small changes in Intro

Manuscript/srDFT_SC.tex

@@ -276,7 +276,7 @@
 
 \begin{document}	
 
-\title{A density-based basis-set correction for weak and strong correlation}
+\title{A density-based basis-set correction for strongly correlated molecular systems}
 
 \author{Emmanuel Giner}
 \email{emmanuel.giner@lct.jussieu.fr}
@@ -296,9 +296,7 @@
 
 
 \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 \alert{enables the use of} RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. To study 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 functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. Specifically, we systematically investigate the dependence of the functional 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. [JT: I don't like this sentence in the abstract in particular because it is not clear what the "effective" spin polarization is.]
-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 potential energy curves along the whole range of distances.
+We extend to strongly correlated molecular 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 enables the use of RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis set. To study 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 functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. Specifically, we investigate the dependence of the functional 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. 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 potential energy curves along the whole range of internuclear distances.
 \end{abstract}
 
 \maketitle
@@ -306,19 +304,19 @@ Quantitatively, we show that the basis-set correction reaches chemical accuracy
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Introduction}
 %%%%%%%%%%%%%%%%%%%%%%%%
-The general goal of quantum chemistry is to provide reliable theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular systems. 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 \alert{to} 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 made to reduce the computational cost of WFT, DFT still remains the workhorse of quantum computational chemistry. 
+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 to 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 made 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) \cite{HatKloKohTew-CR-12} or long range (London dispersion interactions). \cite{AngDobJanGou-BOOK-20} The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining challenge 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 \cite{GorSeiSav-PCCP-08,GagTruLiCarHoyBa-ACR-17} and WFT is king for the treatment of strongly correlated systems. 
 
 In practice, WFT uses a finite one-electron basis set. 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 simultaneously 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, independence of the energy with respect to the spin projection $S_z$) 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 energy (and related properties) with respect to the size of the \alert{one-electron} 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-PRL-19,IrmGru-JCP-2019} 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 distance. 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-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, Refs.~\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}
+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 energy (and related properties) with respect to the size of the one-electron 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-PRL-19,IrmGru-JCP-2019} 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 distance. 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-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, Refs.~\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 $\mu$.
+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 $\mu$.
 
 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 long-range 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-JCTC-20} 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.
+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}. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%