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,
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
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.
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}, 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 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.
Besides the difficulties of accurately describing the electronic structure within a given basis set, a crucial component of the limitations of applicability of WFT concerns the slow convergence of the energies and properties with respect to the quality of the basis set. As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg \textit{et. al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the coulomb interaction at the electron coalescence point, which induces a discontinuity in the first-derivative of the wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete 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 has been developed, either based on the Hylleraas's expansion of the coulomb operator\cite{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 has been proposed by the so-called R12 and F12 methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a function explicitly depending on the interelectronic coordinates ensuring the correct cusp condition in the wave function, and the resulting correlation energies converge much faster than the usual WFT. For instance, using the explicitly correlated version of coupled cluster with single, double and perturbative triple substitution (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to a quintuple-$\zeta$ quality of the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although inherent computational overhead are introduced by the auxiliary basis sets needed to resolve the rather complex three- and four-electron integrals involved in the F12 theory.
An alternative point of view is to leave the short-range correlation effects to DFT and to use WFT to deal only with the long-range and/or strong-correlation effects. A rigorous approach to do so is the range-separated DFT (RSDFT) formalism (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which rely on a splitting of the coulomb interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of such approach is at least two-folds: i) the DFT part deals only with the short-range part of the coulomb interaction, and therefore the usual semi-local approximations to the unknown exchange-correlation functional are more suited to that correlation regime, ii) the WFT part deals with a smooth non divergent interaction, which removes the cusp condition and therefore the basis set convergence is much faster\cite{FraMusLupTou-JCP-15}.
Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches. Nevertheless, there are still some open issues in RSDFT, such as the dependence of the quality of the results on the value of the range separation $\mu$ which can be seen as an empirical parameter, together with the remaining self-interaction errors.
Following this path, a very recent solution to the basis set convergence problem has been proposed by some of the preset authors\cite{GinPraFerAssSavTou-JCP-18} where they proposed to use RSDFT to take into account only the correlation effects outside a given basis set. The key idea in such a work is to realize that as a wave function developed in an incomplete basis set is cusp-less, it could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, the authors proposed a mapping with RSDFT through the introduction of an effective non-divergent interaction representing the usual coulomb interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried recently\cite{LooPraSceTouGin-JCPL-19} together with the first attempt to generalize this approach to excited states\cite{exicted}.
The goal of the present work is to push the development of this new theory toward 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 propose a practical extension for strongly correlated systems. Two key aspect are discussed: the extensivity of the correlation energies together with the $S_z$ independence of the results.
Then in section \ref{sec:results} we discuss the potential energy surfaces (PES) of N$_2$, F$_2$ and H$_{10}$ up to full dissociation as a prototype of strongly correlated problems. Finally, we conclude in section \ref{sec:conclusion}
The theoretical framework of the basis set correction has 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}.
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
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
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
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})
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{LooPraSceTouGin-JCPL-19,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 $\Bas$}
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
$\Gam{pq}{rs}=2\mel*{\wf{}{\Bas}}{\aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated two-body tensor, $\SO{p}{}$ are the spatial orthonormal orbitals,
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
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}
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 $\mu$
