results Ec G2
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@ -189,7 +189,7 @@ Here, we only provide the main working equations.
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We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
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Let us assume we have both the energy $\E{\modY}{\Bas}$ and density $\n{\modZ}{\Bas}$ of a $\Ne$-electron system described by two methods $\modY$ and $\modZ$ (potentially identical) in an incomplete basis set $\Bas$.
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the \alert{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
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\begin{equation}
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\label{eq:e0basis}
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\E{}{}
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@ -214,7 +214,7 @@ Importantly, in the limit of a complete basis set (which we refer to as $\Bas \t
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\end{equation}
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where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the complete basis set.
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In the case $\modY = \FCI$, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
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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 \titou{FCI} energy and density within $\Bas$, respectively.
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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.
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Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
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@ -223,15 +223,15 @@ Because the e-e cusp originates from the divergence of the Coulomb operator at $
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Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction.
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Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
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The first step of the basis set correction consists in obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ which represents the effect of the projection of the Coulomb operator in an incomplete basis set $\Bas$.
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The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$.
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The first step of the present basis set correction consists in obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ which represents the effect of the projection of the Coulomb operator in an incomplete basis set $\Bas$.
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%The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$.
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In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$.
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In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation.
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%=================================================================
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%\subsection{Effective Coulomb operator}
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%=================================================================
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We define the effective operator as
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We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\Bas}{}(\br{1},\br{2}) =
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@ -254,7 +254,7 @@ and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}
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= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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\end{equation}
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and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals.
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Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems do not have any basis set correction.
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Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} \titou{ensures} that one-electron systems do not have any basis set correction.
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%\PFL{I don't agree with this. There must be a correction for one-electron system.
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%However, it does not come from the e-e cusp but from the e-n cusp.}
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With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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@ -267,9 +267,9 @@ Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
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\begin{equation}
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\iint r_{12}^{-1} \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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\alert{it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.}
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries.
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An important quantity to define in the present context is the value of the effective interaction at coalescence of opposite-spin electrons
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A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons
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\begin{equation}
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\label{eq:wcoal}
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\W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
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@ -289,7 +289,7 @@ for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $
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%\subsection{Range-separation function}
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%=================================================================
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Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be straightforwardly linked to RS-DFT which employs smooth long-range operators.
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Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT \titou{which employs smooth long-range operators.}
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Although this choice is not unique, we choose here the range-separation function
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\begin{equation}
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\label{eq:mu_of_r}
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@ -351,7 +351,7 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the following two l
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\end{subequations}
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where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functionals [Eq.~\eqref{eq:ec_md_mu}].
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Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}}$ coincides with $\wf{}{\Bas}$.
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Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \titou{=} \W{\Bas}{}(\br{1},\br{2})$, then $\wf{}{\rsmu{\Bas}{}}$ and $\wf{}{\Bas}$ coincide.
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%The ECMD functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function.
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%This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
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Therefore, following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$.
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@ -362,7 +362,7 @@ The LDA version of $\bE{}{\Bas}[\n{}{}]$ is defined as
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\end{equation}
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where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the short-range reduced (i.e.~per electron) correlation energy of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
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The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
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In order to correct such a defect, we propose here a new ECMD functional inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
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\begin{subequations}
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\begin{gather}
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@ -373,7 +373,7 @@ In order to correct such a defect, we propose here a new ECMD functional inspire
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\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}.
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\end{gather}
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\end{subequations}
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The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\br{}) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}) )$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\n{}{}(\br{})) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}))$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$.
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Therefore, the PBE complementary functional reads
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\begin{equation}
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@ -494,7 +494,7 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
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%%% TABLE II %%%
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\begin{table}
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\caption{
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Statistical analysis (in \kcal) of the G2 correlation energies depicted in Fig.~\ref{fig:G2_AE}.
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Statistical analysis (in \kcal) of the G2 correlation energies depicted in Fig.~\ref{fig:G2_Ec}.
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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 correlation energies.
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CA corresponds to the number of correlation energies (out of 55) obtained with chemical accuracy.
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\label{tab:stats}}
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@ -526,7 +526,7 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
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\caption{
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Deviation (in \kcal) from CCSD(T)/CBS reference correlation energies obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
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The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
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\label{fig:G2_AE}}
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\label{fig:G2_Ec}}
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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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@ -569,6 +569,18 @@ Importantly, the sensitivity with respect to the SR-DFT functional is quite larg
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However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
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Such weak sensitivity to the approximated functionals in the DFT part when reaching large basis sets shows the robustness of the approach.
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We have computed the correlation energies of the G2-1 test sets with $\modY=\CCSDT$, $\modZ=\HF$ and the cc-pVXZ basis.
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The plain CCSD(T) correlation energies as well as the corrected CCSD(T)+LDA and CCSD(T)+PBE values are reported.
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A statistical analysis of these data is also provided in Table \ref{tab:stats}, where one can find the mean absolute deviation (MAD), the root-mean-square deviation (RMSD), and the maximum deviation (MAX) with respect to the CCSD(T)/CBS reference correlation energies.
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From double-$\zeta$ to quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) correlation energies goes down slowly from 14.29 to 1.28 {\kcal}.
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For typical basis like cc-pVTZ, the MAD of CCSD(T) is still 6.06 {\kcal}.
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Applying the basis set correction drastically reduces the basis set incompleteness error.
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Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ levels, the MAD is reduced to 3.24 and 1.96 {\kcal}.
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With the triple-$\zeta$ basis, the CCSD(T)+PBE/cc-pVTZ are already below 1 {\kcal} with 36 cases (out of 55) where we achieve chemical accuracy.
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CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
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Therefore, similarly to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction recover quintuple-$\zeta$ quality coupled-cluster correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Supporting information}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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