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

\begin{document}	

\title{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds}

\begin{abstract}
bla bla bla youpi tralala 
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%
The main goal of quantum chemistry is to propose reliable theoretical tools to describe the rich area of chemistry. 
The accurate computation of the electronic structure of molecular systems plays a central role in the development of methods in quantum chemistry, 
but despite intense developments, no definitive solution to that problem have been found. 
The theoretical challenge to be overcome falls back in the category of the quantum many-body problem due the intrinsic quantum nature 
of the electrons and the coulomb repulsion between them, inducing the so-called electronic correlation problem. 
Tackling this problem translate to solving the Schroedinger equation for a $N$~-~electron system, and two roads have emerged to approximate the solution to this formidably complex mathematical problem: the wave function theory (WFT) and density functional theory (DFT). 
Although both WFT and DFT spring from the same problem, their formalisms are very different as the former deals with the complex 
$N$~-~body wave function whereas the latter handles the much simpler one~-~body density. 
The computational cost of DFT is very appealing as in its Kohn-Sham (KS) formulation it can be recast in a mean-field procedure. 
Therefore, although constant efforts are performed to reduce the computational cost of WFT, DFT remains still the workhorse of quantum chemistry. 

From the theoretician point of view, the complexity of description of a given chemical system can be roughly 
categorized by the strength of the electronic correlation appearing in its electronic structure.  
Weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by the avoidance effects when electron are near the electron coalescence point, which are often called short-range correlation effects, 
or far from each other, typically dispersion forces. The theoretical description of weakly correlated systems is one of the more concrete achievement 
of quantum chemistry, and the main remaining issue for these systems is to push the limit in terms 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 exhibits 
a much more exotic electronic structure. 
Transition metals containing systems, low-spin open shell systems, covalent bond breaking or excited states 
have all in common that they cannot be even qualitatively described by a single electronic configuration. 
It is now clear that the usual approximations in KS-DFT fails in giving an accurate description of these situations and WFT has become 
the standard for the treatment of strongly correlated systems. 
From the theoretical point of view, the complexity of the strong correlation problem is, at least, two-fold: 
i) the presence of near degeneracies and/or strong interactions among a primary set of electronic configurations 
(the size of which can potentially scale exponentially in some cases) determines the qualitative description of the wave function, 
ii) the quantitative description of the systems must take into account weak correlation effects which requires to take into account many 
other electronic configurations with typically much smaller weights in the wave function.  
Fulfilling these two objectives is a rather complicated task, specially if one adds the requirement of size-extensivity and additivity of the computed energy in the case of non interacting fragments, which is a very desirable property for any approximated method. 

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{alex_thom,piotr} or symmetry-broken approaches\cite{scuseria}. 
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 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 (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,ShBuPeyChemPhys78,BuePeyButMolPhys78,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}, and among which 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 stopping criterion. It is important to notice that in the SCI algorithms, neither the SCI or the MRPT are size extensive \text{per se}, but the extensivity property is almost recovered by approaching the FCI limit. 
When the SCI are affordable, their clear advantage are 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. 


A sensible advantage of WFT is its systematically improvable character to tend to the exact solution, which is the so-called full configuration interaction (FCI) in a complete basis set (CBS). 
Such a path can be expressed in two ways which are quite independent one from another: i) improving the description of the wave function in terms of multiple excitations expansion ii) improving the quality of the one-particle basis set. 
When the molecular system 


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%%%
The theoretical framework of the basis set correction have been derived in details in \cite{GinPraFerAssSavTou-JCP-18}, so we recall briefly the main equations involved for the present study. 
First 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 $\Bas$. Then in section \ref{sec:wee} we introduce an effective non divergent interaction in a basis set $\Bas$, which leads us to the definition of an effective range separation parameter varying in space in section \ref{sec:mur}. Thanks to the range separation parameter, we make a mapping with a specific class of RSDFT functionals and propose practical approximations for the unknown density functional complementary to a basis set $\Bas$, for which new approximations for the strong correlation regime are given in section \ref{sec:functional}. 
\subsection{Basic formal equations}
\label{sec:basic}
The exact ground state energy $E_0$ of a $N-$electron system can be obtained by the Levy-Lieb constrained search formalism which is an elegant mathematical framework connecting WFT and DFT
\begin{equation}
 \label{eq:levy}
 E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\},
\end{equation}
where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional 
\begin{equation}
 \label{eq:levy_func}
 F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}.
\end{equation}
The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density. 
As in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$, we assume from thereon that the densities used in the equations belong to $\setdenbasis$. 

Following equation (7) of  \cite{GinPraFerAssSavTou-JCP-18}, we split $F[\denr]$ as
\begin{equation}
 F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr}
\end{equation}
where $\wf{}{\Bas}$ refer to $N-$electron wave functions expanded in $\Bas$, and 
where $\efuncden{\denr}$ is the density functional complementary to the basis set $\Bas$ defined as 
\begin{equation}
 \begin{aligned}
 \efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\ 
                  &- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}. 
 \end{aligned}
\end{equation}
The functional $\efuncden{\denr}$ must therefore recover all physical effects not included in the basis set $\Bas$. 

Assuming that the FCI density $\denFCI$ in $\Bas$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state density (see equations 12-15 of \cite{GinPraFerAssSavTou-JCP-18})
\begin{equation}
 \label{eq:e0approx}
 E_0 = \efci + \efuncbasisFCI
\end{equation}
where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally shown in \cite{GinPraFerAssSavTou-JCP-18} and further emphasized in \cite{G2,excited}, the main role of $\efuncbasisFCI$ is to correct for the basis set incompleteness errors, a large part of which originates from the lack of cusp in any wave function developed in an incomplete basis set. 
The whole purpose of this paper is to determine approximations for $\efuncbasisFCI$ which are suited for treating strong correlation regimes. The two requirement for such conditions are that i) it can be defined for multi-reference wave functions, ii) it must provide size extensive energies, iii) it is invariant of the $S_z$ component of a given spin multiplicity. 

\subsection{Definition of an effective interaction within $\Bas$}
\label{sec:wee}
As it was originally shown by Kato\cite{kato}, the cusp in the exact wave function originates from the divergence of the coulomb interaction at the coalescence point. Therefore, the lack of cusp in any wave function $\wf{}{\Bas}$ could also originate from an effective non-divergent electron-electron interaction. In other words, the incompleteness of a finite basis set can be understood as the removal of the divergence at the electron coalescence point. 

As it was originally derived in \cite{GinPraFerAssSavTou-JCP-18} (see section D and annexes), one can obtain an effective non divergent interaction, here referred as $\wbasis$, which reproduces the expectation value of the coulomb 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, we define the effective interaction associated to a given wave function $\wf{}{\Bas}$ 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 two-body density associated to $\wf{}{\Bas}$ 
\begin{equation}
 \twodmrdiagpsi = \sum_{pqrs} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, 
\end{equation}
$\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated two-body tensor, $\SO{p}{}$ are the spatial orthonormal orbitals,  
\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 $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
With such a definition, one can show that $\wbasis$ satisfies 
\begin{equation}
 \int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \int \int \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}. 
\end{equation}
As it was shown in \cite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessary finite at coalescence for an incomplete basis set, and tends to the regular coulomb interaction in the limit of a complete basis set, that is 
\begin{equation}
 \label{eq:cbs_wbasis}
 \lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}. 
\end{equation}
The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set. 
\subsection{Definition of an range-separation parameter varying in real space}
\label{sec:mur}
As the effective interaction within a basis set $\wbasis$ is non divergent, one can fit such a function with a long-range interaction defined in the framework of RSDFT which depends on the range-separation parameter $\mu$
\begin{equation}
 \label{eq:weelr}
 w_{ee}^{\lr}(\mu;\br{1},\br{2}) = \frac{\text{erf}\big(\mu \,|\br{1}-\br{2}| \big)}{|\br{1}-\br{2}|}. 
\end{equation}
As originally proposed in \cite{GinPraFerAssSavTou-JCP-18}, we introduce a range-separation parameter $\murpsi$ varying in real space
\begin{equation}
 \murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal 
\end{equation}
such that 
\begin{equation}
 w_{ee}^{\lr}(\murpsi;\br{ },\br{ }) = \wbasiscoal. 
\end{equation}
Because of the very definition of $\wbasis$, one has the following properties at the CBS limit (see \eqref{eq:cbs_wbasis}) 
\begin{equation}
 \label{eq:cbs_mu}
  \lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty, 
\end{equation}
which is fundamental to guarantee the good behaviour of the theory at the CBS limit. 

\subsection{Approximation for $\efuncden{\denr}$ : link with RSDFT}
\subsubsection{Generic form of the approximated functionals}
\subsubsection{Introduction of the effective spin-density}
\subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
        \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat.eps}
        \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat_zoom.eps}
        \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_error.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}\\
        \includegraphics[width=\linewidth]{data/N2/DFT_avtzE_error.eps}\\
%       \includegraphics[width=\linewidth]{fig2c}
        \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/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}


\begin{figure}
%        \includegraphics[width=\linewidth]{data/H10/DFT_avdzE_relat.eps}
        \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat_zoom.eps}\\
        \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_error.eps}\\
%       \includegraphics[width=\linewidth]{fig2c}
        \caption{
        H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:H10_vdz}}
\end{figure}


\begin{figure}
        \includegraphics[width=\linewidth]{data/H10/DFT_vtzE_relat_zoom.eps}\\
        \includegraphics[width=\linewidth]{data/H10/DFT_vtzE_error.eps}\\
%       \includegraphics[width=\linewidth]{fig2c}
        \caption{
        H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:H10_vtz}}
\end{figure}


\begin{figure}
        \includegraphics[width=\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}\\
        \includegraphics[width=\linewidth]{data/H10/DFT_vqzE_error.eps}\\
%       \includegraphics[width=\linewidth]{fig2c}
        \caption{
        H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. 
        \label{fig:H10_vqz}}
\end{figure}





\bibliography{srDFT_SC}

\end{document}
