216.3 & 224.8 & 227.2 & 227.8 & \\
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& CCSD(T)+LDA\fnm[1] & 216.3 & 224.8 & 227.2 & 227.8 & \\
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& (FC)CCSD(T)+PBE-val & 225.9 & 226.7 & 227.5 & 227.8 & \\
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& CCSD(T)+PBE\fnm[1] & 225.9 & 226.7 & 227.5 & 227.8 & \\
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\hline
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\\
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%%%%%%%%& ex (FC)FCI+LDA & 216.4 & 223.1 & 227.9 & 228.1 & \\
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%%%%%%%%& ex (FC)FCI+LDA & 216.4 & 223.1 & 227.9 & 228.1 & \\
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%%%%%%%%& ex (FC)FCI+PBE & 225.4 & 225.6 & 228.2 & 227.9 & \\
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%%%%%%%%& ex (FC)FCI+PBE & 225.4 & 225.6 & 228.2 & 227.9 & \\
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%%%%%%%% & exFCI+PBE-on-top& 222.3 & 224.6 & 227.7 & 227.7 & \\
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%%%%%%%% & exFCI+PBE-on-top& 222.3 & 224.6 & 227.7 & 227.7 & \\
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\\
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\\
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\cline{3-6}
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& & \tabc{cc-pCVDZ} & \tabc{cc-pCVTZ} & \tabc{cc-pCVQZ} & \tabc{cc-pCV5Z} & \tabc{$D_e^{Q5Z}$}
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\\
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\\
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%%%%%%%% & ex (FC)FCI & 201.7 & 217.9 & 223.7 & 225.7 & 228.8 \\
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%%%%%%%% & ex (FC)FCI & 201.7 & 217.9 & 223.7 & 225.7 & 228.8 \\
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%%%%%%%%\hline
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%%%%%%%%\hline
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%%%%%%%% & ex (FC)FCI+LDA & 217.5 & 226.2 & 228.4 & 228.5 & \\
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%%%%%%%% & ex (FC)FCI+LDA & 217.5 & 226.2 & 228.4 & 228.5 & \\
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@ -764,70 +766,41 @@ Also, the values obtained with the largest basis sets tends to converge toward a
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%%%%%%%%\hline
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%%%%%%%%\hline
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%%%%%%%% & ex (FC)FCI+PBE & 225.8 & 227.6 & 228.4 & 228.3 & \\
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%%%%%%%% & ex (FC)FCI+PBE & 225.8 & 227.6 & 228.4 & 228.3 & \\
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%%%%%%%% & ex (FC)FCI+PBE-val & 227.5 & 227.7 & 228.4 & 228.0 & \\
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%%%%%%%% & ex (FC)FCI+PBE-val & 227.5 & 227.7 & 228.4 & 228.0 & \\
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\hline
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\ce{N2} & exFCI\fnm[2] & 202.2 & 218.5 & 224.4 & 226.6 & 228.8 \\
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& ex FCI & 202.2 & 218.5 & 224.4 & 226.6 & 228.8 \\
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(cc-pCVXZ) & exFCI+LDA\fnm[2] & 218.0 & 226.8 & 229.1 & 229.4 & \\
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\hline
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%%%%%%%% & ex FCI+LDA-val & 219.1 & 226.9 & 229.0 & 228.9 & \\
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& ex FCI+LDA & 218.0 & 226.8 & 229.1 & 229.4 & \\
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& exFCI+PBE\fnm[2] & 226.4 & 228.2 & 229.1 & 229.2 & \\
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%%%%%%%% & ex FCI+LDA-val & 219.1 & 226.9 & 229.0 & 228.9 & \\
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%%%%%%%% & ex FCI+PBE-val & 228.0 & 228.2 & 229.1 & 228.9 & \\
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\hline
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& ex FCI+PBE & 226.4 & 228.2 & 229.1 & 229.2 & \\
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%%%%%%%% & ex FCI+PBE-val & 228.0 & 228.2 & 229.1 & 228.9 & \\
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\\
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\\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Results from Ref.~\onlinecite{BytLaiRuedenJCP05}.}
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\fnt[2]{Results from Ref.~\onlinecite{PetTouUmr-JCP-12}.}
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\end{table*}
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\ce{O2} & exFCI\fnm[1] & 105.2 & 114.5 & 118.0 &119.1 & 120.0 \\
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\begin{table*}
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\caption{
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\label{tab:diatomics}
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Dissociation energy ($\De$) in kcal/mol of the \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} molecules computed with various methods and basis sets.
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}
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\begin{ruledtabular}
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\begin{tabular}{llddddd}
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\\
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\cline{3-6}
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Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$D_e^{Q5Z}$}
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\\
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\ce{O2} & exFCI & 105.2 & 114.5 & 118.0 &119.1 & 120.0 \\
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\hline
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%%%%%%%% & exFCI+LDA & 111.8 & 117.2 & 120.0 &119.9 & \\
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%%%%%%%% & exFCI+LDA & 111.8 & 117.2 & 120.0 &119.9 & \\
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& exFCI+LDA-val & 112.4 & 118.4 & 120.2 &120.4 & \\
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(cc-pVXZ) & exFCI+LDA\fnm[1] & 112.4 & 118.4 & 120.2 &120.4 & \\
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%%%%%%%%\hline
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%%%%%%%%\hline
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%%%%%%%% & exFCI+PBE & 115.9 & 118.4 & 120.1 &119.9 & \\
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%%%%%%%% & exFCI+PBE & 115.9 & 118.4 & 120.1 &119.9 & \\
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& exFCI+PBE-val & 117.2 & 119.4 & 120.3 &120.4 & \\
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& exFCI+PBE\fnm[1] & 117.2 & 119.4 & 120.3 &120.4 & \\
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\hline
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& CCSD(T)\fnm[1] & 103.9 & 113.6 & 117.1 & 118.6 & 120.0 \\
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\hline
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& CCSD(T)+LDA\fnm[1] & 110.6 & 117.2 & 119.2 & 119.8 & \\
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& (FC)CCSD(T) & 103.9 & 113.6 & 117.1 & 118.6 & 120.0 \\
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& CCSD(T)+PBE\fnm[1] & 115.1 & 118.0 & 119.3 & 119.8 & \\
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& (FC)CCSD(T)+LDA-val & 110.6 & 117.2 & 119.2 & 119.8 & \\
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& (FC)CCSD(T)+PBE-val & 115.1 & 118.0 & 119.3 & 119.8 & \\
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\hline
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%%%%%%%% & exFCI+PBE-on-top & 115.0 & 118.4 & 120.2 & & \\
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%%%%%%%% & exFCI+PBE-on-top & 115.0 & 118.4 & 120.2 & & \\
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%%%%%%%% & exFCI+PBE-on-top-val & 116.1 & 119.4 & 120.5 & & \\
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%%%%%%%% & exFCI+PBE-on-top-val & 116.1 & 119.4 & 120.5 & & \\
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\\
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\\
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Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$D_e^{Q5Z}$}
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\ce{F2} & exFCI\fnm[1] & 26.7 & 35.1 & 37.1 & 38.0 & 39.0 \\
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\\
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\ce{F2} & exFCI & 26.7 & 35.1 & 37.1 & 38.0 & 39.0 \\
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\hline
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%%%%%%%% & exFCI+LDA & 30.8 & 37.0 & 38.7 & 38.7 & \\
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%%%%%%%% & exFCI+LDA & 30.8 & 37.0 & 38.7 & 38.7 & \\
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& exFCI+LDA-val & 30.4 & 37.2 & 38.4 & 38.9 & \\
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(cc-pVXZ) & exFCI+LDA\fnm[1] & 30.4 & 37.2 & 38.4 & 38.9 & \\
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%%%%%%%% \hline
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%%%%%%%% \hline
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%%%%%%%% & exFCI+PBE & 33.3 & 37.8 & 38.8 & 38.7 & \\
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%%%%%%%% & exFCI+PBE & 33.3 & 37.8 & 38.8 & 38.7 & \\
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& exFCI+PBE -val & 33.1 & 37.9 & 38.5 & 38.9 & \\
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& exFCI+PBE\fnm[1] & 33.1 & 37.9 & 38.5 & 38.9 & \\
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%%%%%%%% & exFCI+PBE-on-top& 32.1 & 37.5 & 38.7 & 38.7 & \\
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%%%%%%%% & exFCI+PBE-on-top& 32.1 & 37.5 & 38.7 & 38.7 & \\
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%%%%%%%% & exFCI+PBE-on-top-val & 32.4 & 37.8 & 38.8 & 38.8 & \\
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%%%%%%%% & exFCI+PBE-on-top-val & 32.4 & 37.8 & 38.8 & 38.8 & \\
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\hline
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& CCSD(T)\fnm[1] & 25.7 & 34.4 & 36.5 & 37.4 & 38.2 \\
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\hline
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& CCSD(T)+LDA\fnm[1] & 29.2 & 36.5 & 37.2 & 38.2 & \\
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& (FC)CCSD(T) & 25.7 & 34.4 & 36.5 & 37.4 & 38.2 \\
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& CCSD(T)+PBE\fnm[1] & 31.5 & 37.1 & 37.8 & 38.2 & \\
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& (FC)CCSD(T)+LDA-val & 29.2 & 36.5 & 37.2 & 38.2 & \\
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& (FC)CCSD(T)+PBE-val & 31.5 & 37.1 & 37.8 & 38.2 & \\
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\end{tabular}
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\end{tabular}
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\end{ruledtabular}
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\end{ruledtabular}
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\fnt[1]{Results from Ref.~\onlinecite{BytLaiRuedenJCP05}.}
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\fnt[1]{Fronzen core calculations. Only valence spinorbitals are taken into account in the basis set correction.}
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\fnt[2]{Results from Ref.~\onlinecite{PetTouUmr-JCP-12}.}
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\fnt[2]{``Full'' calculation, i.e., all electrons are correlated. All spinorbitals are taken into account in the basis set correction.}
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\end{table*}
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\end{table*}
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