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\newcommand{\CBS}{\text{CBS}}                                                                                             


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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
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\begin{document}	

\title{A density-based basis-set correction for strong correlation}

\author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr}
\affiliation{\LCT}
\author{Barth\'el\'emy Pradines}
\affiliation{\LCT}
\affiliation{\ISCD}
\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Julien Toulouse}
\email{toulouse@lct.jussieu.fr}
\affiliation{\LCT}
\affiliation{\IUF}


\begin{abstract}
We extend to strongly correlated systems the recently introduced basis-set correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, J. Chem. Phys. \textbf{149}, 194301 (2018)]. This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corresponding to the Coulomb electron-electron interaction projected in the finite basis set, allowing one to use RSDFT-type complementary functionals to recover the dominant part of the short-range correlation effects missing in a finite basis set. Using as test cases the potential energy curves of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit, we systematically explore different approximations for the complementary functionals which are suited to describe strong-correlation regimes and which fulfill two very desirable properties: $S_z$ invariance and size consistency. Specifically, we investigate the dependence of the functionals on different flavours of on-top pair densities and spin polarizations. An important result is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy. 
In the general context of multiconfigurational DFT, this finding shows that one can avoid the effective spin polarization whose mathematical definition is rather \textit{ad hoc} and which can become complex valued. Quantitatively, we show that the basis-set correction reaches chemical accuracy on atomization energies with triple-zeta quality basis sets for most of the systems studied. Also, the present basis-set correction provides smooth curves along the whole potential energy curves.
%We study the potential energy surfaces (PES) of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit using increasing basis sets at near full configuration interaction (FCI) level with and without the present basis-set correction. 
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%
The general goal of quantum chemistry is to provide reliable theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular systems, but despite intense developments, no definitive solution to this problem has been found. The theoretical challenge to tackle belongs to the quantum many-body problem, due the intrinsic quantum nature of the electrons and the Coulomb repulsion between them. This so-called electronic correlation problem corresponds to finding a solution to the Schr\"odinger equation for a $N$-electron system, and two main roads have emerged to approximate this solution: wave-function theory (WFT)~\cite{Pop-RMP-99} and density-functional theory (DFT)~\cite{Koh-RMP-99}. Although both WFT and DFT spring from the same Schr\"odinger equation, they rely on very different formalisms, as the former deals with the complicated $N$-electron wave function whereas the latter focuses on the much simpler one-electron density. In its Kohn-Sham (KS) formulation~\cite{KohSha-PR-65}, the computational cost of DFT is very appealing since it is a simple mean-field procedure. Therefore, although continued efforts have been done to reduce the computational cost of WFT, DFT still remains the workhorse of quantum chemistry. 

The difficulty of obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation appearing in its electronic structure. The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometries, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak correlation effects can be either short range(near the electron-electron coalescence point) or long range (London dispersion interactions). The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining issue for these systems is to push the limit of the size of the chemical systems that can be treated.  The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations and WFT is king for the treatment of strongly correlated systems. 

In practice WFT uses a finite one-particle basis set (here denoted as $\basis$) to project the Schr\"odinger equation. The exact solution within the basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix; ii) the quantitative description of the system requires to also account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing these two objectives is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy and size consistency.

%To tackle this complicated problem, many methods have been proposed and an exhaustive review of the zoology of methods for strong correlation goes beyond the scope and purpose of this article. 

%To tackle this problem, many WFT methods have emerged which can be categorized in two branches: the single-reference (SR) 
%and multi-reference (MR) methods. 
%The SR methods rely on a single electronic configuration as a zeroth-order wave function, typically Hartree-Fock (HF). 
%Then the electron correlation is introduced by increasing the rank of multiple hole-particle excitations, 
%preferably treated in a coupled-cluster (CC) fashion for the sake of compactness of the wave function and extensivity of the computed energies. 
%The advantage of these approaches rely on the rather straightforward way to improve the level of accuracy, 
%which consists in increasing the rank of the excitation operators used to generate the CC wave function. 
%Despite its appealing elegant simplicity, the computational cost of the CC methods increase drastically with the rank of the excitation 
%operators, even if promising alternative approaches have been proposed using stochastic techniques\cite{Thom-PRL-10,ScoTho-JCP-17,SpeNeuVigFraTho-JCP-18,DeuEmiShePie-PRL-17,DeuEmiMagShePie-JCP-18,DeuEmiYumShePie-JCP-19} or symmetry-broken approaches\cite{QiuHenZhaScu-JCP-17,QiuHenZhaScu-JCP-18,GomHenScu-JCP-19}. 
%In the MR approaches, the zeroth order wave function consists in a linear combination of Slater determinants which are supposed to concentrate most of strong interactions and near degeneracies inherent in the structure of the Hamiltonian for a strongly correlated system. The usual approach to build such a zeroth-order wave function is to perform a complete active space self consistent field (CASSCF) whose variational property prevent any divergence, and which can provide extensive energies. Of course, the choice of the active space is rather a subtle art and the CASSCF results might strongly depend on the level of chemical/physical knowledge of the user.  
%On top of this zeroth-order wave function, weak correlation is introduced by the addition of other configurations through either configuration interaction\cite{WerKno-JCP-88,KnoWer-CPL-88} (MRCI) or perturbation theory (MRPT) and even coupled cluster (MRCC), which have their strengths and weaknesses, 
%The advantage of MRCI approaches rely essentially in their simple linear parametrisation for the wave function together with the variational property of their energies, whose inherent drawback is the lack of size extensivity of their energies unless reaching the FCI limit. On the other hand, MRPT and MRCC can provide extensive energies but to the price of rather complicated formalisms, and these approaches might be subject to divergences and/or convergence problems due to the non linearity of the parametrisation for MRCC or a too poor choice of the zeroth-order Hamiltonian. 
%A natural alternative is to combine MRCI and MRPT, which falls in the category of selected CI (SCI) which goes back to the late 60's and who has received a revival of interest and applications during the last decade \cite{BenErn-PhysRev-1969,WhiHac-JCP-1969,HurMalRan-1973,EvaDauMal-ChemPhys-83,Cim-JCP-1985,Cim-JCC-1987,IllRubRic-JCP-88,PovRubIll-TCA-92,BunCarRam-JCP-06,AbrSheDav-CPL-05,MusEngels-JCC-06,BytRue-CP-09,GinSceCaf-CJC-13,CafGinScemRam-JCTC-14,GinSceCaf-JCP-15,CafAplGinScem-arxiv-16,CafAplGinSce-JCP-16,SchEva-JCP-16,LiuHofJCTC-16,HolUmrSha-JCP-17,ShaHolJeaAlaUmr-JCTC-17,HolUmrSha-JCP-17,SchEva-JCTC-17,PerCle-JCP-17,OhtJun-JCP-17,Zim-JCP-17,LiOttHolShaUmr-JCP-2018,ChiHolOttUmrShaZim-JPCA-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,GarSceGinCaffLoo-JCP-18,SceGarCafLoo-JCTC-18,GarGinMalSce-JCP-16,LooBogSceCafJac-JCTC-19}. 
%Among the SCI algorithms, the CI perturbatively selected iteratively (CIPSI) can be considered as a pioneer. The main idea of the CIPSI and other related SCI algorithms is to iteratively select the most important Slater determinants thanks to perturbation theory in order to build a MRCI zeroth-order wave function which automatically concentrate the strongly interacting part of the wave function. On top of this MRCI zeroth-order wave function, a rather simple MRPT approach is used to recover the missing weak correlation and the process is iterated until reaching a given convergence criterion. It is important to notice that in the SCI algorithms, neither the SCI or the MRPT are size extensive \textit{per se}, but the extensivity property is almost recovered by approaching the FCI limit. 
%When the SCI are affordable, their clear advantage are that they provide near FCI wave functions and energies, whatever the level of knowledge of the user on the specific physical/chemical problem considered. The drawback of SCI is certainly their \textit{intrinsic} exponential scaling due to their linear parametrisation. Nevertheless, such an exponential scaling is lowered by the smart selection of the zeroth-order wave function together with the MRPT calculation. 

Beside the difficulties of accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energies and properties with respect to the size of the basis set. As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg and coworkers~\textit{et al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the Coulomb electron-electron interaction at the coalescence point, which induces a discontinuity in the first derivative of the exact wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete one-electron basis set is impossible and, as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques have been developed, either based on a partial-wave expansion analysis~\cite{HelKloKocNog-JCP-97,HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis-set convergence problem is provided by the so-called R12 and F12 explicitly correlated methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a geminal function explicitly depending on the interelectronic distances ensuring the correct cusp condition in the wave function, and lead to a much faster convergence of the correlation energies than usual WFT methods. For instance, using the explicitly correlated version of coupled cluster with singles, doubles, and perturbative triples (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to using a quintuple-$\zeta$ quality basis set with the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although a computational overhead is introduced by the auxiliary basis set needed to compute the three- and four-electron integrals involved in F12 theory. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires non-trivial developments for adapting it to a new method. For strongly correlated systems, several multi-reference methods have been extended to explicitly correlation (see for instance Ref.~\onlinecite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on the so-called universal F12 theory which are potentially applicable to any electronic-structure computational methods~\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}. 

An alternative way to improve the convergence towards the complete-basis-set (CBS) limit is to treat the short-range correlation effects within DFT and to use WFT methods to deal only with the long-range and/or strong-correlation effects. A rigorous approach achieving this mixing of DFT and WFT is range-separated DFT (RSDFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which relies on a splitting of the Coulomb electron-electron interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of this approach is at least two-fold: i) the DFT part deals primarily with the short-range part of the Coulomb interaction, and consequently the usual semilocal density-functional approximations are more accurate than for standard KS DFT; ii) the WFT part deals only with a smooth non-divergent interaction, and consequently the wave function has no electron-electron cusp\cite{GorSav-PRA-06} and the basis-set convergence is much faster\cite{FraMusLupTou-JCP-15}. A number of approximate RSDFT schemes have been developed involving single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15,KalTou-JCP-18,KalMusTou-JCP-19} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT methods. Nevertheless, there are still some open issues in RSDFT, such as remaining fractional-charge and fractional-spin errors in the short-range density functionals~\cite{MusTou-MP-17} or the dependence of the quality of the results on the value of the range-separation parameter $\mu$.
% which can be seen as an empirical parameter. 

Building on the development of RSDFT, a possible solution to the basis-set convergence problem has been recently proposed by some of the present authors~\cite{GinPraFerAssSavTou-JCP-18} where RSDFT functionals are used to recover only the correlation effects outside a given basis set. The key point here is to realize that a wave function developed in an incomplete basis set is cuspless and could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, a mapping with RSDFT can be introduced through the introduction of an effective non-divergent interaction representing the usual Coulomb electron-electron interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried out~\cite{LooPraSceTouGin-JCPL-19}, together with extensions of this approach to the calculations of excitation energies~\cite{GinSceTouLoo-JCP-19} and ionization potentials~\cite{LooPraSceGinTou-ARX-19}. The goal of the present work is to further develop this approach for the description of strongly correlated systems. The paper is organized as follows. In Section \ref{sec:theory} we recall the mathematical framework of the basis-set correction and we present the extension for strongly correlated systems. In particular, we focus on imposition of two important formal properties: size-consistency and spin-multiplet degeneracy.
Then, in Section \ref{sec:results} we apply the method to the calculation of the potential energy curves of the C$_2$, N$_2$, O$_2$, F$_2$, and H$_{10}$ molecules up to the dissociation limit, representing prototypes of strongly correlated systems. Finally, we conclude in Section \ref{sec:conclusion}. 

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%
As the theory of the basis-set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Section \ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$. In Section \ref{sec:wee} we introduce the effective non-divergent interaction in the basis set $\Bas$, which leads us to the definition of the effective local range-separation parameter in Section \ref{sec:mur}. Then, in Section \ref{sec:functional} we expose the new approximate complementary functionals based on RSDFT. The generic form of such functionals is exposed in Section \ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Section \ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Section \ref{sec:requirements}. Finally, the actual form of the functionals used in this work are introduced in Section  \ref{sec:final_def_func}. 

\subsection{Basic formal equations}
\label{sec:basic}

The exact ground-state energy $E_0$ of a $N$-electron system can in principle be obtained in DFT by a minimization over $N$-electron density $\denr$
\begin{equation}
 \label{eq:levy}
 E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\},
\end{equation}
where $v_{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the universal Levy-Lieb density functional written with the constrained search formalism as~\cite{Lev-PNAS-79,Lie-IJQC-83}
\begin{equation}
 \label{eq:levy_func}
 F[\den] = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi},
\end{equation}
where the notation $\Psi \rightarrow \den$ means that the wave function $\Psi$ yields the density $n$. The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', i.e. densities coming from wave functions expandable in the Hilbert space generated by the basis set $\Bas$. In the following, we always implicitly consider only such densities representable in $\Bas$. With this restriction, Eq.~\eqref{eq:levy} gives then an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a fast convergence with the size of the basis set, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is a very good approximation to the exact ground-state energy: $E_0^\Bas \approx E_0$.

In the present context, it is important to notice that in the definition of Eq.~\eqref{eq:levy_func} the wave functions $\Psi$ involved have no restriction to a finite basis set, i.e. they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then propose to decompose $F[\den]$ as
\begin{equation}
 \label{eq:def_levy_bas}
 F[\den] = \min_{\wf{}{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den}, 
\end{equation}
where $\wf{}{\Bas}$ are wave functions expandable in the Hilbert space generated by $\basis$, and $\efuncden{\den}$ is the density functional complementary to the basis set $\Bas$ defined as
\begin{equation}
 \begin{aligned}
 \efuncden{\den} = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi}  
                  - \min_{\Psi^{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}. 
 \end{aligned}
\end{equation}
Introducing the decomposition in Eq. \eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy}, we arrive at the following expression for $E_0^\Bas$
\begin{eqnarray}
 \label{eq:E0basminPsiB}
 E_0^\Bas &=& \min_{\Psi^{\Bas}} \bigg\{ \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}} 
\nonumber\\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\},
\end{eqnarray}
where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density extracted from $\wf{}{\Bas}$. Therefore, with Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional accounting for the correlation effects not included in the basis set.

As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy (see Eqs. (12)-(15) of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation}
 \label{eq:e0approx}
 E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI,
\end{equation}
where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation regimes. Two requirements on the approximations for this purpose are i) size consistency and ii) spin-multiplet degeneracy.

\subsection{Definition of an effective interaction within $\Bas$}
\label{sec:wee}
As originally shown by Kato\cite{Kat-CPAM-57}, the cusp in the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. In other words, the impact of the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction at the coalescence point. 

As originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (Section D and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
\begin{equation}
 \label{eq:wbasis}
 \wbasis =
    \begin{cases}
      \fbasis /\twodmrdiagpsi,    & \text{if $\twodmrdiagpsi \ne 0$,}
\\
       \infty,                                                                                          & \text{otherwise,}
    \end{cases}
\end{equation}
where $\twodmrdiagpsi$ is the opposite-spin pair density associated with $\wf{}{\Bas}$ 
\begin{equation}
 \twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, 
\end{equation}
and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, and $\fbasis$ is
\begin{equation}
        \label{eq:fbasis}
        \fbasis
        = \sum_{pqrstu \in \Bas}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation}
with the usual two-electron Coulomb integrals $\V{pq}{rs}=\langle pq | rs \rangle$.
With such a definition, one can show that $\wbasis$ satisfies 
\begin{eqnarray}
 \frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi =
\nonumber\\
\frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}. 
\end{eqnarray}
As shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, i.e.
\begin{equation}
 \label{eq:cbs_wbasis}
 \lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|},\quad \forall\,\psibasis. 
\end{equation}
The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the correct behavior of the theory in the CBS limit.

\subsection{Definition of a local range-separation parameter}
\label{sec:mur}
\subsubsection{General definition}
As the effective interaction within a basis set, $\wbasis$, is non divergent, it ressembles the long-range interaction used in RSDFT
\begin{equation}
 \label{eq:weelr}
 w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}, 
\end{equation}
where $\mu$ is the range-separation parameter. As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondance between these two interactions by using the local range-separation parameter $\murpsi$
\begin{equation}
 \label{eq:def_mur}
 \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal, 
\end{equation}
such that the interactions coincide at the electron-electron colescence point for each $\br{}$
\begin{equation}
 w_\text{ee}^{\lr}(\murpsi;0) = \wbasiscoal, \quad \forall \, \br{}. 
\end{equation}
Because of the very definition of $\wbasis$, one has the following property in the CBS limit (see Eq.~\eqref{eq:cbs_wbasis}) 
\begin{equation}
 \label{eq:cbs_mu}
  \lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis, 
\end{equation}
which is again fundamental to guarantee the correct behavior of the theory in the CBS limit. 

\subsubsection{Frozen-core approximation}
As all WFT calculations in this work are performed within the frozen-core approximation, we use the valence-only version of the various quantities needed for the complementary basis functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as
\begin{equation}
 \label{eq:def_mur_val}
 \murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},  
\end{equation}
where $\wbasisval$ is the valence-only  effective interaction defined as 
\begin{equation}
 \label{eq:wbasis_val}
 \wbasisval =
    \begin{cases}
      \fbasisval /\twodmrdiagpsival,    & \text{if $\twodmrdiagpsival \ne 0$,}
\\
       \infty,                                                                                          & \text{otherwise,}
    \end{cases}
\end{equation}
where $\fbasisval$ and $\twodmrdiagpsival$ are defined as 
\begin{equation}
        \label{eq:fbasis_val}
        \fbasisval
        = \sum_{pq\in \Bas} \sum_{rstu \in \BasFC}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation}
and
\begin{equation}
 \label{eq:twordm_val}
 \twodmrdiagpsival = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}. 
\end{equation}
Notice the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}. 
It is noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.

\subsection{Generic form and properties of the approximations for $\efuncden{\den}$ }
\label{sec:functional}

\subsubsection{Generic form of the approximate functionals}
\label{sec:functional_form}

As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}~\cite{TouGorSav-TCA-05}. Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarisation $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form
\begin{equation}
 \begin{aligned}
 \label{eq:def_ecmdpbebasis}
 &\efuncdenpbe{\argebasis} =  \;\;\;\;\;\;\;\; \\ &\int  \d\br{} \,\denr   \ecmd(\argrebasis),
 \end{aligned}
\end{equation}
where $\ecmd(\argecmd)$ is the correlation energy per particle taken as
\begin{equation}
 \label{eq:def_ecmdpbe}
 \ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe) \; \mu^3},
\end{equation}
with 
\begin{equation}
 \label{eq:def_beta}
 \beta(\argepbe) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den}, 
\end{equation}
where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle~\cite{PerBurErn-PRL-96}. Before introducing the different flavors of approximate functionals that we will use here (see Section~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form. 

The functional form of $\ecmd(\argecmd)$ in Eq.~\ref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional
\begin{equation}
   \lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe),   
\end{equation}
which is relevant in the weak-correlation (or high-density) limit. In the large-$\mu$ limit, it behaves as
\begin{equation}
 \label{eq:lim_mularge}
    \ecmd(\argecmd) \isEquivTo{\mu\to\infty} \frac{2\sqrt{\pi}(1 - \sqrt{2})}{3 \mu^3}  \frac{\ntwo}{n},
\end{equation}
which is the exact large-$\mu$ behavior of the exact ECMD correlation energy~\cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18}. Of course, for a specific system, the large-$\mu$ behavior will be exact only if one uses for $n_2$ the \textit{exact} on-top pair density of this system. This large-$\mu$ limit in Eq.~\eqref{eq:lim_mularge} is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in Eq. \eqref{eq:def_ecmdpbe} plays indeed a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime have been also recently acknowledged by Gagliardi and coworkers~\cite{CarTruGag-JPCA-17} and Pernal and coworkers\cite{GritMeePer-PRA-18}.

Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes 
\begin{equation}
 \label{eq:lim_n2}
  \lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0,
\end{equation}
which is expected for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ molecule which is the archetype of strongly correlated systems.  Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$ like all RSDFT short-range functionals \begin{equation}
 \label{eq:lim_muinf}
  \lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0. 
\end{equation}

\subsubsection{Properties of approximated functionals}
\label{sec:functional_prop}
Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximated complementary basis set functionals $\efuncdenpbe{\argecmd}$ satisfies two important properties. 
Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncdenpbe{\argecmd}$ vanishes when reaching the complete basis set limit, whatever the wave function $\psibasis$ used to define the range separation parameter $\mu_{\Psi^{\basis}}$: 
\begin{equation}
 \label{eq:lim_ebasis}
 \lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argecmd} = 0\quad \forall\, \psibasis, 
\end{equation}
which guarantees an unaltered limit when reaching the CBS limit. 
Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system. 
Such a property is guaranteed independently by i) the definition of the effective interaction  $\wbasis$ (see equation \eqref{eq:wbasis}) together with the condition \eqref{eq:lim_muinf}, ii) the fact that the $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see equation \eqref{eq:lim_n2}). 

\subsection{Requirements for the approximated functionals in the strong correlation regime}
\label{sec:requirements}
\subsubsection{Requirements: separability of the energies and $S_z$ invariance}
An important requirement for any electronic structure method is the extensivity of the energy, \textit{i. e.} the additivity of the energies in the case of non interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. 
When two subsystems $A$ and $B$ dissociate in closed shell systems, as in the case of weak interactions for instance, a simple RHF wave function leads to extensive energies. 
When the two subsystems dissociate in open shell systems, such as in covalent bond breaking, it is well known that the RHF approach fail and an alternative is to use a CASSCF wave function which, provided that the active space has been properly chosen, leads to additives energies.  
Another important requirement is the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function.   
Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components.

\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance}
A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which in the case of $\ecmd(\argecmd)$ is the spin polarisation $\zeta(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see equation \eqref{eq:def_ecmdpbe}). 
As originally shown by Perdew and co-workers\cite{PerSavBur-PRA-95}, the dependence on the spin polarisation in the KS-DFT framework can be removed by the rewriting the spin polarisation of a single Slater determinant with only the on-top pair density and the total density. In other terms, the spin density dependence usually introduced in the correlation functionals of KS-DFT tries to mimic the effect of the on-top pair density. 
Based on this reasoning, a similar approach has been used in the context of multi configurational DFT in order to remove the $S_z$ dependency. 
In practice, these approaches introduce the effective spin polarisation 
\begin{equation}
 \label{eq:def_effspin}
 \tilde{\zeta}(n,\ntwo^{\psibasis}) =  
%   \begin{cases}
                       \sqrt{ n^2 - 4 \ntwo^{\psibasis} }
%                              0              & \text{otherwise.}
%   \end{cases}
\end{equation}
which uses the on-top pair density $\ntwo^{\psibasis}$ of a given wave function $\psibasis$. 

The advantages of this approach are at least two folds: i) the effective spin  polarisation $\tilde{\zeta}$ is $S_z$ invariant, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation. 
Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo^{\psibasis}<0$ and also 
the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo^{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions.  

An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional. 

\subsubsection{Conditions on $\psibasis$ for the extensivity}
In the case of the present basis set correction, as $\efuncdenpbe{\argebasis}$ is an integral over $\mathbb{R}^3$ of local quantities, in the case of non overlapping fragments $A\ldots B$ it can be written as the sum of two local contributions: one coming from the integration over the region of the sub-system $A$ and the other one from the region of the sub-system $B$. 
Therefore, a sufficient condition for the extensivity is that these quantities coincide in the isolated systems and in the subsystem of the super system $A\ldots B$. 
As $\efuncdenpbe{\argebasis}$ depends only on quantities which are properties of the wave function $\psibasis$, a sufficient condition for the extensivity of these quantities is that the function factorise in the limit of non-interacting fragments, that is $\Psi_{A\ldots B}^{\basis} = \Psi_A^{\basis} \Psi_B^{\basis}$. 
In the case where the two subsystems $A$ and $B$ dissociate in closed shell systems, a simple HF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CASSCF wave function is sufficient to recover this property. 
The condition for the active space involved in the CASSCF wave function is that it has to lead to extensive energies in the limit of dissociated fragments. 


\subsection{Different types of approximations for the functional}
\label{sec:final_def_func}
\subsubsection{Definition of the protocol to design functionals}
As the present work focusses on the strong correlation regime, we propose here to investigate only approximated functionals which are $S_z$ invariant and size extensive in the case of covalent bond breaking. Therefore, the wave function $\psibasis$ used throughout this paper are of CASSCF type in order to ensure extensivity of all density related quantities. 
The difference between the different flavours of functionals are only on i) the type of on-top pair density used, and ii) the type of spin polarisation used. 

Regarding the spin polarisation that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$ invariant formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in equation \eqref{eq:def_effspin}, and iii) a \textit{zero} spin polarization. 

Regarding the approximation to the \textit{exact} on-top pair density entering in equation \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads 
\begin{equation}
 \label{eq:def_n2ueg}
 \ntwo^{\text{UEG}}(n,\zeta,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big)
\end{equation}
where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in equation \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of equation \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density. 

Another approach to approximate of the exact on top pair density consists in taking advantage of the on-top pair density of the wave function $\psibasis$. Following the work of some of the previous authors\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
\begin{equation}
 \label{eq:def_n2extrap}
 \ntwoextrap(\ntwo^{\psibasis},\mu,\br{}) = \ntwo^{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
\end{equation}
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. 
When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ot". 


\subsubsection{Definition of functionals with good formal properties}
\label{sec:def_func}
We define the following functionals: 
i) The PBE-UEG-$\tilde{\zeta}$ which uses the UEG-like on-top pair density defined in equation \eqref{eq:def_n2ueg}, the effective spin polarization of equation \eqref{eq:def_effspin} and which reads
\begin{equation}
 \label{eq:def_pbeueg}
 \begin{aligned}
 \pbeuegXi =  &\int  d\br{} \,\denr  \\ & \ecmd(\argrpbeuegXi), 
 \end{aligned}
\end{equation}
ii) the PBE-ot-$\tilde{\zeta}$ where the on-top pair density of equation \eqref{eq:def_n2extrap} is used and which reads 
\begin{equation}
 \label{eq:def_pbeueg}
 \begin{aligned}
 \pbeontXi =  &\int  d\br{} \,\denr  \\ & \ecmd(\argrpbeontXi),  
 \end{aligned}
\end{equation}
iii) and the PBE-ot-$0{\zeta}$ where no spin polarization is used and which therefore uses only the total density and the on-top pair density of equation \eqref{eq:def_n2extrap} and which reads 
\begin{equation}
 \label{eq:def_pbeueg}
 \begin{aligned}
 \pbeontns  =  &\int  d\br{} \,\denr  \\ & \ecmd(\argrpbeontns).  
 \end{aligned}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results for the C$_2$, N$_2$, O$_2$, F$_2$ and H$_{10}$ potential energy curves}
\label{sec:results}
\subsection{Computational details}
The purpose of the present paper being the study of the basis set correction in the regime of strong correlation, we propose to study the potential energy surfaces (PES) until dissociation of an equally distant H$_{10}$ chain, together with the C$_2$, N$_2$, O$_2$ and F$_2$ molecules. 
In a given basis set, to compute the approximation of the exact ground state energy using equation \eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the complementary basis set energy functional $\efuncbasisFCI$. 

In the case of C$_2$, N$_2$, O$_2$ and F$_2$, the approximation to the FCI energies are obtained using converged frozen-core (1s orbitals are kept frozen) CIPSI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.} 
(see Refs \onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the Quantum Package software\cite{QP2}. The estimated exact PES are obtained from Ref. \onlinecite{LieCle-JCP-74a}. 
For all geometry and basis sets, the error with respect to actual FCI energies are estimated to be below 0.5 mH. 
In the case of H$_{10}$, the approximation to $\efci$ together with the estimated exact curves are obtained from the data from of Ref. \onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space). 

Regarding the complementary basis set energy functional, we use a full valence CASSCF wave functions computed with the GAMESS-US software\cite{gamess} to obtain the wave functions $\psibasis$. Therefore, all density related quantities (such as the total densities, different flavors of spin polarizations and on-top pair densities) together with the $\murpsi$ of equation \eqref{eq:def_mur} are obtained at full valence CASSCF level. 
These CASSCF wave functions correspond to the following active spaces: ten electrons in ten orbitals for H$_{10}$, 8 electrons in 8 electrons for C$_2$, 10 electrons in 8 orbitals for N$_2$, twelve electrons in eight orbitals for O$_2$ and forteen electrons in eight orbitals for F$_2$. 

Also, as the frozen core approximation is used in all near FCI calculations, we use the corresponding valence-only complementary functionals. Therefore, all density related quantities exclude any contribution from the core $1s$ orbitals, and the range-separation parameter is taken as the one defined in equation \eqref{eq:def_mur_val}. 

\subsection{Dissociation of equally distant H$_{10}$ chains}
The study of equally distant H$_{10}$ chains is a good prototype for the study of strong correlation regime as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations can be performed at near CBS values can be obtained (see Ref. \onlinecite{h10_prx} for detailed study of that problem). 

We report in figures \ref{fig:H10_vdz}, \ref{fig:H10_vtz}, \ref{fig:H10_vqz}  the PES computed using the cc-pVXZ (X=D,T,Q) basis sets of H$_{10}$, for different levels of approximations. 
The computation of the atomization energies $D_0$ at each level of theory used here is reported in table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy surfaces are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no divergence of bizarre behaviour are found when stretching the bonds, which shows that the functionals are robust when reaching the strong correlation regime.  

More quantitatively, the values of $D_0$ are within the chemical accuracy (\textit{i. e.} an error below 1.4 mH) from the cc-pVTZ basis set when using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, whereas such accuracy is not reached at the cc-pVQZ basis set using MRCI+Q. 

Regarding in more details the performance of the different types of approximated functionals, the results show that the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference being 0.3 mH on $D_0$), and they give slightly more accurate than the PBE-UEG-$\tilde{\zeta}$. 
These observations bring two important clues on the role of the different physical ingredients used in the functionals: 
i) the explicit use of the on-top pair density coming from the CASSCF wave function (see equation \eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see equation \eqref{eq:def_n2ueg}),
ii) removing the dependence on any kind of spin polarizations does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. The point ii) is important as it shows that the use of the spin-polarization in density functional approximations (DFA) essentially plays the role of the effect of the on-top pair density.  

\subsection{Dissociation of C$_2$, N$_2$, O$_2$ and F$_2$}
The C$_2$, N$_2$, O$_2$ and F$_2$ molecules are complementary to the H$_{10}$ system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of H$_{10}$, but also these systems exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of $D_0$, while the shape of the curve far from the equilibrium geometry is governed by dispersion forces which are medium to long-range weak correlation effects. 
Also, O$_2$ exhibit a triplet ground state and is therefore a good check for the performance of the dependence on the spin polarization of various types of functionals proposed here.

We report in figures \ref{fig:C2_avdz}, \ref{fig:N2_avdz}, \ref{fig:O2_avdz} and \ref{fig:F2_avdz} (\ref{fig:C2_avtz},  \ref{fig:N2_avtz}, \ref{fig:O2_avtz}  and  \ref{fig:F2_avtz}) the potential energy curves computed using the aug-cc-pVDZ (aug-cc-pVTZ) basis sets of C$_2$, N$_2$, O$_2$ and N$_2$, respectively, for different levels of computations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in table \ref{tab:d0}. 


Just as the case of H$_{10}$, the quality of $D_0$ are globally improved by adding the basis set correction and it is remarkable that the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals  give very similar results. 
The latter observation confirms that the dependence on the on-top pair density allows to remove the dependence of any kind of spin polarizations for a quite wide spread of electron density and also for high spin systems as O$_2$. 
More quantitatively, an error below 1.0 mH on the estimated exact valence-only $D_0$ is found for N$_2$, O$_2$ and F$_2$ in aug-cc-pVTZ with the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at near FCI level. 
In the case of C$_2$ in the aug-cc-pVTZ basis set, an error of about 5.5 mH is found with respect to the estimated exact $D_0$. Such an error is remarkably large with respect to the other diatomic molecules studied here and might be associated to the level of strong correlation of the C$_2$ molecule. 

Regarding now the performance of the basis set correction along the whole PES, it is interesting to notice that it fails to provide a noticeable improvement of the PES far from the equilibrium geometry. 
Acknowledging that the weak correlation effects in these regions are dominated by dispersion forces which are long-range effects, the failure of the present approximations for the complementary basis set functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics near the electron-electron cusp: the $\murpsi$ is designed by looking at the electron coalescence point and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis set correction to describe dispersion forces can be considered as a good behaviour. 

\begin{table*}
\label{tab:d0}
\caption{$D_0$ in mH and associated error with respect to the estimated exact values computed at different levels in various basis sets.   \\
        $^a$: The MRCI+Q and estimated exact curves are obtained from Ref. \onlinecite{h10_prx}.  \\
        $^b$: The estimated exact $D_0$ are obtained from the extrapolated valence-only non relativistic calculations of Ref. \onlinecite{BytLaiRuedenJCP05}. 
        }
\begin{ruledtabular}
\begin{tabular}{lcccc}

   System/basis             &  MRCI+Q$^a$                &   (MRCI+Q)+$\pbeuegXi$   &   (MRCI+Q)+$\pbeontXi$   &     (MRCI+Q)+$\pbeontns$    \\
\hline                                                                                                                 
H$_{10}$, cc-pvdz           &  622.1$/$43.3             &   642.6$/$22.8           &   649.2$/$16.2           &     649.5$/$15.9            \\
H$_{10}$, cc-pvtz           &  655.2$/$10.2             &   661.9$/$3.5            &   666.0$/$-0.6           &     666.0$/$-0.6            \\
H$_{10}$, cc-pvqz           &  661.2$/$4.2              &   664.1$/$1.3            &   666.4$/$-1.0           &     666.5$/$-1.1            \\
 \hline                                                                
 & \multicolumn{4}{c}{Estimated exact$^a$} \\
 & \multicolumn{4}{c}{665.4 } \\
\hline                                                                                                                 
   System/basis             &  FCI                    &   FCI+$\pbeuegXi$   &   FCI+$\pbeontXi$   &     FCI+$\pbeontns$    \\
\hline                                                                                                                 
C$_2$, aug-cc-pvdz          &   204.6$/$29.5          &   218.0$/$16.1      &   217.4$/$16.7      &     217.0$/$17.1            \\
C$_2$, aug-cc-pvtz          &   223.4$/$10.9          &   228.1$/$6.0       &   228.6$/$5.5       &     226.5$/$5.6             \\
 \hline                                                                
 & \multicolumn{4}{c}{Estimated exact$^b$} \\
 & \multicolumn{4}{c}{234.1 } \\
\hline                                                                                                                 
   System/basis             &  FCI                    &   FCI+$\pbeuegXi$   &   FCI+$\pbeontXi$   &     FCI+$\pbeontns$    \\
\hline                                                                                                                 
N$_2$, aug-cc-pvdz          &   321.9$/ $40.8            &   356.2$/$6.5            &   355.5$/$7.2            &     354.6$/$ 8.1            \\
N$_2$, aug-cc-pvtz          &   348.5$/$14.2             &   361.5$/$1.2            &   363.5$/$-0.5           &     363.2$/$-0.3            \\
 \hline                                                                
 & \multicolumn{4}{c}{Estimated exact$^b$} \\
 & \multicolumn{4}{c}{362.7 } \\
\hline                                                                                                                 
   System/basis             &  FCI                    &   FCI+$\pbeuegXi$   &   FCI+$\pbeontXi$   &     FCI+$\pbeontns$    \\
\hline                                                                                                                 
O$_2$, aug-cc-pvdz          &   171.4$/$20.5          &   187.6$/$4.3       &   187.6$/$4.3       &     187.1$/$4.8            \\
O$_2$, aug-cc-pvtz          &   184.5$/$7.4           &   190.3$/$1.6       &   191.2$/$0.7       &     191.0$/$0.9             \\
 \hline                                                                
 & \multicolumn{4}{c}{Estimated exact$^b$} \\
 & \multicolumn{4}{c}{191.9 } \\
\hline                                                                                                                 
F$_2$, aug-cc-pvdz          &   49.6$/$12.6           &    54.8$/$7.4       &   54.9$/$7.3        &     54.8$/$7.4              \\
F$_2$, aug-cc-pvtz          &   59.3$/$2.9            &    61.2$/$1.0       &   61.5$/$0.7        &     61.5$/$0.7              \\
 \hline                                                                
 & \multicolumn{4}{c}{Estimated exact$^b$} \\
 & \multicolumn{4}{c}{62.2 } \\
\end{tabular}
\end{ruledtabular}

\label{tab:extensiv_closed}
\end{table*}


\begin{figure}
        \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat.eps}\\
        \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat_zoom.eps}
        \caption{
        H$_{10}$, cc-pvdz: Comparison between MRCI+Q and corrected MRCI+Q energies and the estimated exact one. 
        The MRCI+Q and estimated exact values are obtained from Ref. \onlinecite{h10_prx}. 
        \label{fig:H10_vdz}}
\end{figure}


\begin{figure}
        \includegraphics[width=\linewidth]{data/H10/DFT_vtzE_relat.eps}\\
        \includegraphics[width=\linewidth]{data/H10/DFT_vtzE_relat_zoom.eps}
%       \includegraphics[width=\linewidth]{fig2c}
        \caption{
        H$_{10}$, cc-pvtz: Comparison between MRCI+Q and corrected MRCI+Q energies and the estimated exact one. 
        The MRCI+Q and estimated exact values are obtained from Ref. \onlinecite{h10_prx}. 
        \label{fig:H10_vtz}}
\end{figure}


\begin{figure}
        \includegraphics[width=\linewidth]{data/H10/DFT_vqzE_relat.eps}\\
        \includegraphics[width=\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}
%       \includegraphics[width=\linewidth]{fig2c}
        \caption{
        H$_{10}$, cc-pvqz: Comparison between MRCI+Q and corrected MRCI+Q energies and the estimated exact one. 
        The MRCI+Q and estimated exact values are obtained from Ref. \onlinecite{h10_prx}. 
        \label{fig:H10_vqz}}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
        \includegraphics[width=\linewidth]{data/C2/DFT_avdzE_relat.eps}
        \includegraphics[width=\linewidth]{data/C2/DFT_avdzE_relat_zoom.eps}
        \caption{
        C$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:C2_avdz}}
\end{figure}

\begin{figure}
        \includegraphics[width=\linewidth]{data/C2/DFT_avtzE_relat.eps}
        \includegraphics[width=\linewidth]{data/C2/DFT_avtzE_relat_zoom.eps}
        \caption{
        C$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:C2_avtz}}
\end{figure}


\begin{figure}
        \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat.eps}
        \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat_zoom.eps}
        \caption{
        N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:N2_avdz}}
\end{figure}

\begin{figure}
        \includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat.eps}
        \includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps}
        \caption{
        N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:N2_avtz}}
\end{figure}

\begin{figure}
        \includegraphics[width=\linewidth]{data/O2/DFT_avdzE_relat.eps}
        \includegraphics[width=\linewidth]{data/O2/DFT_avdzE_relat_zoom.eps}
        \caption{
        O$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:O2_avdz}}
\end{figure}

%\begin{figure}
%        \includegraphics[width=\linewidth]{data/O2/DFT_avtzE_relat.eps}
%        \includegraphics[width=\linewidth]{data/O2/DFT_avtzE_relat_zoom.eps}
%        \caption{
%        F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
%        \label{fig:O2_avtz}}
%\end{figure}

\begin{figure}
        \includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat.eps}
        \includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat_zoom.eps}
%       \includegraphics[width=\linewidth]{data/F2/DFT_avdzE_error.eps}\\
%       \includegraphics[width=\linewidth]{fig2c}
        \caption{
        F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:F2_avdz}}
\end{figure}

\begin{figure}
        \includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat.eps}
        \includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat_zoom.eps}
%       \includegraphics[width=\linewidth]{data/F2/DFT_avtzE_error.eps}\\
%       \includegraphics[width=\linewidth]{fig2c}
        \caption{
        F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:F2_avtz}}
\end{figure}


\section{Conclusion}
\label{sec:conclusion}
In the present paper we have extended the recently proposed DFT-based basis set correction to strongly correlated systems. 
We studied the H$_{10}$, C$_2$, N$_2$, O$_2$ and F$_2$ linear molecules up to full dissociation limits at near FCI level in increasing basis sets, and investigated how the basis set correction affects the convergence toward the CBS limits of the PES of these molecular systems. 

The DFT-based basis set correction relies on three aspects: 
i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the coulomb interaction projected into an incomplete basis set $\basis$, 
ii) the fitting of such effective interaction with a long-range interaction used in RS-DFT, 
iii) the use of complementary correlation functional of RS-DFT. 
In the present paper, we investigated points i) and iii) in the context of strong correlation and focussed on PES and atomization energies. 
More precisely, we proposed a new scheme to design functionals fulfilling a) $S_z$ invariance, b) size extensivity. To achieve such requirements we proposed to use CASSCF wave functions leading to extensive energies, and to develop functionals using only $S_z$ invariant density-related quantities. 

The development of new $S_z$ invariant and size extensive functionals has lead us to investigate the role of two related quantities: the spin-polarization and the on-top pair density. 
One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can avoid dependence to any form of spin-polarization without loss of accuracy. 
This avoids the commonly used effective spin-polarization with multi-configurational wave function whose mathematical definition originally proposed by Perdrew and co-workers in Ref. \cite{PerSavBur-PRA-95} has only a clear mathematical ground for a single Slater determinant and can be become complex-valued in the case of multi-configurational wave functions. From a more fundamental aspect, this shows that the spin-polarization in DFT-related frameworks only mimic's the role of the on-top density. 

Regarding the results of the present approach, the basis set correction systematically improves the near FCI calculation in a given basis set. More quantitatively, it is shown that the atomization energy $D_0$ is within the chemical accuracy for all systems but C$_2$ within a triple zeta quality basis set, whereas the near FCI values are far from that accuracy within the same basis set. 
In the case of C$_2$, an error of 5.5 mH is obtained with respect to the estimated exact $D_0$, and we leave for further study the detailed investigation of the reasons of this relatively unusual poor performance of the basis set correction. 

Also, it is shown that the basis set correction gives substantial differential contribution along the PES only close to the equilibrium geometry, meaning that it cannot recover the dispersion forces missing because the incompleteness of the basis set. Although it can be looked as a failure of the basis set correction, in our context such behaviour is actually preferable as the dispersion forces are long-range effects and the present approach was designed to recover electronic correlation effects near the electron coalescence.  

Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary materials) that it is minor with respect to WFT methods for all systems and basis set studied here. We believe that such approach is a significant step towards calculations near the CBS limit for strongly correlated systems. 

\bibliography{srDFT_SC,biblio}

\end{document}
