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 Coulomb electron-electron interaction projected in the finite basis set. This allows to use RSDFT-type complementary functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. Using as test cases the potential energy curves of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit, we explore various approximations of complementary functionals suited to describe strong correlation. These short-range correlation functionals fulfill two very desirable properties: invariance with respect to the spin operator $S_z$ and size consistency. Specifically, we systematically investigate the dependence of the functionals on different flavors 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. 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 surfaces.
The general goal of quantum chemistry is to provide reliable theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular systems, but despite intense developments, no definitive solution to this problem has been found. The theoretical challenge to tackle belongs to the quantum many-body problem, due the intrinsic quantum nature of the electrons and the Coulomb repulsion between them. This so-called electronic correlation problem corresponds to finding a solution to the Schr\"odinger equation for a $N$-electron system, and two main roads have emerged to approximate this solution: wave-function theory (WFT)~\cite{Pop-RMP-99} and density-functional theory (DFT)~\cite{Koh-RMP-99}. Although both WFT and DFT spring from the same Schr\"odinger equation, they rely on very different formalisms, as the former deals with the complicated $N$-electron wave function whereas the latter focuses on the much simpler one-electron density. In its Kohn-Sham (KS) formulation~\cite{KohSha-PR-65}, the computational cost of DFT is very appealing since it is a simple mean-field procedure. Therefore, although continued efforts have been done to reduce the computational cost of WFT, DFT still remains the workhorse of quantum chemistry.
The difficulty of obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation appearing in its electronic structure. The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometries, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak correlation effects can be either short range(near the electron-electron coalescence point) or long range (London dispersion interactions). The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining issue for these systems is to push the limit of the size of the chemical systems that can be treated. The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations and WFT is king for the treatment of strongly correlated systems.
In practice WFT uses a finite one-particle basis set (here denoted as $\basis$) to project the Schr\"odinger equation. The exact solution within the basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix; ii) the quantitative description of the system requires to also account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing these two objectives is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy and size consistency.
%To tackle this complicated problem, many methods have been proposed and an exhaustive review of the zoology of methods for strong correlation goes beyond the scope and purpose of this article.
%To tackle this problem, many WFT methods have emerged which can be categorized in two branches: the single-reference (SR)
%and multi-reference (MR) methods.
%The SR methods rely on a single electronic configuration as a zeroth-order wave function, typically Hartree-Fock (HF).
%Then the electron correlation is introduced by increasing the rank of multiple hole-particle excitations,
%preferably treated in a coupled-cluster (CC) fashion for the sake of compactness of the wave function and extensivity of the computed energies.
%The advantage of these approaches rely on the rather straightforward way to improve the level of accuracy,
%which consists in increasing the rank of the excitation operators used to generate the CC wave function.
%Despite its appealing elegant simplicity, the computational cost of the CC methods increase drastically with the rank of the excitation
%operators, even if promising alternative approaches have been proposed using stochastic techniques\cite{Thom-PRL-10,ScoTho-JCP-17,SpeNeuVigFraTho-JCP-18,DeuEmiShePie-PRL-17,DeuEmiMagShePie-JCP-18,DeuEmiYumShePie-JCP-19} or symmetry-broken approaches\cite{QiuHenZhaScu-JCP-17,QiuHenZhaScu-JCP-18,GomHenScu-JCP-19}.
%In the MR approaches, the zeroth order wave function consists in a linear combination of Slater determinants which are supposed to concentrate most of strong interactions and near degeneracies inherent in the structure of the Hamiltonian for a strongly correlated system. The usual approach to build such a zeroth-order wave function is to perform a complete active space self consistent field (CASSCF) whose variational property prevent any divergence, and which can provide extensive energies. Of course, the choice of the active space is rather a subtle art and the CASSCF results might strongly depend on the level of chemical/physical knowledge of the user.
%On top of this zeroth-order wave function, weak correlation is introduced by the addition of other configurations through either configuration interaction\cite{WerKno-JCP-88,KnoWer-CPL-88} (MRCI) or perturbation theory (MRPT) and even coupled cluster (MRCC), which have their strengths and weaknesses,
%The advantage of MRCI approaches rely essentially in their simple linear parametrisation for the wave function together with the variational property of their energies, whose inherent drawback is the lack of size extensivity of their energies unless reaching the FCI limit. On the other hand, MRPT and MRCC can provide extensive energies but to the price of rather complicated formalisms, and these approaches might be subject to divergences and/or convergence problems due to the non linearity of the parametrisation for MRCC or a too poor choice of the zeroth-order Hamiltonian.
%A natural alternative is to combine MRCI and MRPT, which falls in the category of selected CI (SCI) which goes back to the late 60's and who has received a revival of interest and applications during the last decade \cite{BenErn-PhysRev-1969,WhiHac-JCP-1969,HurMalRan-1973,EvaDauMal-ChemPhys-83,Cim-JCP-1985,Cim-JCC-1987,IllRubRic-JCP-88,PovRubIll-TCA-92,BunCarRam-JCP-06,AbrSheDav-CPL-05,MusEngels-JCC-06,BytRue-CP-09,GinSceCaf-CJC-13,CafGinScemRam-JCTC-14,GinSceCaf-JCP-15,CafAplGinScem-arxiv-16,CafAplGinSce-JCP-16,SchEva-JCP-16,LiuHofJCTC-16,HolUmrSha-JCP-17,ShaHolJeaAlaUmr-JCTC-17,HolUmrSha-JCP-17,SchEva-JCTC-17,PerCle-JCP-17,OhtJun-JCP-17,Zim-JCP-17,LiOttHolShaUmr-JCP-2018,ChiHolOttUmrShaZim-JPCA-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,GarSceGinCaffLoo-JCP-18,SceGarCafLoo-JCTC-18,GarGinMalSce-JCP-16,LooBogSceCafJac-JCTC-19}.
%Among the SCI algorithms, the CI perturbatively selected iteratively (CIPSI) can be considered as a pioneer. The main idea of the CIPSI and other related SCI algorithms is to iteratively select the most important Slater determinants thanks to perturbation theory in order to build a MRCI zeroth-order wave function which automatically concentrate the strongly interacting part of the wave function. On top of this MRCI zeroth-order wave function, a rather simple MRPT approach is used to recover the missing weak correlation and the process is iterated until reaching a given convergence criterion. It is important to notice that in the SCI algorithms, neither the SCI or the MRPT are size extensive \textit{per se}, but the extensivity property is almost recovered by approaching the FCI limit.
%When the SCI are affordable, their clear advantage are that they provide near FCI wave functions and energies, whatever the level of knowledge of the user on the specific physical/chemical problem considered. The drawback of SCI is certainly their \textit{intrinsic} exponential scaling due to their linear parametrisation. Nevertheless, such an exponential scaling is lowered by the smart selection of the zeroth-order wave function together with the MRPT calculation.
Beside the difficulties of accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energies and properties with respect to the size of the basis set. As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg and coworkers~\textit{et al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the Coulomb electron-electron interaction at the coalescence point, which induces a discontinuity in the first derivative of the exact wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete one-electron basis set is impossible and, as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques have been developed, either based on a partial-wave expansion analysis~\cite{HelKloKocNog-JCP-97,HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis-set convergence problem is provided by the so-called R12 and F12 explicitly correlated methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a geminal function explicitly depending on the interelectronic distances ensuring the correct cusp condition in the wave function, and lead to a much faster convergence of the correlation energies than usual WFT methods. For instance, using the explicitly correlated version of coupled cluster with singles, doubles, and perturbative triples (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to using a quintuple-$\zeta$ quality basis set with the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although a computational overhead is introduced by the auxiliary basis set needed to compute the three- and four-electron integrals involved in F12 theory. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires non-trivial developments for adapting it to a new method. For strongly correlated systems, several multi-reference methods have been extended to explicitly correlation (see for instance Ref.~\onlinecite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on the so-called universal F12 theory which are potentially applicable to any electronic-structure computational methods~\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}.
An alternative way to improve the convergence towards the complete-basis-set (CBS) limit is to treat the short-range correlation effects within DFT and to use WFT methods to deal only with the long-range and/or strong-correlation effects. A rigorous approach achieving this mixing of DFT and WFT is range-separated DFT (RSDFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which relies on a splitting of the Coulomb electron-electron interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of this approach is at least two-fold: i) the DFT part deals primarily with the short-range part of the Coulomb interaction, and consequently the usual semilocal density-functional approximations are more accurate than for standard KS DFT; ii) the WFT part deals only with a smooth non-divergent interaction, and consequently the wave function has no electron-electron cusp\cite{GorSav-PRA-06} and the basis-set convergence is much faster\cite{FraMusLupTou-JCP-15}. A number of approximate RSDFT schemes have been developed involving single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15,KalTou-JCP-18,KalMusTou-JCP-19} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT methods. Nevertheless, there are still some open issues in RSDFT, such as remaining fractional-charge and fractional-spin errors in the short-range density functionals~\cite{MusTou-MP-17} or the dependence of the quality of the results on the value of the range-separation parameter $\mu$.
Building on the development of RSDFT, a possible solution to the basis-set convergence problem has been recently proposed by some of the present authors~\cite{GinPraFerAssSavTou-JCP-18} where RSDFT functionals are used to recover only the correlation effects outside a given basis set. The key point here is to realize that a wave function developed in an incomplete basis set is cuspless and could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, a mapping with RSDFT can be introduced through the introduction of an effective non-divergent interaction representing the usual Coulomb electron-electron interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried out~\cite{LooPraSceTouGin-JCPL-19}, together with extensions of this approach to the calculations of excitation energies~\cite{GinSceTouLoo-JCP-19} and ionization potentials~\cite{LooPraSceGinTou-ARX-19}. The goal of the present work is to further develop this approach for the description of strongly correlated systems. The paper is organized as follows. In Section \ref{sec:theory} we recall the mathematical framework of the basis-set correction and we present the extension for strongly correlated systems. In particular, we focus on imposition of two important formal properties: size-consistency and spin-multiplet degeneracy.
Then, in Section \ref{sec:results} we apply the method to the calculation of the potential energy curves of the C$_2$, N$_2$, O$_2$, F$_2$, and H$_{10}$ molecules up to the dissociation limit, representing prototypes of strongly correlated systems. Finally, we conclude in Section \ref{sec:conclusion}.
As the theory of the basis-set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Section \ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$. In Section \ref{sec:wee} we introduce the effective non-divergent interaction in the basis set $\Bas$, which leads us to the definition of the effective local range-separation parameter in Section \ref{sec:mur}. Then, in Section \ref{sec:functional} we expose the new approximate complementary functionals based on RSDFT. The generic form of such functionals is exposed in Section \ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Section \ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Section \ref{sec:requirements}. Finally, the actual form of the functionals used in this work are introduced in Section \ref{sec:final_def_func}.
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}
where the notation $\Psi\rightarrow\den$ means that the wave function $\Psi$ yields the density $n$. The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', i.e. densities coming from wave functions expandable in the Hilbert space generated by the basis set $\Bas$. In the following, we always implicitly consider only such densities representable in $\Bas$. With this restriction, Eq.~\eqref{eq:levy} gives then an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a fast convergence with the size of the basis set, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is a very good approximation to the exact ground-state energy: $E_0^\Bas\approx E_0$.
In the present context, it is important to notice that in the definition of Eq.~\eqref{eq:levy_func} the wave functions $\Psi$ involved have no restriction to a finite basis set, i.e. they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then propose to decompose $F[\den]$ as
where $\wf{}{\Bas}$ are wave functions expandable in the Hilbert space generated by $\basis$, and $\efuncden{\den}$ is the density functional complementary to the basis set $\Bas$ defined as
where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density extracted from $\wf{}{\Bas}$. Therefore, with Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional accounting for the correlation effects not included in the basis set.
As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy (see Eqs. (12)-(15) of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18})
where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation regimes. Two requirements on the approximations for this purpose are i) size consistency and ii) spin-multiplet degeneracy.
As originally shown by Kato\cite{Kat-CPAM-57}, the cusp in the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. In other words, the impact of the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction at the coalescence point.
As originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (Section D and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
and $\Gam{pq}{rs}=2\mel*{\wf{}{\Bas}}{\aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, and $\fbasis$ is
As shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, i.e.
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 $\murpsi$
As all WFT calculations in this work are performed within the frozen-core approximation, we use the valence-only version of the various quantities needed for the complementary basis functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas=\Cor\bigcup\BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}~\cite{TouGorSav-TCA-05}. Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{})=\nabla\denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv\ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur\equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form
where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle~\cite{PerBurErn-PRL-96}. Before introducing the different flavors of approximate functionals that we will use here (see Section~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form.
The functional form of $\ecmd(\argecmd)$ in Eq.~\ref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to0$ limit, it reduces to the usual PBE correlation functional
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
which is expected for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ molecule which is the archetype of strongly correlated systems. Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu\rightarrow\infty$ like all RSDFT short-range functionals \begin{equation}
Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary basis functional $\efuncdenpbe{\argebasis}$ satisfies two important properties.
First, thanks to the properties in Eq.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, whatever the wave function $\psibasis$ used to define the local range-separation parameter $\mu(\br{})$,
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}, ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see Eq.~\eqref{eq:lim_n2}).
\subsection{Requirements for the approximate functionals in the strong-correlation regime}
\subsubsection{Requirements: size-consistency and spin-multiplet degeneracy}
An important requirement for any electronic-structure method is size-consistency, i.e. the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems $A$ and $B$ dissociate in closed-shell systems, as in the case of weak 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 (CAS) wave function which, provided that the active space has been properly chosen, leads to additive energies.
Another important requirement is spin-multiplet degeneracy, i.e. 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 $A+B$ is generally low spin while the ground states of the fragments $A$ and $B$ are high spin and can have multiple $S_z$ components.
\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain spin-multiplet degeneracy}
A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependency on the spin polarization $\zeta(\br{})$ use the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}).
It has been proposed to replace in functionals 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})
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 independent from $S_z$. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents the disadvantage that since this expression was derived for a single-determinant wave function, and it does not appear well justified to use it for a multideterminant wave function as well. In particular, for a multideterminant wave function, it may happen that $1-2\; n_{2}/n^2 < 0$ and thus in this cas Eq.~\eqref{eq:def_effspin} gives a complex value of $\tilde{\zeta}$~\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 $\zeta=0$, i.e. to always use the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, i.e. for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for $\mu$ sufficiently large, 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 Eq.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}). The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Therefore, we propose here to also test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures a $S_z$ independence and, as will be numerically shown, very weakly affects the accuracy of the functional.
Since $\efuncdenpbe{\argebasis}$ is a single integral over $\mathbb{R}^3$ of local quantities ($n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$,$\mu(\br{})$), in the case of non-overlapping fragments $A\ldots B$ it can be written as the sum of two local contributions: one coming from the integration over the region of the subsystem $A$ and the other one from the region of the subsystem $B$. Therefore, a sufficient condition for the size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem $A\ldots B$. 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, i.e. $\Psi_{A\ldots B}^{\basis}=\Psi_A^{\basis}\Psi_B^{\basis}$. In the case where the two subsystems $A$ and $B$ dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CAS wave function is sufficient to recover this property. The condition for the active space involved in the CAS wave function is that it has to lead to size-consistent energies in the limit of dissociated fragments.
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 CAS wave functions in order to ensure size consistency of all local quantities. The difference between the different flavors of functionals are only on i) the type of spin polarization used, and ii) the type of on-top pair density used.
Regarding the spin polarization that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin} and calculated from the CAS wave function, and ii) a \textit{zero} spin polarization.
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
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 CAS wave function through $\tilde{\zeta}$.
Another approach to approximate the exact on top pair density consists in using directly the on-top pair density of the CAS wave function. Following the work of some of the previous authors~\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
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''.
i) PBE-UEG-$\tilde{\zeta}$ which uses the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg} and the effective spin polarization of Eq.~\eqref{eq:def_effspin}
Potential energy curves of the H$_{10}$ chain with equally distant atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the 1) cc-pVDZ, 2) cc-pVTZ, and 3) cc-pVQZ basis sets. The MRCI+Q energies and the estimated exact energies are from Ref. \onlinecite{h10_prx}.
The purpose of the present paper being the study of the basis-set correction in the regime of strong correlation, we study the potential energy curves up to the dissociation limit of a H$_{10}$ chain with equally distant atoms and the C$_2$, N$_2$, O$_2$, and F$_2$ molecules. 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 C$_2$, N$_2$, O$_2$, and F$_2$ molecules, approximations to the FCI energies are obtained using converged frozen-core (1s orbitals are kept frozen) CIPSI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.} (see Refs. \onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the Quantum Package software~\cite{QP2}. The estimated exact potential energy curves are obtained from Ref. \onlinecite{LieCle-JCP-74a}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be below 0.5 mH. In the case of the H$_{10}$ 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 basis 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. These CASSCF wave functions correspond to the following active spaces: 10 electrons in 10 orbitals for H$_{10}$, 8 electrons in 8 electrons for C$_2$, 10 electrons in 8 orbitals for N$_2$, 12 electrons in 8 orbitals for O$_2$, and 14 electrons in 8 orbitals for F$_2$.
Also, as the frozen core approximation is used in all our CIPSI calculations, we use the corresponding valence-only complementary functionals. Therefore, all density-like quantities exclude any contribution from the core 1s orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}.
\caption{Atomization energies $D_0$ (in mH) and associated errors (in square brackets) with respect to the estimated exact values computed at different approximation levels with different basis sets.}
The study of the H$_{10}$ 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 Figure \ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X=D,T,Q) basis sets for different levels of approximations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy curves are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no bizarre behaviors are found when stretching the bonds, which shows that the functionals are robust when reaching the strong correlation regime.
More quantitatively, the values of $D_0$ are within chemical accuracy (i.e., an error below 1.4 mH) from the cc-pVTZ basis set when using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, whereas such an accuracy is not reached at the cc-pVQZ basis set using standard MRCI+Q.
Regarding in more details the performance of the different types of approximate functionals, the results show that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference on $D_0$ being 0.3 mH), and they give slightly more accurate results than PBE-UEG-$\tilde{\zeta}$. These findings bring two important clues on the role of the different physical ingredients used in the functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function (see Eq.~\eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see Eq.~\eqref{eq:def_n2ueg}); ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. Point ii) is important as it shows that spin polarization in density-functional approximations essentially plays the same role as that of the on-top pair density.
Potential energy curves of the C$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}.
Potential energy curves of the N$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}.
Potential energy curves of the O$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}.
Potential energy curves of the F$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref. \onlinecite{LieCle-JCP-74a}.
The C$_2$, N$_2$, O$_2$ and F$_2$ molecules are complementary to the H$_{10}$ system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of H$_{10}$, but 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. Also, O$_2$ has a triplet ground state and is therefore a good check for the performance of the dependence on the spin polarization of the different types of functionals proposed here.
We report in Figure \ref{fig:C2}, \ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves computed using the aug-cc-pVDZ and aug-cc-pVTZ basis sets of C$_2$, N$_2$, O$_2$, and N$_2$, respectively, for different approximation levels. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}.
Just as in the case of H$_{10}$, the quality of $D_0$ are globally improved by adding the basis-set correction and it is remarkable that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ give 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 a high-spin system like O$_2$. More quantitatively, an error below 1.0 mH on the estimated exact valence-only $D_0$ is found for N$_2$, O$_2$, and F$_2$ with the aug-cc-pVTZ basis set using the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at the near-FCI level. In the case of C$_2$ with the aug-cc-pVTZ basis set, an error of about 5.5 mH is found with respect to the estimated exact $D_0$. Such an error is remarkably large with respect to the other diatomic molecules studied here and might be associated to the level of strong correlation in the C$_2$ molecule.
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 basis functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron cusp: the local range-separation function $\mu(\br{})$ is designed by looking at the electron-electron coalescence point and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is expected.
In the present paper we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We studied the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit at near-FCI level in increasing 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 fitting of this effective interaction with the long-range interaction used in RS-DFT, iii) the use of a complementary correlation functional of RS-DFT. In the present paper, we investigated points 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 a) spin-multiplet degeneracy, and b) 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 avoid dependence to any form of spin polarization without loss of accuracy. This avoids the commonly used effective spin polarization calculated from a multideterminant wave function originally proposed in Ref. \onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly become complex-valued for some multideterminant wave functions. From a more fundamental aspect, this shows that the spin polarization in DFT-related frameworks only mimics the role of 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 but C$_2$, whereas the uncorrected near-FCI results are far from that accuracy within the same basis set. In the case of C$_2$, an error of 5.5 mH is obtained with respect to the estimated exact $D_0$, and we leave for further study the detailed investigation of the reasons of this relatively unusual poor performance of the basis-set correction.
Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long internuclear distances it cannot recover the dispersion interactions missing because of the incompleteness of the basis set. This behaviour is actually expected as the dispersion interactions are long-range correlation effects and the present approach was designed to only recover electron correlation effects near the electron-electron coalescence.
Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary information) that it is minor with respect to WFT methods for all systems and basis sets studied here. We thus believe that this approach is a significant step towards calculations near the CBS limit for strongly correlated systems.
