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Manuscript/G2-srDFT.tex
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@ -89,7 +89,7 @@
% basis sets % basis sets
\newcommand{\Bas}{\mathcal{B}} \newcommand{\Bas}{\mathcal{B}}
\newcommand{\BasFC}{\Bar{\mathcal{B}}} \newcommand{\BasFC}{\mathcal{A}}
\newcommand{\FC}{\text{FC}} \newcommand{\FC}{\text{FC}}
\newcommand{\occ}{\text{occ}} \newcommand{\occ}{\text{occ}}
\newcommand{\virt}{\text{virt}} \newcommand{\virt}{\text{virt}}
@ -119,6 +119,9 @@
\title{A Density-Based Basis Set Correction For Wave Function Theory} \title{A Density-Based Basis Set Correction For Wave Function Theory}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Bath\'elemy Pradines} \author{Bath\'elemy Pradines}
\affiliation{\LCT} \affiliation{\LCT}
\affiliation{\ISCD} \affiliation{\ISCD}
@ -127,9 +130,6 @@
\author{Julien Toulouse} \author{Julien Toulouse}
\email{toulouse@lct.jussieu.fr} \email{toulouse@lct.jussieu.fr}
\affiliation{\LCT} \affiliation{\LCT}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Emmanuel Giner} \author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr} \email{emmanuel.giner@lct.jussieu.fr}
\affiliation{\LCT} \affiliation{\LCT}
@ -205,7 +205,7 @@ Importantly, in the limit of a complete basis set (which we refer to as $\Bas \t
\lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E, \lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E,
\end{equation} \end{equation}
where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the complete basis set. where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the complete basis set.
In the case $\modY = \FCI$ \manu{in \eqref{eq:limitfunc}}, we have a strict equality as $\E{\FCI}{} = \E{}{}$. In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$.% for the FCI energy and density within $\Bas$, respectively. Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$.% for the FCI energy and density within $\Bas$, respectively.
The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
@ -239,7 +239,7 @@ where
\n{2}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2})
= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation} \end{equation}
and $\Gam{pq}{rs} = \manu{1/2\mel*{\wf{}{\Bas}}{ \aic{r\downarrow}\aic{s\uparrow}\ai{p\uparrow}\ai{q\downarrow} + \aic{r\uparrow}\aic{s\downarrow}\ai{p\downarrow}\ai{q\uparrow} }{\wf{}{\Bas}}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor (respectively), $\SO{p}{}$ is a \trashMG{spinorbital}\manu{spatial orbital}, and $\Gam{pq}{rs} =\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor (respectively), $\SO{p}{}$ is a molecular orbital (MO),
\begin{equation} \begin{equation}
\label{eq:fbasis} \label{eq:fbasis}
\f{\Bas}{}(\br{1},\br{2}) \f{\Bas}{}(\br{1},\br{2})
@ -377,8 +377,8 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
%\subsection{Valence approximation} %\subsection{Valence approximation}
%================================================================= %=================================================================
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of spinorbitals. As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of MOs.
We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core \trashMG{spinorbitals} \manu{spatial orbitals}, and define the FC version of the effective interaction as We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core MOs, and define the FC version of the effective interaction as
\begin{equation} \begin{equation}
\W{\Bas}{\FC}(\br{1},\br{2}) = \W{\Bas}{\FC}(\br{1},\br{2}) =
\begin{cases} \begin{cases}
@ -488,7 +488,6 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
\end{figure*} \end{figure*}
We begin our investigation of the performance of the basis set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis sets (X $=$ D, T, Q and 5). We begin our investigation of the performance of the basis set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis sets (X $=$ D, T, Q and 5).
\titou{In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.}
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11} \ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets. In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) \titou{and can be considered as a representative set for typical quantum chemical calculations on small organic molecules}. This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) \titou{and can be considered as a representative set for typical quantum chemical calculations on small organic molecules}.
@ -503,7 +502,7 @@ RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17} For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory. Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
Frozen-core calculations are defined as such: an \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}. Frozen-core calculations are defined as such: an \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
In the context of the basis set correction, the set of spinorbitals $\BasFC$ involved in the definition of the effective interaction refers to the non-frozen spinorbitals. In the context of the basis set correction, the set of MOs $\BasFC$ involved in the definition of the effective interaction refers to the non-frozen MOs.
The FC density-based correction is set consistently when the FC approximation was applied in WFT methods. The FC density-based correction is set consistently when the FC approximation was applied in WFT methods.
In order to estimate the complete basis set (CBS) limit for each model, \manu{following the work of Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98}}, In order to estimate the complete basis set (CBS) limit for each model, \manu{following the work of Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98}},
we employ the two-point extrapolation for the correlation energies \manu{for each model in quadruple- and quintuple-$\zeta$ basis sets, which is refered to as $\CBS$ correlation energies, and we add the HF energies in the largest basis sets (\textit{i.e.} quintuple-$\zeta$ quality basis sets) to the CBS correlation energies to estimate the CBS FCI energies.} we employ the two-point extrapolation for the correlation energies \manu{for each model in quadruple- and quintuple-$\zeta$ basis sets, which is refered to as $\CBS$ correlation energies, and we add the HF energies in the largest basis sets (\textit{i.e.} quintuple-$\zeta$ quality basis sets) to the CBS correlation energies to estimate the CBS FCI energies.}

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Manuscript/SI/G2_srDFT-SI.tex
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@ -118,16 +118,18 @@
\title{Supplementary Information for ``A Density-Based Basis Set Correction For Wave Function Theory''} \title{Supplementary Information for ``A Density-Based Basis Set Correction For Wave Function Theory''}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Bath\'elemy Pradines} \author{Bath\'elemy Pradines}
\affiliation{\LCPQ} \affiliation{\LCPQ}
\author{Anthony Scemama} \author{Anthony Scemama}
\affiliation{\LCPQ} \affiliation{\LCPQ}
\author{Julien Toulouse} \author{Julien Toulouse}
\email{toulouse@lct.jussieu.fr}
\affiliation{\LCT} \affiliation{\LCT}
\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Emmanuel Giner} \author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr}
\affiliation{\LCT} \affiliation{\LCT}
\begin{abstract} \begin{abstract}
@ -139,51 +141,51 @@
\begin{table*} \begin{table*}
\caption{ \caption{
\label{tab:diatomics} \label{tab:diatomics}
Atomization energies (in {\kcal}) of \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} computed with various methods and basis sets. Frozen-core atomization energies (in {\kcal}) of \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} computed with various methods and basis sets.
The deviations with respect to the corresponding CBS values are reported in parenthesis. The deviations with respect to the corresponding CBS values are reported in parenthesis.
} }
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llddddd} \begin{tabular}{llddddd}
& & \mc{4}{c}{Dunning's basis set} & & \mc{4}{c}{Dunning's basis set cc-pVXZ}
\\ \\
\cline{3-6} \cline{3-6}
Molecule & Method & \tabc{$\X = \D$} & \tabc{$\X = \T$} & \tabc{$\X = \Q$} & \tabc{$\X = 5$} & \tabc{CBS} Molecule & Method & \tabc{$\X = \D$} & \tabc{$\X = \T$} & \tabc{$\X = \Q$} & \tabc{$\X = 5$} & \tabc{CBS}
\\ \\
\hline \hline
\ce{C2} & exFCI\fnm[1] & 132.0 (-13.7 ) & 140.3 (-5.4 ) & 143.6 (-2.1 ) & 144.7 (-1.0 ) & 145.7 \\ \ce{C2} & exFCI & 132.0 (-13.7 ) & 140.3 (-5.4 ) & 143.6 (-2.1 ) & 144.7 (-1.0 ) & 145.7 \\
(cc-pVXZ) & exFCI+LDA\fnm[1] & 141.3 (-4.4 ) & 145.1 (-0.6 ) & 146.4 (+0.7 ) & 146.3 (+0.6 ) & \\ & exFCI+LDA & 141.3 (-4.4 ) & 145.1 (-0.6 ) & 146.4 (+0.7 ) & 146.3 (+0.6 ) & \\
& exFCI+PBE\fnm[1] & 145.7 (+0.0 ) & 145.7 (+0.0 ) & 146.3 (+0.6 ) & 146.2 (+0.5 ) & \\ & exFCI+PBE & 145.7 (+0.0 ) & 145.7 (+0.0 ) & 146.3 (+0.6 ) & 146.2 (+0.5 ) & \\
& CCSD(T)\fnm[1] & 129.2 (-16.2 ) & 139.1 (-6.3 ) & 143.0 (-2.4 ) & 144.2 (-1.2 ) & 145.4 \\ & CCSD(T) & 129.2 (-16.2 ) & 139.1 (-6.3 ) & 143.0 (-2.4 ) & 144.2 (-1.2 ) & 145.4 \\
& CCSD(T)+LDA\fnm[1] & 139.1 (-6.3 ) & 143.7 (-1.7 ) & 145.9 (+0.5 ) & 145.9 (+0.5 ) & \\ & CCSD(T)+LDA & 139.1 (-6.3 ) & 143.7 (-1.7 ) & 145.9 (+0.5 ) & 145.9 (+0.5 ) & \\
& CCSD(T)+PBE\fnm[1] & 142.8 (-2.6 ) & 144.2 (-1.2 ) & 145.9 (+0.5 ) & 145.8 (+0.4 ) & \\ \\ & CCSD(T)+PBE & 142.8 (-2.6 ) & 144.2 (-1.2 ) & 145.9 (+0.5 ) & 145.8 (+0.4 ) & \\ \\
\ce{C2} & exFCI\fnm[2] & 131.0 (-16.1 ) & 141.5 (-5.6 ) & 145.1 (-2.0 ) & 146.1 (-1.0 ) & 147.1 \\ % \ce{C2} & exFCI\fnm[2] & 131.0 (-16.1 ) & 141.5 (-5.6 ) & 145.1 (-2.0 ) & 146.1 (-1.0 ) & 147.1 \\
(cc-pCVXZ) & exFCI+LDA\fnm[2] & 141.4 (-5.7 ) & 146.7 (-0.4 ) & 147.8 (+0.7 ) & 147.6 (+0.5 ) & \\ % (cc-pCVXZ) & exFCI+LDA\fnm[2] & 141.4 (-5.7 ) & 146.7 (-0.4 ) & 147.8 (+0.7 ) & 147.6 (+0.5 ) & \\
& exFCI+PBE\fnm[2] & 145.1 (-2.0 ) & 147.0 (-0.1 ) & 147.7 (+0.6 ) & 147.5 (+0.4 ) & \\ \\ % & exFCI+PBE\fnm[2] & 145.1 (-2.0 ) & 147.0 (-0.1 ) & 147.7 (+0.6 ) & 147.5 (+0.4 ) & \\ \\
\ce{N2} & exFCI\fnm[1] & 201.1 (-26.7 ) & 217.1 (-10.7 ) & 223.5 (-4.3 ) & 225.7 (-2.1 ) & 227.8 \\ \ce{N2} & exFCI & 201.1 (-26.7 ) & 217.1 (-10.7 ) & 223.5 (-4.3 ) & 225.7 (-2.1 ) & 227.8 \\
(cc-pVXZ) & exFCI+LDA\fnm[1] & 217.9 (-9.9 ) & 225.9 (-1.9 ) & 228.0 (+0.2 ) & 228.6 (+0.8 ) & \\ & exFCI+LDA & 217.9 (-9.9 ) & 225.9 (-1.9 ) & 228.0 (+0.2 ) & 228.6 (+0.8 ) & \\
& exFCI+PBE\fnm[1] & 227.7 (-0.1 ) & 227.8 (+0.0 ) & 228.3 (+0.5 ) & 228.5 (+0.7 ) & \\ & exFCI+PBE & 227.7 (-0.1 ) & 227.8 (+0.0 ) & 228.3 (+0.5 ) & 228.5 (+0.7 ) & \\
& CCSD(T)\fnm[1] & 199.9 (-27.3 ) & 216.3 (-10.9 ) & 222.8 (-4.4 ) & 225.0 (-2.2 ) & 227.2 \\ & CCSD(T) & 199.9 (-27.3 ) & 216.3 (-10.9 ) & 222.8 (-4.4 ) & 225.0 (-2.2 ) & 227.2 \\
& CCSD(T)+LDA\fnm[1] & 216.3 (-10.9 ) & 224.8 (-2.4 ) & 227.2 (-0.0 ) & 227.8 (+0.6 ) & \\ & CCSD(T)+LDA & 216.3 (-10.9 ) & 224.8 (-2.4 ) & 227.2 (-0.0 ) & 227.8 (+0.6 ) & \\
& CCSD(T)+PBE\fnm[1] & 225.9 (-1.3 ) & 226.7 (-0.5 ) & 227.5 (+0.3 ) & 227.8 (+0.6 ) & \\ \\ & CCSD(T)+PBE & 225.9 (-1.3 ) & 226.7 (-0.5 ) & 227.5 (+0.3 ) & 227.8 (+0.6 ) & \\ \\
\ce{N2} & exFCI\fnm[2] & 202.2 (-26.6 ) & 218.5 (-10.3 ) & 224.4 (-4.4 ) & 226.6 (-2.2 ) & 228.8 \\ % \ce{N2} & exFCI\fnm[2] & 202.2 (-26.6 ) & 218.5 (-10.3 ) & 224.4 (-4.4 ) & 226.6 (-2.2 ) & 228.8 \\
(cc-pCVXZ) & exFCI+LDA\fnm[2] & 218.0 (-10.8 ) & 226.8 (-2.0 ) & 229.1 (+0.3 ) & 229.4 (+0.6 ) & \\ % (cc-pCVXZ) & exFCI+LDA\fnm[2] & 218.0 (-10.8 ) & 226.8 (-2.0 ) & 229.1 (+0.3 ) & 229.4 (+0.6 ) & \\
& exFCI+PBE\fnm[2] & 226.4 (-2.4 ) & 228.2 (-0.6 ) & 229.1 (+0.3 ) & 229.2 (+0.4 ) & \\ \\ % & exFCI+PBE\fnm[2] & 226.4 (-2.4 ) & 228.2 (-0.6 ) & 229.1 (+0.3 ) & 229.2 (+0.4 ) & \\ \\
\ce{O2} & exFCI\fnm[1] & 105.2 (-14.8 ) & 114.5 (-5.5 ) & 118.0 (-2.0 ) & 119.1 (-0.9 ) & 120.0 \\ \ce{O2} & exFCI & 105.2 (-14.8 ) & 114.5 (-5.5 ) & 118.0 (-2.0 ) & 119.1 (-0.9 ) & 120.0 \\
(cc-pVXZ) & exFCI+LDA\fnm[1] & 112.4 (-7.6 ) & 118.4 (-1.6 ) & 120.2 (+0.2 ) & 120.4 (+0.4 ) & \\ & exFCI+LDA & 112.4 (-7.6 ) & 118.4 (-1.6 ) & 120.2 (+0.2 ) & 120.4 (+0.4 ) & \\
& exFCI+PBE\fnm[1] & 117.2 (-2.8 ) & 119.4 (-0.6 ) & 120.3 (+0.3 ) & 120.4 (+0.4 ) & \\ & exFCI+PBE & 117.2 (-2.8 ) & 119.4 (-0.6 ) & 120.3 (+0.3 ) & 120.4 (+0.4 ) & \\
& CCSD(T)\fnm[1] & 103.9 (-16.1 ) & 113.6 (-6.0 ) & 117.1 (-2.5 ) & 118.6 (-1.0 ) & 119.6 \\ & CCSD(T) & 103.9 (-16.1 ) & 113.6 (-6.0 ) & 117.1 (-2.5 ) & 118.6 (-1.0 ) & 119.6 \\
& CCSD(T)+LDA\fnm[1] & 110.6 (-9.0 ) & 117.2 (-2.4 ) & 119.2 (-0.4 ) & 119.8 (+0.2 ) & \\ & CCSD(T)+LDA & 110.6 (-9.0 ) & 117.2 (-2.4 ) & 119.2 (-0.4 ) & 119.8 (+0.2 ) & \\
& CCSD(T)+PBE\fnm[1] & 115.1 (-4.5 ) & 118.0 (-1.6 ) & 119.3 (-0.3 ) & 119.8 (+0.2 ) & \\ \\ & CCSD(T)+PBE & 115.1 (-4.5 ) & 118.0 (-1.6 ) & 119.3 (-0.3 ) & 119.8 (+0.2 ) & \\ \\
\ce{F2} & exFCI\fnm[1] & 26.7 (-12.3 ) & 35.1 (-3.9 ) & 37.1 (-1.9 ) & 38.0 (-1.0 ) & 39.0 \\ \ce{F2} & exFCI & 26.7 (-12.3 ) & 35.1 (-3.9 ) & 37.1 (-1.9 ) & 38.0 (-1.0 ) & 39.0 \\
(cc-pVXZ) & exFCI+LDA\fnm[1] & 30.4 (-8.6 ) & 37.2 (-1.8 ) & 38.4 (-0.6 ) & 38.9 (-0.1 ) & \\ & exFCI+LDA & 30.4 (-8.6 ) & 37.2 (-1.8 ) & 38.4 (-0.6 ) & 38.9 (-0.1 ) & \\
& exFCI+PBE\fnm[1] & 33.1 (-5.9 ) & 37.9 (-1.1 ) & 38.5 (-0.5 ) & 38.9 (-0.1 ) & \\ & exFCI+PBE & 33.1 (-5.9 ) & 37.9 (-1.1 ) & 38.5 (-0.5 ) & 38.9 (-0.1 ) & \\
& CCSD(T)\fnm[1] & 25.7 (-12.5 ) & 34.4 (-3.8 ) & 36.5 (-1.7 ) & 37.4 (-0.8 ) & 38.2 \\ & CCSD(T) & 25.7 (-12.5 ) & 34.4 (-3.8 ) & 36.5 (-1.7 ) & 37.4 (-0.8 ) & 38.2 \\
& CCSD(T)+LDA\fnm[1] & 29.2 (-9.0 ) & 36.5 (-1.7 ) & 37.2 (-1.0 ) & 38.2 (+0.0 ) & \\ & CCSD(T)+LDA & 29.2 (-9.0 ) & 36.5 (-1.7 ) & 37.2 (-1.0 ) & 38.2 (+0.0 ) & \\
& CCSD(T)+PBE\fnm[1] & 31.5 (-6.7 ) & 37.1 (-1.1 ) & 37.8 (-0.4 ) & 38.2 (+0.0 ) & \\ & CCSD(T)+PBE & 31.5 (-6.7 ) & 37.1 (-1.1 ) & 37.8 (-0.4 ) & 38.2 (+0.0 ) & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Frozen core calculations. Only valence spinorbitals are taken into account in the basis set correction.} % \fnt[1]{ calculations. Only valence orbitals are taken into account in the basis set correction.}
\fnt[2]{``Full'' calculation, i.e., all electrons are correlated. All spinorbitals are taken into account in the basis set correction.} % \fnt[2]{``Full'' calculation, i.e., all electrons are correlated. All spinorbitals are taken into account in the basis set correction.}
\end{table*} \end{table*}
@ -192,11 +194,11 @@
\begin{table} \begin{table}
\caption{ \caption{
\label{tab:AE} \label{tab:AE}
Deviation from the reference CBS correlation energies (in {\kcal}) for various methods and basis sets.} Deviation from the reference CBS atomization energies (in {\kcal}) for various methods and basis sets.}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lddddddddddd} \begin{tabular}{lddddddddddd}
& &
& \mc{10}{c}{Deviation from CBS correlation energies} \\ & \mc{10}{c}{Deviation from CBS atomization energies} \\
\cline{3-12} \cline{3-12}
& & \mc{4}{c}{CCSD(T)} & \mc{3}{c}{CCSD(T)+LDA} & \mc{3}{c}{CCSD(T)+PBE} \\ & & \mc{4}{c}{CCSD(T)} & \mc{3}{c}{CCSD(T)+LDA} & \mc{3}{c}{CCSD(T)+PBE} \\
\cline{3-6} \cline{7-9} \cline{10-12} \cline{3-6} \cline{7-9} \cline{10-12}