\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,xspace}
\usepackage{mathpazo,libertine}

\usepackage{natbib}
\bibliographystyle{achemso}
\AtBeginDocument{\nocite{achemso-control}}

\newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\juju}[1]{\textcolor{purple}{#1}}
\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}}
\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}}
\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}}
\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}

\usepackage{hyperref}
\hypersetup{
    colorlinks=true,
    linkcolor=blue,
    filecolor=blue,      
    urlcolor=blue,
    citecolor=blue
}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\SI}{\textcolor{blue}{supporting information}}
\newcommand{\QP}{\textsc{quantum package}}

% second quantized operators
\newcommand{\ai}[1]{\hat{a}_{#1}}
\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}

% units
\newcommand{\IneV}[1]{#1 eV}
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\InAA}[1]{#1 \AA}
\newcommand{\kcal}{kcal/mol}

% methods
\newcommand{\D}{\text{D}}
\newcommand{\T}{\text{T}}
\newcommand{\Q}{\text{Q}}
\newcommand{\X}{\text{X}}
\newcommand{\UEG}{\text{UEG}}
\newcommand{\HF}{\text{HF}}
\newcommand{\LDA}{\text{LDA}}
\newcommand{\PBE}{\text{PBE}}
\newcommand{\FCI}{\text{FCI}}
\newcommand{\CBS}{\text{CBS}}
\newcommand{\exFCI}{\text{exFCI}}
\newcommand{\CCSDT}{\text{CCSD(T)}}
\newcommand{\lr}{\text{lr}}
\newcommand{\sr}{\text{sr}}

\newcommand{\Ne}{N}
\newcommand{\Nb}{N_{\Bas}}
\newcommand{\Ng}{N_\text{grid}}
\newcommand{\nocca}{n_{\text{occ}^{\alpha}}}
\newcommand{\noccb}{n_{\text{occ}^{\beta}}}

\newcommand{\n}[2]{n_{#1}^{#2}}
\newcommand{\Ec}{E_\text{c}}
\newcommand{\E}[2]{E_{#1}^{#2}}
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}}
\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
\newcommand{\bec}[1]{\Bar{e}^{#1}}
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
\newcommand{\W}[2]{W_{#1}^{#2}}
\newcommand{\w}[2]{w_{#1}^{#2}}
\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
\newcommand{\V}[2]{V_{#1}^{#2}}
\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}

\newcommand{\modX}{\mathcal{X}}
\newcommand{\modY}{\mathcal{Y}}

% basis sets 
\newcommand{\Bas}{\mathcal{B}}
\newcommand{\Basval}{\mathcal{B}_\text{val}}
\newcommand{\Val}{\mathcal{V}}
\newcommand{\Cor}{\mathcal{C}}

% operators
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
\newcommand{\f}[2]{f_{#1}^{#2}}
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}

% coordinates
\newcommand{\br}[1]{\mathbf{r}_{#1}}
%\newcommand{\br}[1]{\mathbf{x}_{#1}}
\newcommand{\dbr}[1]{d\br{#1}}
%\newcommand{\dbr}[1]{d\br{#1}}

\newcommand{\ra}{\rightarrow}
\newcommand{\De}{D_\text{e}}

\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}

\begin{document}	

\title{A Density-Based Basis Set Correction For Wave Function Theory}

\author{Bath\'elemy Pradines}
\affiliation{\LCPQ}
\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Julien Toulouse}
\affiliation{\LCT}
\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Emmanuel Giner}
\affiliation{\LCT}

\begin{abstract}
We report a universal density-based basis set incompleteness correction that can be applied to any wave function method.
The present correction relies on short-range correlation functionals (with multi-determinant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis set incompleteness error.
Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range-separated \textit{parameter} $\mu$, the key ingredient here is a basis-dependent, range-separated \textit{function} $\mu(\bf{r})$ which is dynamically determined to catch the missing short-range correlation due to the lack of electron-electron cusp in standard wave function methods.
As illustrative examples, we show how this density-based correction allows to obtain near-complete basis set CCSD(T) atomization energies for the G2 set of molecules with compact Gaussian basis sets. 
For example, CCSD(T)+LDA/cc-pVTZ and CCSD(T)+PBE/cc-pVTZ return mean absolute deviations of \titou{X.XX} and \titou{X.XX} kcal/mol, respectively, compared to CBS atomization energies.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%
Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
Although both spring from the same Schr\"odinger equation, each of these philosophies has its own advantages and shortcomings.

WFT is attractive as it exists a well-defined path for systematic improvement and powerful tools, such as perturbation theory, to guide the development of new attractive WFT models.
The coupled cluster (CC) family of methods are a typical example of the WFT philosophy for the description of weakly correlated systems and is well regarded as the gold standard of quantum chemistry. 
By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration interaction (FCI) limit, although the computational cost associated with such improvement is usually pricey.
One of the most fundamental drawback of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set.
This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
To palliate this, in the Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ as a basis function. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94}
The resulting F12 methods yields a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12}
For example, at the CCSD(T) level, it is advertised that one can obtain quintuple-zeta quality correlation energies with a triple-zeta basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals.

Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65}
DFT's attractivity originates from its very favorable cost/efficient ratio as it can provide accurate energies and properties at a relatively low computational cost.
Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
To obtain accurate results within DFT, one only requires an exchange and correlation functionals, which can be classified in various families depending on their physical input quantities. \cite{Bec-JCP-14}
Although there is no clear way on how to systematically improve density-functional approximations (DFAs), climbing the Jacob's ladder of DFT is potentially the most satisfactory way forward (or upward in that case). \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
%The local-density approximation (LDA) sits on the first rung of the Jacob's ladder and only uses as input the electron density $n$. \cite{Dir-PCPRS-30, VosWilNus-CJP-80}
%The generalized-gradient approximation (GGA) corresponds to the second rung and adds the gradient of the electron density $\nabla n$ as an extra ingredient.\cite{Bec-PRA-88, PerWan-PRA-91, PerBurErn-PRL-96}
In the present context, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}

Progress toward unifying these two approaches are on-going.
Using accurate and rigorous WFT methods, some of us have developed radical generalizations of DFT that are free of the well-known limitations of conventional DFT. 
In that respect range-separated DFT (RS-DFT) is particularly promising as it allows to perform multi-configurational DFT calculations within a rigorous mathematical framework. 
Range-separated hybrids, i.e.~single-determinant approximations of RS-DFT, correct for the wrong long-range behavior of the usual hybrid approximations thanks to the inclusion of the long-range part of the Hartree-Fock (HF) exchange.  

Other basis set corrections are cool too, \cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019} but not as cool as ours.

%The present manuscript is organized as follows.
Unless otherwise stated, atomic used are used.

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%%%
The present basis set correction relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set. 
Here, we only provide the main working equations. 
We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.  

%=================================================================
%\subsection{Correcting the basis set error of a general WFT model}
%=================================================================
Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Ne$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$. 
According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
\begin{equation}
	\label{eq:e0basis}
	\E{}{} 
	\approx \E{\modX}{\Bas} 
	+ \bE{}{\Bas}[\n{\modY}{\Bas}],
\end{equation}
where
\begin{equation}
	\label{eq:E_funcbasis}
	 \bE{}{\Bas}[\n{}{}]
	= \min_{\wf{}{} \to \n{}{}}  \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}} 
	- \min_{\wf{}{\Bas} \to \n{}{}}  \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
\end{equation}
is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
Both wave functions yield the same target density $\n{}{}$. 

%\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}

Importantly, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
\begin{equation}
  \label{eq:limitfunc}
        \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{\infty} \approx E,
\end{equation}
where  $\E{\modX}{\infty}$ is the energy associated with the method $\modX$ in the complete basis set.
In the case $\modX = \FCI$, we have as strict equality as $E_{\FCI}^\infty = E$.
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 $\modX$ and $\modY$ for the FCI energy and density within $\Bas$.

%Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$. 
%As any wave function model is necessary an approximation to the FCI model, one can write 
%\begin{equation}
% \efci \approx \emodel
%\end{equation}
%and 
%\begin{equation}
% \denfci \approx \denmodel
%\end{equation}
%and by defining the energy provided by the model $\model$ in the complete basis set 
%\begin{equation}
% \emodelcomplete = \lim_{\Bas \rightarrow \infty} \emodel\,\, , 
%\end{equation}
%we can then write 
%\begin{equation}
% \emodelcomplete \approx \emodel + \ecompmodel
%\end{equation}
%which verifies the correct limit since 
%\begin{equation}
% \lim_{\Bas \rightarrow \infty} \ecompmodel = 0\,\, .
%\end{equation}


%=================================================================
%\subsection{Basis set correction for the CIPSI algorithm and the CCSD(T) ansatz}
%=================================================================
%In this work we propose to apply the basis set correction to a selected CI algorithm, namely the CIPSI algorithm, and to the CCSD(T) ansatz in %order to speed-up the basis set convergence of these models. 

%=================================================================
%\subsubsection{Basis set correction for the CCSD(T) energy}
%=================================================================
%The CCSD(T) method is a very popular WFT approach which is known to provide very good estimation of the correlation energies for weakly correlated systems, whose wave function are dominated by the HF Slater determinant. 
%Defining $\ecc$ as the CCSD(T) energy obtained in $\Bas$, in the present notations we have 
%\begin{equation}
% \emodel = \ecc \,\, .
%\end{equation}
%In the context of the basis set correction, one needs to choose a density as the density of the model $\denmodel$, and we chose here the HF density
%\begin{equation}
% \denmodel = \denhf \,\, .
%\end{equation}
%Such a choice can be motivated by the fact that the correction to the HF density brought by the excited Slater determinants are at least of second-order in perturbation theory. 
%Therefore, we approximate the complete basis set CCSD(T) energy $\ecccomplete$ by
%\begin{equation}
% \ecccomplete \approx \ecc + \efuncden{\denhf} \,\, .
%\end{equation}

%=================================================================
%\subsubsection{Correction of the CIPSI algorithm}
%=================================================================
%The CIPSI algorithm approximates the FCI wave function through an iterative selected CI procedure, and the FCI energy through a second-order multi-reference perturbation theory.
%The CIPSI algorithm belongs to the general class of methods build upon selected CI\cite{bender,malrieu,buenker1,buenker-book,three_class_CIPSI,harrison,hbci}
%which have been successfully used to converge to FCI correlation energies, one-body properties, and nodal surfaces.\cite{three_class_CIPSI,Rubio198698,cimiraglia_cipsi,cele_cipsi_zeroth_order,Angeli2000472,canadian,atoms_3d,f2_dmc,atoms_dmc_julien,GinTewGarAla-JCTC-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,LooBogSceCafJAc-JCTC-19}
%The CIPSI algorithm used in this work uses iteratively enlarged selected CI spaces and
%Epstein--Nesbet\cite{epstein,nesbet} multi-reference perturbation theory. Within a basis set $\Bas$, the CIPSI energy is
%\begin{align}
%  E_\mathrm{CIPSI}^{\Bas} &= E_\text{v} + E^{(2)} \,\,,
%\end{align}
%where $E_\text{v}$ is the variational energy
%\begin{align}
% E_\text{v} &= \min_{\{ c_{\rm I}\}} \frac{\elemm{\Psi^{(0)}}{\hat{H}}{\Psi^{(0)}}  }{\ovrlp{\Psi^{(0)}}{\Psi^{(0)}}}\,\,,
%\end{align}
%where the reference wave function $\ket{\Psi^{(0)}} = \sum_{{\rm I}\,\in\,\mathcal{R}} \,\,c_{\rm I} \,\,\ket{\rm I}$ is expanded in Slater determinants I within the CI reference space $\mathcal{R}$, and $E^{(2)}$ is the second-order energy correction
%\begin{align}
%  E^{(2)} &= \sum_{\kappa} \frac{|\elemm{\Psi^{(0)}}{\hat{H}}{\kappa}|^2}{E_\text{v} - \elemm{\kappa}{H}{\kappa}} = \sum_{\kappa} \,\, e_{\kappa}^{(2)} \,\, ,
%\end{align}
%where $\kappa$ denotes a determinant outside $\mathcal{R}$.
%To reduce the cost of the evaluation of the second-order energy correction, the semi-stochastic multi-reference approach
%of Garniron \textit{et al.} \cite{stochastic_pt_yan} was used, adopting the technical specifications recommended in that work.
%The CIPSI energy is systematically refined by doubling the size of the CI reference space at each iteration, selecting
%the determinants $\kappa$ with the largest $\vert e_{\kappa}^{(2)} \vert$. 
%In order to reach a faster convergence of the estimation of the FCI energy, we use the extrapolated FCI energy (exFCI) proposed by Holmes \textit{et al}\cite{HolUmrSha-JCP-17} which we refer here as $\EexFCIbasis$. 
%
%In the context of the basis set correction, we use the following conventions
%\begin{equation}
% \emodel = \EexFCIbasis
%\end{equation}
%\begin{equation}
% \denmodelr = \dencipsir
%\end{equation}
%where the density $\dencipsir$ is defined as
%\begin{equation}
% \dencipsi = \sum_{ij \in \Bas} \elemm{\Psi^{(0)}}{\aic{i}\ai{j}}{\Psi^{(0)}} \phi_i(\bfrb{} ) \phi_j(\bfrb{} ) \,\, ,
%\end{equation}
%and $\phi_i(\bfrb{} )$ are the spin orbitals in the MO basis evaluated at $\bfrb{}$. As it was shown in \onlinecite{GinPraFerAssSavTou-JCP-18} that the CIPSI density converges rapidly with the size of $\Psi^{(0)}$ for weakly correlated systems, $\dencipsir$ can be thought as a reasonable approximation of the FCI density $\denfci$. 
%
%Finally, we approximate complete basis set exFCI energy $\EexFCIinfty$ as
%\begin{equation}
% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
%\end{equation}

Rigorously speaking,  the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 
Nevertheless, from the physical point of view $\bE{}{\Bas}[\n{}{}]$ plays a quite universal role as it aims at fixing the main 
consequence of the incompleteness of $\Bas$, which is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points. 
As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$. 
Therefore, as we shall do later on, it feels natural to  approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, the spatial inhomogeneity of $\Bas$ forces us to define a range-separated \textit{function} $\rsmu{}{}(\br{})$ as the value of $\rsmu{}{}$ must be known at any point in space.

The first step of our basis set correction consists in obtaining an effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ which i) is finite at the e-e coalescence point as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set. 
In a second step, we shall link $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ to $\rsmu{}{}(\br{})$.
In the final step, we employ $\rsmu{}{}(\br{})$ within short-range density functionals. \cite{TouGorSav-TCA-05}
%Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}]. 
%First, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .   
%Second, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}). 


%=================================================================
%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
%=================================================================


%=================================================================
%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$}
%\label{sec:weff}
%=================================================================
%=================================================================
%\subsection{Effective Coulomb operator}
%=================================================================
We define the effective operator $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation}
	\label{eq:def_weebasis}
	\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{}(\br{1},\br{2})/\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})
\end{equation}
where  $\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})$ is the opposite-spin two-body density associated with $\wf{}{\Bas}$
\begin{equation}
	\label{eq:n2basis}
	\n{\wf{}{\Bas}}{(2)}(\br{1},\br{2})
	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}, 
\end{equation}
$\Gam{pq}{rs}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ is the opposite-spin two-body density tensor of $\wf{}{\Bas}$, $\SO{i}{}$ are spinorbitals, $\f{\wf{}{\Bas}}{}(\br{1},\br{2})$ is defined as
\begin{multline}
	\label{eq:fbasis}
	\f{\wf{}{\Bas}}{}(\br{1},\br{2}) 
	\\
	=  \sum_{pqrstu \in \Bas}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[\wf{}{\Bas}] \SO{t}{1} \SO{u}{2},
\end{multline}
and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals. 
With such a definition, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ verifies 
\begin{equation}
	\label{eq:int_eq_wee}
	\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
\end{equation}
where the $\hWee{}$ contains only the alpha-beta component of the two-electron interaction, and \eqref{eq:int_eq_wee} can be rewritten as
\begin{equation}
  \iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2} =  \iint \W{\wf{}{\Bas}}{}(\br{1},\br{2}) \n{\wf{}{\Bas}}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
\end{equation}
which intuitively motivates $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. 
As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries. 
An important quantity to define is $\W{\wf{}{\Bas}}{}(\br{})$ which is the value of the effective interaction at $\br{}$ for opposite spins at coalescence
\begin{equation}
  \label{eq:wcoal}
  \W{\wf{}{\Bas}}{}(\br{}) =  \W{\wf{}{\Bas}}{}(\br{},{\br{}})
\end{equation}
and which is necessarily \textit{finite} at for an \textit{incomplete} basis set. 

Of course, there exists \textit{a priori} an infinite set of functions in ${\rm I\!R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its very definition one can show (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that 
\begin{equation}
	\label{eq:lim_W}
	\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\br{1},\br{2}) = r_{12}^{-1}\quad \forall \,\,(\br{1},\br{2}),  
\end{equation}
which therefore guarantees a physically satisfying limit for all $\wf{}{\Bas}$. An important point here is that, with the present definition of $\W{\wf{}{\Bas}}{}(\br{1},\br{2})$, one can see the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.  

%=================================================================
%\subsection{Range-separation function}
%=================================================================
As we can map the Coulomb operator within a basis set $\Bas$ with a non divergent two-electron interaction, we can link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators. 
To do so, we choose a range-separation \textit{function} $\rsmu{\wf{}{\Bas}}{}(\br{})$
\begin{equation}
	\label{eq:mu_of_r}
        \rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{}),
\end{equation}
such that the long-range interaction $\w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2})$
\begin{equation}
        \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\wf{}{\Bas}}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\wf{}{\Bas}}{}(\br{2}) r_{12}]}{ r_{12}} }
\end{equation}
coincides with the effective interaction $\W{\wf{}{\Bas}}{}(\br{})$ for all points in ${\rm I\!R}^3$  
\begin{equation}
        \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\br{},\br{}) = \W{\wf{}{\Bas}}{}(\br{})\quad \forall \,\, \br{}. 
\end{equation}


%More precisely, if we define the value of the interaction at coalescence as 
%\begin{equation}
% \label{eq:def_wcoal}
% \wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
%\end{equation}
%where $(\br{},\Bar{\br{}})$ means a couple of anti-parallel spins at the same position $\br{}$, 
%we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
%\begin{equation}
%  \wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{})
%\end{equation}
%where the long-range-like interaction is defined as
%\begin{equation}
%	\w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }.  
%\end{equation}
%Equation \eqref{eq:def_wcoal} is equivalent to the following condition
%\begin{equation}
% \label{eq:mu_of_r}
% 	\rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{})
%\end{equation}
%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
%\begin{equation}
% \label{eq:mu_of_r_val}
% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
%\end{equation}
%An important point to notice is that, in the limit of a complete basis set $\Bas$, as 
%\begin{equation}
%\label{eq:lim_W}
%	\lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1}  \quad \forall  (\br{1},\br{2})
%% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , 
%\end{equation}
%one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$
%% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, 
%and therefore 
%\begin{equation}
%\label{eq:lim_mur}
%	\lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty
%%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
%\end{equation}



%=================================================================
%\subsection{Complementary functional}
%=================================================================
%\label{sec:ecmd}

Once defined the range-separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$, we can use the functionals defined in the field of RS-DFT to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline}
  \label{eq:ec_md_mu}
  \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
  \\
  - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
\end{multline}
where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
\begin{equation}
\label{eq:argmin}
	\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation}
with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
%and
%\begin{equation}
%\label{eq:erf}
%	\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
%\end{equation}
%is the long-range part of the Coulomb operator.
The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
\begin{subequations}
\begin{align}
	\label{eq:large_mu_ecmd}
	\lim_{\mu \to \infty}  \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] & = 0,
	\\
	\label{eq:small_mu_ecmd}
	\lim_{\mu \to 0}  \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] & = \Ec[\n{}{}(\br{})],
\end{align}
\end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. 
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.
This makes them particularly well adapted to the present context where one aims at correcting a general WFT method. 

%--------------------------------------------
%\subsubsection{Local density approximation}
%--------------------------------------------
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as 
\begin{equation}
	\label{eq:def_lda_tot}
        \bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
\end{equation}
where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. 
%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.

%--------------------------------------------
%\subsubsection{New PBE functional}
%--------------------------------------------
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 $\mu$. 
In order to correct such a defect, we propose here a new ECMD functional inspired by the recent functional proposed by some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
\begin{subequations}
\begin{gather}
	\label{eq:epsilon_cmdpbe}
	\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \rsmu{}{})\rsmu{}{3} },
	\\
	\label{eq:beta_cmdpbe}
	\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
\end{gather}
\end{subequations}
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 of the system $\n{}{(2)}$ by its UEG version, i.e.~$\n{}{(2)} \approx \n{\UEG}{(2)} =  \n{}{2} g_0(\n{}{})$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. 
This represents a major computational saving without loss of performance as we eschew the computation of $\n{}{(2)}$.
Therefore, the PBE complementary functional reads
\begin{equation}
	\label{eq:def_pbe_tot}
        \bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}.
\end{equation}
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ where $\rsmu{\wf{}{\Bas}}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.

%The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$  labelled by ECMD-$\mathcal{X}$. 
%Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.  
%A general scheme to approximate  $\ecompmodel$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and to evaluate it at the density defined by the model $\denmodel$
%\begin{equation}
% \label{eq:approx_ecfuncbasis}
% \ecompmodel \approx \ecmuapproxmurmodel
%\end{equation}
%Therefore, any approximated ECMD can be used to estimate $\ecompmodel$. 
%It is important to notice that in the limit of a complete basis set, according to equations \eqref{eq:lim_mur} and \eqref{eq:large_mu_ecmd} one has 
%\begin{equation}
% \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
%\end{equation}
%for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD. 
The main computational source of the present approach is the computation of $\W{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) for $\Gam{rs}{tu}[\wf{}{\Bas}]$ and therefore the computational bottleneck reduces to the evaluation at each quadrature grid point of 
\begin{equation}
	\label{eq:fcoal}
        f_{\text{HF}}^{\Bas}(\br{}) = \sum_{p,q\in\Bas} \sum_{i\in \nocca} \sum_{j\in \noccb} \V{pq}{ij} \SO{p}{} \SO{q}{} \SO{i}{} \SO{j}{}
\end{equation}
which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly parallel. Within the present formulation, the bottleneck is the four-index transformation to obtaine the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}, but this step has in general to be performed before a correlated WFT calculations. 
When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$  but where one can take advantage of the sparsity atomic-orbital-based algorithms to reach a linear scaling method.  


%=================================================================
%\subsection{Valence effective interaction}
%=================================================================
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals. 
We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.

%According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. 
Accounting solely for the valence electrons, we define 
\begin{multline}
	\label{eq:fbasisval}
	\f{\wf{}{\Bas}}{\Val}(\br{1},\br{2}) 
	\\
	=  \sum_{pq \in \Bas} \sum_{rstu \in \Val}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu}[{\wf{}{\Bas}}] \SO{t}{1} \SO{u}{2},
\end{multline}
and the valence part of the effective interaction is
\begin{subequations}
\begin{gather}
	\label{eq:Wval}
	\W{\wf{}{\Bas}}{\Val}(\br{1},\br{2}) = \f{\wf{}{\Bas}}{\Val}(\br{1},\br{2})/\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2}),
	\\
	\label{eq:muval}
	\rsmu{\wf{}{\Bas}}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{\Val}(\br{},\Bar{\br{}}),
\end{gather}
\end{subequations}
where 
\begin{equation}
	\n{\wf{}{\Bas},\Val}{(2)}(\br{1},\br{2})
	= \sum_{pqrs \in \Val}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs}[\wf{}{\Bas}] \SO{r}{1} \SO{s}{2}
\end{equation}
is the two body density associated to the valence electrons.
%\begin{equation}
% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
%\end{equation}
%It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$,  and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$. 
It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.

%We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models. 
%Defining the valence one-body spin density matrix as
%\begin{equation}
% \begin{aligned}
% \onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\
%                      & = 0 \qquad \text{in other cases}
% \end{aligned}
%\end{equation}
%then one can define the valence density as:
%\begin{equation}
%  \denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r}) 
%\end{equation}
%Therefore, we propose the following valence-only approximations for the complementary functional 
%\begin{equation}
% \label{eq:def_lda_tot}
%  \ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,,
%\end{equation}
%\begin{equation}
% \label{eq:def_lda_tot}
%  \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
%\end{equation}
Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.

To conclude this theory session, it is important to notice that in the limit of a complete basis set, because of the behaviour of the effective interaction [see \eqref{eq:lim_W}], the range separation function $\rsmu{\wf{}{\Bas}}{}(\br{})$ tends to infinity. Therefore, in the limit of a complete basis set, according to equation \eqref{eq:large_mu_ecmd}, the complementary functional tends to zero \textit{whatever the choice of functional}. 
Thefore in the limit of a complete basis set one recovers the correct limit of the WFT model whatever approximations are made in the DFT part, just like in equation \eqref{eq:limitfunc}. 

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
%%%%%%%%%%%%%%%%%%%%%%%%

%%% TABLE I %%%
\begin{table*}
	\caption{
		\label{tab:diatomics}
		Dissociation energy ($\De$) 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.
		}
		\begin{ruledtabular}
			\begin{tabular}{llddddd}
							&					&	\mc{4}{c}{Dunning's basis set}													
				\\
													\cline{3-6}
                  Molecule	&	Method				&	\tabc{$\X = \D$}	&	\tabc{$\X = \T$}	&	\tabc{$\X = \Q$}	&	\tabc{$\X = 5$}		&	\tabc{CBS}	
				\\
				\hline
				\ce{C2}		&	exFCI\fnm[1]		&	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+PBE\fnm[1]	&	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)+LDA\fnm[1]	&	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	)	&				\\	\\
				\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	)	&					\\	
							&	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       	\\
				(cc-pVXZ)	&	exFCI+LDA\fnm[1]	&	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	)	&					\\	
                            &	CCSD(T)\fnm[1]		&	199.9	(-27.9	)	&	216.3	(-11.5	)	&	222.8	(-5.0	)	&	225.0	(-2.8	)	&	227.2			\\
							&	CCSD(T)+LDA\fnm[1]	&	216.3	(-11.5	)	&	224.8	(-3.0	)	&	227.2	(-0.6	)	&	227.8	(+0.0	)	&					\\	
							&	CCSD(T)+PBE\fnm[1]	&	225.9	(-1.9	)	&	226.7	(-1.1	)	&	227.5	(-0.3	)	&	227.8	(+0.0	)	&				\\	\\
				\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	)	&					\\
							&	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       	\\
				(cc-pVXZ)	&	exFCI+LDA\fnm[1]	&	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	)	&					\\	
                            &	CCSD(T)\fnm[1]		&	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)+PBE\fnm[1]	&	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			\\
				(cc-pVXZ)	&	exFCI+LDA\fnm[1]	&	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	)	&					\\	
                            &	CCSD(T)\fnm[1]		&	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)+PBE\fnm[1]	&	31.5 	(-6.7	)	&	37.1 	(-1.1	)	&	37.8 	(-0.4	)	&	38.2 	(+0.0	)	&					\\	
		\end{tabular}
	\end{ruledtabular}
	\fnt[1]{Fronzen core calculations. Only valence 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*}

%%% TABLE II %%%
\begin{table}
	\caption{
	Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_AE}. 
	Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference data.
	\label{tab:stats}}
	\begin{ruledtabular}
		\begin{tabular}{lddd}
			Method					&	\tabc{MAD}	&	\tabc{RMSD}	&	\tabc{MAX}	\\
			\hline
			CCSD(T)/cc-pVDZ			&	22.81		&	25.82			&	72.08	\\
			CCSD(T)/cc-pVTZ			&	7.95		&	8.99			&	25.99	\\
			CCSD(T)/cc-pVQZ			&	3.24		&	3.67			&	11.66	\\
			CCSD(T)/cc-pV5Z			&	1.39		&	1.54			&	3.46	\\
			\\
			CCSD(T)+LDA/cc-pVDZ		&	11.75		&	13.99			&	54.88	\\
			CCSD(T)+LDA/cc-pVTZ		&	3.11		&	3.94			&	16.77	\\
			CCSD(T)+LDA/cc-pVQZ		&	0.87		&	1.36			&	6.22	\\
			\\
			CCSD(T)+PBE/cc-pVDZ		&	8.68		&	10.92			&	45.81	\\
			CCSD(T)+PBE/cc-pVTZ		&	2.66		&	3.52			&	14.73	\\
			CCSD(T)+PBE/cc-pVQZ		&	0.88		&	1.36			&	5.96	\\
		\end{tabular}
	\end{ruledtabular}
\end{table}

%%% FIGURE 1 %%%
\begin{figure*}
	\includegraphics[width=\linewidth]{VDZ}
	\includegraphics[width=\linewidth]{VTZ}
	\includegraphics[width=\linewidth]{VQZ}
	\caption{
	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.	
	\label{fig:G2_AE}}
\end{figure*}

%\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
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).
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, whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11} 
In a second time, we compute the entire atomization energies of the G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family.
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,PetTouUmr-JCP-12,FelPet-JCP-13,KesSylKohTewMar-JCP-18}).
%The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
As a method $\modX$ we employ either CCSD(T) or exFCI.
Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
In the case of the CCSD(T) calculations, we have $\modY = \HF$ as we use the Restricted Open Shel Hartree-Fock (ROHF) one-electron density to compute the complementary energy, and for the CIPSI calculations we use the density of a converged variational wave function. 
For the definition of the interaction, we use a single Slater determinant in the ROHF basis. 
The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}
RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
\titou{For the quadrature grid, we employ ... radial and angular points.}
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{B} to \ce{Mg}, while a \ce{Ne} core is frozen from \ce{Al} to \ce{Xe}.
In the context of the basis set correction, the set of valence spinorbitals $\Val$ involved in the definition of the effective interaction [see Eq.~\eqref{eq:Wval}] refers to the non-frozen spinorbitals.
The ``valence'' correction was used consistently when the FC approximation was applied.
In order to estimate the complete basis set (CBS) limit for each model, we employed the two-point extrapolation proposed in Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98} for the correlation energies. 
We refer to these atomization energies as $\CBS$. 

%\subsection{Convergence of the atomization energies with the WFT models }
As the exFCI calculations were converged with a precision of about 0.1 {\kcal}, we can consider these atomization energies as near-FCI values.
They will be our references for a given system in a given basis. 
The results for four diatomics mentioned above are reported in Table \ref{tab:diatomics}. 
As one can see, the convergence of the exFCI atomization energies is, as expected, slow with respect to the basis set: chemical accuracy (error below 1 {\kcal}) is barely reached for \ce{C2}, \ce{O2} and \ce{F2} even with cc-pV5Z. 
Also, the atomization energies are consistently underestimated, reflecting that, in a given basis, the atom is always better described than the molecule due to the larger number of interacting electron pairs in the molecular system. 
A similar trend holds for CCSD(T).
%, and one can notice that the atomization energies of the CCSD(T) are always slightly underestimated with respect to the CIPSI ones, showing that the CCSD(T) ansatz is better suited for the atoms than for the molecule. 

%\subsection{The effect of the basis set correction within the LDA and PBE approximation}
Regarding the effect of the basis set correction, several general observations can be made for both exFCI and CCSD(T). 
First, in a given basis set, the basis set correction systematically improves the result (both at the LDA and PBE level).
A small overestimation can occur compared to the CBS values by a few tenths of a {\kcal} (the largest deviation being 0.6 {\kcal} for \ce{N2} at the CCSD(T)+PBE/cc-pV5Z level). 
Nevertheless, the deviation observed for the largest basis set is typically within the extrapolation error of the CBS atomization energies, which is highly satisfactory knowing the marginal computation cost of the present correction.
%Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values. 
Importantly, the sensitivity with respect to the SR-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better. 
However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting information}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for raw data associated with the G2 atomization energies.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The authors would like to thank... nobody.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{G2-srDFT,G2-srDFT-control}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
