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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 weak and 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 incompleteness correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, \href{https://doi.org/10.1063/1.5052714}{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 electron-electron Coulomb interaction projected in the finite basis set. This allows to use RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. To model both weak and strong correlation regimes we consider the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit, and we explore various approximations of complementary density functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$) and size consistency. Specifically, we systematically investigate the functional dependence on different types of on-top pair densities and spin polarizations. The key result of this study 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 in certain cases. 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 here. Also, the present basis-set incompleteness correction provides smooth curves along the whole potential energy surface.
\end{abstract}

\maketitle

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\section{Introduction}
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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. 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 computational 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. The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, 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 points) 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 chemical system size that can be treated.  The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is 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. 
%\PFL{I think we should add some references in the paragraph above.}

In practice, WFT uses a finite one-particle basis set (here denoted as $\basis$). The exact solution of the Schr\"odinger equation within this basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra eigenvalue 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 also to account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing these two issues 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 (\ie, invariance with respect to the spin operator $S_z$) and size consistency.

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 electronic energy (and related 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, \cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94} the main convergence problem originates from the divergence of the electron-electron Coulomb 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 are strongly affected. To alleviate 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 explicitly correlated F12 (or R12) 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 depending explicitly on the interelectronic distances. This ensures a correct representation of the Coulomb correlation hole around the electron-electron coalescence point, and leads to a much faster convergence of the energy 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-$\zeta$ basis set is equivalent to using a quintuple-$\zeta$ 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. \cite{BarLoo-JCP-17} 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 explicit correlation (see, for example, 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 decomposition of the electron-electron Coulomb 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} and 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} in which 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 originate from a Hamiltonian with a non-divergent electron-electron interaction. Therefore, a mapping with RSDFT can be performed through the introduction of an effective non-divergent interaction representing the usual electron-electron Coulomb 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 Sec.~\ref{sec:theory}, we recall the mathematical framework of the basis-set correction and we present its extension for strongly correlated systems. In particular, our focus is primarily set on imposing two key formal properties: spin-multiplet degeneracy and size-consistency.
Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit. Finally, we conclude in Sec.~\ref{sec:conclusion}. 

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\section{Theory}
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%
As the theory behind the present 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 Secs.~\ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Sec.~\ref{sec:basic}, we recall the basic mathematical framework of the present theory by introducing the complementary density functional to a basis set $\Bas$. Section \ref{sec:wee} introduces the effective non-divergent interaction in the basis set $\Bas$, which leads us to the definition of the effective \textit{local} range-separation parameter in Sec.~\ref{sec:mur}. Then, Sec.~\ref{sec:functional} exposes the new approximate RSDFT-based complementary correlation functionals. The generic form of such functionals is exposed in Sec.~\ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Sec.~\ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Sec.~\ref{sec:requirements}. Finally, the actual form of the functionals used in this work are introduced in Sec.~\ref{sec:def_func}. 

\subsection{Basic 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 \titou{one-electron densities} $\denr$
\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 \to \den} \mel{\Psi}{\kinop +\weeop}{\Psi},
\end{equation}
where $\kinop$ and $\weeop$ are the kinetic and electron-electron Coulomb operators, and the notation $\Psi \to \den$ means that the wave function $\Psi$ yields the density $\den$. 
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 $\Bas$'', \ie, 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 representable-in-$\Bas$ densities. With this restriction, Eq.~\eqref{eq:levy} then gives an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an acceptable approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$.

In the present context, it is important to notice that the wave functions $\Psi$ defined in Eq.~\eqref{eq:levy_func} are not restricted to a finite basis set, \ie, they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then proposed to decompose $F[\den]$ as
\begin{equation}
 \label{eq:def_levy_bas}
 F[\den] = \min_{\wf{}{\Bas} \to \den} \mel*{\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 
\begin{equation}
 \begin{aligned}
 \efuncden{\den} = \min_{\Psi \to \den} \mel*{\Psi}{\kinop +\weeop }{\Psi} 
                  - \min_{\Psi^{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} 
 \end{aligned}
\end{equation}
is the complementary density functional to the basis set $\Bas$.
Introducing the decomposition in Eq.~\eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy} yields 
\begin{multline}
 \label{eq:E0basminPsiB}
 E_0^\Bas = \min_{\Psi^{\Bas}} \bigg\{ \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}} 
	\\
	+ \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\},
\end{multline}
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 generated from $\wf{}{\Bas}$. Therefore, thanks to Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional (hereafter referred to as complementary density functional) accounting for the correlation effects that are 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 Refs.~\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. Two key requirements for this purpose are i) size consistency, and ii) spin-multiplet degeneracy.

\subsection{Effective interaction in a finite basis}
\label{sec:wee}
As originally shown by Kato, \cite{Kat-CPAM-57} the electron-electron cusp of 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 basis set incompleteness can be understood as the removal of the divergence of the usual electron-electron Coulomb interaction.

As originally derived in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see Sec.~II 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 electron-electron Coulomb 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 $\{\SO{p}{}\}$, 
\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}
and $\V{pq}{rs}= \braket{pq}{rs}$ are the usual two-electron Coulomb integrals.
With such a definition, one can show that $\wbasis$ satisfies 
\begin{multline}
	\frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi =
	\\
	\frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{\abs{\br{1}-\br{2}}}. 
\end{multline}
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 $\psibasis$, \ie,
\begin{equation}
 \label{eq:cbs_wbasis}
 \lim_{\Bas \to \text{CBS}} \wbasis = \frac{1}{\abs{\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{Local range-separation parameter}
\label{sec:mur}
\subsubsection{General definition}
As the effective interaction within a finite basis, $\wbasis$ is bounded and resembles 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 correspondence between these two interactions by using the local range-separation parameter 
\begin{equation}
 \label{eq:def_mur}
 \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal, 
\end{equation}
such that the two interactions coincide at the electron-electron coalescence 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 \to \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 a valence-only version of the various quantities needed for the complementary density functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as \titou{$\Bas = \Cor \bigcup \BasFC$, 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 
\begin{equation}
 \label{eq:wbasis_val}
 \wbasisval =
    \begin{cases}
      \fbasisval /\twodmrdiagpsival,    & \text{if $\twodmrdiagpsival \ne 0$,}
\\
       \infty,                                                                                          & \text{otherwise,}
    \end{cases}
\end{equation}
is the valence-only effective interaction and
\begin{gather}
	\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},
	\\
	\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{gather}
One would note the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}. 
It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.

\subsection{Complementary density functionals}
\label{sec:functional}

\subsubsection{Generic form}
\label{sec:functional_form}

As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary density 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 one-electron density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$ (where $n_\uparrow(\br{})$ and $n_\downarrow(\br{})$ are the spin-up and spin-down densities), 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, $\efuncden{\den}$ has the following generic form
\begin{multline}
 \label{eq:def_ecmdpbebasis}
 \efuncdenpbe{\argebasis} =  
 \\ 
 \int  \d\br{} \,\denr   \ecmd(\argrebasis),
\end{multline}
where 
\begin{equation}
 \label{eq:def_ecmdpbe}
 \ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe,\ntwo) \; \mu^3},
\end{equation}
is the correlation energy per particle, with 
\begin{equation}
 \label{eq:def_beta}
 \beta(\argepbe,\ntwo) = \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 Sec.~\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, \ie,
\begin{equation}
   \lim_{\mu \to 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, \ie,
\begin{equation}
 \label{eq:lim_n2}
  \lim_{\ntwo \to 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 $\mu \to \infty$ like all RSDFT short-range functionals, \ie,
\begin{equation}
 \label{eq:lim_muinf}
  \lim_{\mu \to \infty} \ecmd(\argecmd) = 0. 
\end{equation}

\subsubsection{Properties}
\label{sec:functional_prop}

Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary density functional $\efuncdenpbe{\argebasis}$ satisfies two important properties.

First, thanks to the properties in Eqs.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, independently of the type of wave function $\psibasis$ used to define the local range-separation parameter $\mu(\br{})$ in a given basis set $\Bas$, 
\begin{equation}
 \label{eq:lim_ebasis}
 \lim_{\basis \to \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis, 
\end{equation}

Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density guarantees the correct limit for one-electron systems and for the stretched H$_2$ molecule. This property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ [see Eq.~\eqref{eq:wbasis}] together with the condition in Eq.~\eqref{eq:lim_muinf}, and ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}]. 

\subsection{Requirements for strong correlation}
\label{sec:requirements}

An important requirement for any electronic-structure method is size-consistency, \ie, the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space self-consistent-field (CASSCF) wave function which, provided that the active space has been properly chosen, leads to additive energies.  

Another important requirement is spin-multiplet degeneracy, \ie, 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 supersystem $\ce{A + B}$ is generally of lower spin than the corresponding ground states of the fragments (\ce{A} and \ce{B}) which can have multiple $S_z$ components.

\subsubsection{Spin-multiplet degeneracy}

A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$. In the case of the functional $\ecmd(\argecmd)$, this means removing the dependency on the spin polarization $\zeta(\br{})$ originating from the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ [see Eq.~\eqref{eq:def_ecmdpbe}]. 

To do so, it has been proposed to substitute the dependency on the spin polarization by the dependency on the on-top pair density. Most often, it is done by introducing an effective spin polarization~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96,MieStoSav-MP-97,TakYamYam-CPL-02,TakYamYam-IJQC-04,GraCre-MP-05,TsuScuSav-JCP-10,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15,GarBulHenScu-PCCP-15,CarTruGag-JCTC-15,GagTruLiCarHoyBa-ACR-17} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01}) \manu{is zero if square roots is complex}
\begin{equation}
 \label{eq:def_effspin}
 \tilde{\zeta}(n,n_{2}) =  
%   \begin{cases}
                       \sqrt{ 1 - 2 \; n_{2}/n^2 },
%                              0              & \text{otherwise.}
%   \end{cases}
\end{equation}
expressed as a function of the density $n$ and the on-top pair density $n_2$, calculated from a given wave function. The advantage of this approach is that this effective spin  polarization $\tilde{\zeta}$ is independent from $S_z$, since the on-top pair density is $S_z$-independent. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents some disadvantages since this expression was derived for a single-determinant wave function. Hence, it does not appear justified to use it for a multideterminant wave function. More particularly, it may happen, in the multideterminant case, that $1 - 2 \; n_{2}/n^2 < 0 $ which results in a complex-valued spin polarization [see Eq.~\eqref{eq:def_effspin}]. \cite{BecSavSto-TCA-95}

%The advantage of this approach are at least two folds: i) the effective spin  polarization $\tilde{\zeta}$ is independent from $S_z$ since the on-top pair density is independent from $S_z$, 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 $1 - 2 \; n_{2}/n^2 < 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 way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, \ie, to resort to the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for sufficiently large $\mu$, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density, \cite{PerSavBur-PRA-95} but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density [see Eqs.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}]. The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Consequently, we propose here to test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures its $S_z$ invariance and, as will be numerically demonstrated, very weakly affects the complementary density functional accuracy.


\subsubsection{Size consistency}

Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments \ce{A\bond{...}B}, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem \ce{A\bond{...}B}. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$.
We refer the interested reader to the {\SI} for a detailed demonstration of the latter statement. 
In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function is sufficient to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.


\subsection{Complementary density functional approximations}
\label{sec:def_func}
%\subsubsection{Definition of the protocol to design functionals}

As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CASSCF wave functions in order to ensure size consistency of all local quantities. The difference between two flavors of functionals are only due to the type of i) spin polarization, and ii) on-top pair density.

Regarding the spin polarization that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are considered: i) \titou{the \textit{effective} spin polarization $\tilde{\zeta}$} defined in Eq.~\eqref{eq:def_effspin} and calculated from the CASSCF wave function, and ii) a \textit{zero} spin polarization. \titou{In the latter case, the functional is referred as to ``SU'' which stands for ``spin unpolarized''}. 

Regarding the on-top pair density entering in Eq.~\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) \approx n^2\big(1-\zeta^2\big)g_0(n),
\end{equation}
where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function through $\tilde{\zeta}$. When using $\ntwo^{\text{UEG}}(n,\tilde{\zeta})$ in a functional, we will refer to it as ``UEG''.

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

We then define four functionals:

\begin{itemize}

\item[i)] $\pbeuegXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin}  and  the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg}:
\begin{multline}
	\label{eq:def_pbeueg_i}
	\bar{E}^\Bas_{\pbeuegXi} 
	\\
	=  \int  \d\br{} \,\denr  \ecmd(\argrpbeuegXi), 
\end{multline}
\item[ii)] $\pbeontXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin} and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}:
\begin{equation}
	\label{eq:def_pbeueg_ii}
	\bar{E}^\Bas_{\pbeontXi} =  \int  \d\br{} \,\denr  \ecmd(\argrpbeontXi),  
\end{equation}
\item[iii)] $\pbeuegns$ which combines a zero spin polarization and the UEG on-top pair density of Eq.~\eqref{eq:def_n2ueg}:
\begin{equation}
	\label{eq:def_pbeueg_iii}
	\bar{E}^\Bas_{\pbeuegns}  =  \int  \d\br{} \,\denr  \ecmd(\argrpbeuegns),
\end{equation}
\item[iv)] $\pbeontns$ which combines a zero spin polarization and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}:
\begin{equation}
	\label{eq:def_pbeueg_iv}
	\bar{E}^\Bas_{\pbeontns}  =  \int  \d\br{} \,\denr  \ecmd(\argrpbeontns).  
\end{equation}
\end{itemize}
The performance of each of these four functionals is tested below.


%DFT: BLACK BOX and not CASSCF 
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\label{sec:results}
\begin{figure*}
        \includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat.eps}
        \includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat_zoom.eps}
        \includegraphics[width=0.45\linewidth]{data/H10/DFT_vtzE_relat.eps}
        \includegraphics[width=0.45\linewidth]{data/H10/DFT_vtzE_relat_zoom.eps}
        \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat.eps}
        \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}
        \caption{
        Potential energy curves of the H$_{10}$ chain with equally-spaced atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top), cc-pVTZ (middle), and cc-pVQZ (bottom) basis sets. 
        The MRCI+Q energies and the estimated exact energies have been extracted from Ref.~\onlinecite{h10_prx}.
        \label{fig:H10}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Computational details}

The purpose of the present paper being the study of the basis-set correction in regimes of both weak and/or strong correlation, we study the potential energy curves up to the dissociation limit of a \ce{H10} chain with equally-spaced atoms and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics. In a given basis set, in order to compute the approximation of the exact ground-state energy using Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$. 

In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVXZ (X=D,T), approximations to the FCI energies are obtained using converged frozen-core ($1s$ orbitals are kept frozen) selected CI 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 latest version of \textsc{QUANTUM PACKAGE} \cite{QP2} (exFCI), and the correlation energy extrapolation by intrinsic scaling\cite{BytNagGorRue-JCP-07} (CEEIS) in the case of \ce{F2} for the cc-pVXZ (X=D,T,Q) basis set. The estimated exact potential energy curves are obtained from experimental data in Ref.~\onlinecite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from extrapolated CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be on the order of $0.5$ mHa. For the \ce{N2}, \ce{O2}, and \ce{F2} molecules, we also performed single-point exFCI calculations in the aug-cc-pVQZ basis set at the equilibrium geometry to obtain estimations of the FCI dissociation energies in these basis sets. 
In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data 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 density functional, we first perform full-valence complete-active-space self-consistent-field (CASSCF) calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-like quantities involved in the functional [density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. The CASSCF calculations have been performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}. 

Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only complementary density functionals. Therefore, all density-like quantities exclude any contribution from the $1s$ core orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}. 

Regarding the computational cost of the present approach, it should be stressed (see supplementary information) that the basis set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any of the selected CI calculations. We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*}
\label{tab:d0}
\caption{Atomization energies $D_0$ (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different approximation levels with various basis sets.}
\begin{ruledtabular}
\begin{tabular}{lrdddd}

   System 	&  \tabc{Basis set}        &  \tabc{MRCI+Q\fnm[1]}              &   \tabc{(MRCI+Q)+$\pbeuegXi$}   &   \tabc{(MRCI+Q)+$\pbeontXi$}   &     \tabc{(MRCI+Q)+$\pbeontns$}    \\
\hline                                                                                                                 
\ce{H10} & cc-pVDZ           &  622.1 [43.3]            &   642.6 [22.8]         &   649.2  [16.2]          &     649.5  [15.9]           \\
         & cc-pVTZ           &  655.2 [10.2]            &   661.9 [3.5]          &   666.0  [-0.6]          &     666.0  [-0.6]           \\
         & cc-pVQZ           &  661.2 [4.2]             &   664.1 [1.3]          &   666.4  [-1.0]          &     666.5  [-1.1]           \\[0.1cm]
%\cline{2-6}
  	&	&\multicolumn{4}{c}{Estimated exact:\fnm[1] 665.4} \\[0.2cm]
\hline
%    		&         		& \tabc{exFCI}                    &   \tabc{exFCI+$\pbeuegXi$}   &   \tabc{exFCI+$\pbeontXi$}   &     \tabc{exFCI+$\pbeontns$}\\
%\hline
%\ce{C2} &  aug-cc-pVDZ          &   204.6  [29.5]         &   218.0  [16.1]     &   217.4  [16.7]     &     217.0  [17.1]           \\
%      & aug-cc-pVTZ          &   223.4  [10.9]         &   228.1  [6.0]      &   228.6  [5.5]      &     226.5  [5.6]            \\[0.1cm]
%\cline{2-6}
% 	&	 & \multicolumn{4}{c}{Estimated exact:\fnm[2] 234.1} \\[0.2cm]
%\hline                                                                                                                 
    		&         		& \tabc{exFCI}                    &   \tabc{exFCI+$\pbeuegXi$}   &   \tabc{exFCI+$\pbeontXi$}   &     \tabc{exFCI+$\pbeontns$}\\
\hline                                                                                                                 
\ce{N2} & aug-cc-pVDZ          &   321.9  [40.8]         &   356.2  [6.5]           &   355.5  [7.2]           &     354.6  [8.1]           \\
        & aug-cc-pVTZ          &   348.5  [14.2]         &   361.5  [1.2]           &   363.5  [-0.5]          &     363.2  [-0.3]           \\[0.1cm]
        & aug-cc-pVQZ          &   356.6  [6.1 ]         &   362.8  [-0.1]          &   364.2  [-1.5]          &     364.3  [-1.6]           \\[0.1cm]
 	&	 & \multicolumn{4}{c}{Estimated exact:\fnm[2] 362.7} \\[0.2cm]
\hline                                                                                                                 
    		&         		& \tabc{exFCI}                    &   \tabc{exFCI+$\pbeuegXi$}   &   \tabc{exFCI+$\pbeontXi$}   &     \tabc{exFCI+$\pbeontns$}\\
\hline                                                                                                                 
\ce{O2} & aug-cc-pVDZ          &   171.4  [20.5]         &   187.6  [4.3]      &   187.6  [4.3]      &     187.1  [4.8]           \\
        & aug-cc-pVTZ          &   184.5  [7.4]          &   190.3  [1.6]      &   191.2  [0.7]      &     191.0  [0.9]            \\[0.1cm]
        & aug-cc-pVQZ          &   188.3  [3.6]          &   190.3  [1.6]      &   191.0  [0.9]      &     190.9  [1.0]            \\[0.1cm]
 	&	 & \multicolumn{4}{c}{Estimated exact:\fnm[2] 191.9} \\[0.2cm]
\hline                                                                                                                 
    		&         		& \tabc{exFCI}                    &   \tabc{exFCI+$\pbeuegXi$}   &   \tabc{exFCI+$\pbeontXi$}   &     \tabc{exFCI+$\pbeontns$}\\
\hline                                                                                                                 
\ce{F2} & aug-cc-pVDZ          &   49.6  [12.6]          &    54.8  [7.4]      &   54.9  [7.3]       &     54.8  [7.4]             \\
        & aug-cc-pVTZ          &   59.3  [2.9]           &    61.2  [1.0]      &   61.5  [0.7]       &     61.5  [0.7]             \\[0.1cm]
        & aug-cc-pVQZ          &   60.1  [   ]           &    61.0  [1.2]      &   61.3  [0.9]       &     61.3  [0.9]             \\[0.1cm]
\hline                                                                                                                 
    		&         		& \tabc{CEEIS\fnm[3]}                    &   \tabc{CEEIS\fnm[3]+$\pbeuegXi$}   &   \tabc{CEEIS\fnm[3]+$\pbeontXi$}   &     \tabc{CEEIS\fnm[3]+$\pbeontns$}\\
\hline                                                                                                                 
\ce{F2} &     cc-pVDZ          &   43.7  [18.5]          &    51.0  [11.2]     &   51.0 [11.2]       &     50.7  [11.5]            \\
        &     cc-pVTZ          &   56.3  [5.9]           &    59.2  [3.0]      &   59.6 [2.6]        &     59.5  [2.7]             \\
        &     cc-pVQZ          &   59.9  [2.3]           &    61.3  [0.9]      &   61.6 [0.6]        &     61.6  [0.6]             \\[0.1cm]
 	&	& \multicolumn{4}{c}{Estimated exact:\fnm[2] 62.2} \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{From Ref.~\onlinecite{h10_prx}.}
\fnt[2]{From the extrapolated valence-only non-relativistic calculations of Ref.~\onlinecite{BytLaiRuedenJCP05}.}
\fnt[3]{CEEIS calculations obtained from non-relativistic calculations of Ref.~\onlinecite{BytNagGorRue-JCP-07}.}
\label{tab:extensiv_closed}
\end{table*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{H$_{10}$ chain}

The study of the \ce{H10} chain with equally distant atoms is a good prototype of strongly-correlated systems 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 at near-CBS values can be obtained (see Ref.~\onlinecite{h10_prx} for a detailed study of this problem). 

We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation. The computation of the atomization energies $D_0$ for each level of theory 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 curves are globally improved by adding the basis-set correction, independently of the approximation level of $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\textit{i.e.} the determination of the range-separation parameter and the definition of the functionals) is robust when reaching the strong-correlation regime.  
In other words, smooth potential energy surfaces are obtained with the present basis-set correction.
More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, an error below $1.4$ mHa) from the cc-pVTZ basis set when using the $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not even reached at the standard MRCI+Q/cc-pVQZ level of theory. 

Analyzing more carefully the performance of the different types of approximate density functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on $D_0$ being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provide two important clues on the role of the different physical ingredients used in these functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function [see Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [see Eq.~\eqref{eq:def_n2ueg}] which is somehow understandable, and ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it shows that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density.
This could have significant implications for the construction of more robust families of density-functional approximations within DFT.
%\PFL{Why can't we see the effect of dispersion in that system?}

\subsection{Dissociation of diatomics}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure*}
%        \includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat.eps}
%        \includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat_zoom.eps}
%        \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat.eps}
%        \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat_zoom.eps}
%        \caption{
%        Potential energy curves of the \ce{C2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
%        \label{fig:C2}}
%\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}
        \includegraphics[width=0.45\linewidth]{data/N2/DFT_avdzE_relat.eps}
        \includegraphics[width=0.45\linewidth]{data/N2/DFT_avdzE_relat_zoom.eps}
        \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.eps}
        \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps}
        \caption{
        Potential energy curves of the \ce{N2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are  based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
        \label{fig:N2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}
        \includegraphics[width=0.45\linewidth]{data/O2/DFT_avdzE_relat.eps}
        \includegraphics[width=0.45\linewidth]{data/O2/DFT_avdzE_relat_zoom.eps}
%        \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.eps}
%        \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.eps}
        \caption{
        Potential energy curves of the \ce{O2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and \titou{aug-cc-pVTZ (bottom)} basis sets. The estimated exact energies are  based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
        \label{fig:O2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}
        \includegraphics[width=0.45\linewidth]{data/F2/DFT_vdzE_relat.eps}
        \includegraphics[width=0.45\linewidth]{data/F2/DFT_vdzE_relat_zoom.eps}
        \includegraphics[width=0.45\linewidth]{data/F2/DFT_vtzE_relat.eps}
        \includegraphics[width=0.45\linewidth]{data/F2/DFT_vtzE_relat_zoom.eps}
        \includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat.eps}
        \includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat_zoom.eps}
        \caption{
        Potential energy curves of the \ce{F2} molecule calculated with CEEIS$^1$ and basis-set corrected CEEIS$^1$ using the cc-pVDZ (top), cc-pVTZ (middle) and cc-pVQZ (bottom) basis sets. The estimated exact energies are  based on fit on valence-only extrapolated CEEIS data obtained from Ref.~\onlinecite{BytNagGorRue-JCP-07}. \\
$^1$: CEEIS calculations obtained from non-relativistic calculations of Ref.~\onlinecite{BytNagGorRue-JCP-07}.
        \label{fig:F2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study as the level of strong correlation in these diatomics also increases while stretching the bond similarly to the case of \ce{H10}. In addition, these molecules 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 interactions which are medium to long-range weak-correlation effects. The dispersion forces in \ce{H10} play a much minor role in the PES due to the much smaller number of near-neighboring electron pairs compared to  \ce{N2}, \ce{O2} or \ce{F2}. Also, \ce{O2} has a triplet ground state and is therefore a good candidate for checking the spin-polarization dependence of the various functionals proposed here.

We report in Figs.~\ref{fig:N2} and \ref{fig:O2} the potential energy curves of \ce{N2}, \ce{O2}, and computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets, and in Fig~\ref{fig:F2} the potential energy surface of \ce{F2} using the cc-pVXZ (X=D,T,Q) basis set. The computation of the atomization energies $D_0$ at each level of theory is reported in Table \ref{tab:d0}. 

Just as in \ce{H10}, the quality of $D_0$ is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for open-shell systems like \ce{O2}. More quantitatively, an error below 1.0 mHa on the estimated exact valence-only $D_0$ is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the presence of diffuse functions in for double- and triple-zeta types basis sets strongly improves the results, which is somehow understandable due to the strong breathing-orbital effect in this molecule induced by the ionic valence bond forms. \cite{HibHumByrLen-JCP-94} 
It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the quality of $D_0$ slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI $D_0$ and very close the to chemical accuracy. 

Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary density functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is designed thanks to the universal condition provided by the electron-electron cusp and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected and predictable.
We hope to report further on this in the near future.

\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 \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit at near-FCI level in increasing-large basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves 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 fit of this effective interaction with the long-range interaction used in RS-DFT, and iii) the use of a short-range, complementary correlation functional from RS-DFT. In the present paper, we investigated i) and iii) in the context of strong correlation and focused on potential energy curves  and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling i) spin-multiplet degeneracy, and ii) size consistency. To fulfil such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and to develop functionals using only $S_z$-independent density-like quantities. 

The development of new $S_z$-independent and size-consistent 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 eschew its spin polarization dependency without loss of accuracy. This avoids the commonly used effective spin polarization originally proposed in Ref.~\onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly becoming complex-valued in the multideterminant case. From a more fundamental aspect, this shows that, in a DFT framework, the spin polarization mimics the role of the on-top pair density. 
Consequently, we believe that one could potentially develop new families of density functional approximations where the spin polarization is abondonned and replaced by the on-top pair density.

Regarding the results of the present approach, the basis-set correction systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple-$\zeta$ quality basis sets chemically accurate atomization energies, $D_0$, are obtained for all systems whereas the uncorrected near-FCI results are far from this accuracy within the same basis set. 

Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long internuclear distances it cannot recover the dispersion energy missing because of the basis set incompleteness. This behaviour is actually expected as dispersion has a long-range correlation nature and the present approach is designed to recover only short-range correlation effects.

\bibliography{srDFT_SC}

\end{document}
