almost converged

Manuscript/srDFT_SC.bbl

@@ -6,7 +6,7 @@
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@@ -166,6 +166,13 @@
   {\bibfield  {journal} {\bibinfo  {journal} {Phys. Chem. Chem. Phys.}\
   }\textbf {\bibinfo {volume} {9}},\ \bibinfo {pages} {1921} (\bibinfo {year}
   {2007})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Ten-no}(2007)}]{Ten-CPL-07}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
+  {Ten-no}},\ }\href {\doibase https://doi.org/10.1016/j.cplett.2007.09.006}
+  {\bibfield  {journal} {\bibinfo  {journal} {Chemical Physics Letters}\
+  }\textbf {\bibinfo {volume} {447}},\ \bibinfo {pages} {175 } (\bibinfo {year}
+  {2007})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Shiozaki}\ and\ \citenamefont
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   \BibitemOpen
@@ -176,6 +183,31 @@
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   {https://doi.org/10.1063/1.3489000} \BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Shiozaki}, \citenamefont {Knizia},\ and\
+  \citenamefont {Werner}(2011)}]{TorKniWer-JCP-11}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
+  {Shiozaki}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Knizia}}, \
+  and\ \bibinfo {author} {\bibfnamefont {H.-J.}\ \bibnamefont {Werner}},\
+  }\href {\doibase 10.1063/1.3528720} {\bibfield  {journal} {\bibinfo
+  {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo {volume}
+  {134}},\ \bibinfo {pages} {034113} (\bibinfo {year} {2011})},\ \Eprint
+  {http://arxiv.org/abs/https://doi.org/10.1063/1.3528720}
+  {https://doi.org/10.1063/1.3528720} \BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Demel}\ \emph {et~al.}(2012)\citenamefont {Demel},
+  \citenamefont {Kedžuch}, \citenamefont {Švaňa}, \citenamefont {Ten-no},
+  \citenamefont {Pittner},\ and\ \citenamefont
+  {Noga}}]{DemStanMatTenPitNog-PCCP-12}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {O.}~\bibnamefont
+  {Demel}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Kedžuch}},
+  \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Švaňa}}, \bibinfo
+  {author} {\bibfnamefont {S.}~\bibnamefont {Ten-no}}, \bibinfo {author}
+  {\bibfnamefont {J.}~\bibnamefont {Pittner}}, \ and\ \bibinfo {author}
+  {\bibfnamefont {J.}~\bibnamefont {Noga}},\ }\href {\doibase
+  10.1039/C2CP23198K} {\bibfield  {journal} {\bibinfo  {journal} {Phys. Chem.
+  Chem. Phys.}\ }\textbf {\bibinfo {volume} {14}},\ \bibinfo {pages} {4753}
+  (\bibinfo {year} {2012})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Guo}\ \emph {et~al.}(2017)\citenamefont {Guo},
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@@ -189,6 +221,51 @@
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+  {\bibinfo  {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo
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+  \BibitemOpen
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+  \BibitemOpen
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+  {NoStop}%
+\bibitem [{\citenamefont {Booth}\ \emph {et~al.}(2012)\citenamefont {Booth},
+  \citenamefont {Cleland}, \citenamefont {Alavi},\ and\ \citenamefont
+  {Tew}}]{BooCleAlaTew-JCP-12}%
+  \BibitemOpen
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+  \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Alavi}}, \ and\ \bibinfo
+  {author} {\bibfnamefont {D.~P.}\ \bibnamefont {Tew}},\ }\href {\doibase
+  10.1063/1.4762445} {\bibfield  {journal} {\bibinfo  {journal} {The Journal of
+  Chemical Physics}\ }\textbf {\bibinfo {volume} {137}},\ \bibinfo {pages}
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+\bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
+  {Savin}(2006{\natexlab{a}})}]{GorSav-PRA-06}%
+  \BibitemOpen
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 \bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
-  {Savin}(2006{\natexlab{a}})}]{GoriSav-PRA-06}%
+  {Savin}(2006{\natexlab{b}})}]{GoriSav-PRA-06}%
   \BibitemOpen
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 \bibitem [{\citenamefont {Paziani}\ \emph {et~al.}(2006)\citenamefont
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Manuscript/srDFT_SC.tex

@@ -321,15 +321,15 @@ Fulfilling these two objectives is a rather complicated task for a given approxi
 %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. 
 
-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. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires a quite involved mathematical development to adapt to a new theory. In the context of strong correlation, several multi-reference methods have been extended to explicitly correlation (see for instance Ref. \cite{Ten-CPL-07,TorVal-JCP-09,ShiWer-JCP-10,KedDemPitTenNog-CPL-11,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,HauMaoMukKlo-CPL-12,GuoSivValNee-JCP-17}), including so-called universal approaches potentially applicable to any electronic structure approaches\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}. 
+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. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires a quite involved mathematical development to adapt to a new theory. In the context of strong correlation, several multi-reference methods have been extended to explicitly correlation (see for instance Ref. \cite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on so-called universal F12 which are potentially applicable to any electronic structure approaches\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}. 
 
-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) as the WFT part deals with a smooth non divergent interaction, the exact wave function has no cusp removed and therefore the basis set convergence is much faster\cite{FraMusLupTou-JCP-15}. 
+An alternative point of view to improve the convergence towards the CBS limit 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 mix DFT and WFT 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) as the WFT part deals with a smooth non divergent interaction, the exact wave function has no cusp\cite{GorSav-PRA-06} 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, and 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{GinSceTouLoo-JCP-19}. 
 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. Within the present development, two important formal properties are imposed: 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}. 
+Then in section \ref{sec:results} we discuss the potential energy surfaces (PES) of the C$_2$, N$_2$, O$_2$, F$_2$ and H$_{10}$ molecules up to full dissociation as a prototype of strongly correlated problems. Finally, we conclude in section \ref{sec:conclusion}. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Theory}
@@ -804,19 +804,25 @@ F$_2$, aug-cc-pvtz          &   59.3$/$2.9            &    61.2$/$1.0       &
 In the present paper we have extended the recently proposed DFT-based basis set correction to strongly correlated systems. 
 We studied the H$_{10}$, C$_2$, N$_2$, O$_2$ and F$_2$ linear molecules up to full dissociation limits at near FCI level in increasing basis sets, and investigated how the basis set correction affect the convergence toward the CBS limits of the PES of these molecular systems. 
 
-The DFT-based basis set correction rely 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 regular coulomb interaction projected into an incomplete basis set $\basis$, ii) the fitting of such effective interaction with a long-range interaction used in RS-DFT, iii) the use of complementary correlation functional of RS-DFT. 
+The DFT-based basis set correction rely on three aspects: 
-In the present paper, we investigated points i) and iii) in order to  study atomization energies. 
+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 $\basis$, 
-In this context, we proposed a new scheme to design functionals fulfilling a) $S_z$ invariance, b) size extensivity. To achieve such requirements we proposed to use CASSCF wave functions leading to extensive energies, and to develop functionals using only $S_z$ invariant density-related quantities. 
+ii) the fitting of such effective interaction with a long-range interaction used in RS-DFT, 
+iii) the use of complementary correlation functional of RS-DFT. 
+In the present paper, we investigated points i) and iii) in the context of strong correlation and focussed on atomization energies. 
+More precisely, we proposed a new scheme to design functionals fulfilling a) $S_z$ invariance, b) size extensivity. To achieve such requirements we proposed to use CASSCF wave functions leading to extensive energies, and to develop functionals using only $S_z$ invariant density-related quantities. 
 
 The development of new $S_z$ invariant and size extensive 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 loos of accuracy. 
-From a practical point of view, this allows to remove the use of the effective spin-polarization\cite{PerSavBur-PRA-95} which has only a clear mathematical ground for single Slater determinant and can be become complex-valued in the case of multi-configurational wave functions. From a more fundamental aspect, this shows that the spin-polarization in DFT-related frameworks only mimic's the role of the on-top density. 
+From a practical point of view, this allows to avoid the commonly used effective spin-polarization\cite{PerSavBur-PRA-95} with multi-configurational wave function, 
+which is satisfying as the definition originally proposed by Perdrew and co-workers in Ref. \cite{PerSavBur-PRA-95} has only a clear mathematical ground for single Slater determinant and can be become complex-valued in the case of multi-configurational wave functions. From a more fundamental aspect, this shows that the spin-polarization in DFT-related frameworks only mimic's the role of the on-top density. 
 
 Regarding the results of the present approach, the basis set correction systematically improves the near FCI calculation in a given basis set. More quantitatively, it is shown that the atomization energy $D_0$ is within the chemical accuracy for all systems but C$_2$ within a triple zeta quality basis set, whereas the near FCI values 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 along the PES only close to the equilibrium geometry, meaning that it cannot recover the dispersion forces missing because the incompleteness of the basis set. Although it can be looked as a failure of the basis set correction, in our context such behaviour is actually preferable as the dispersion forces are long-range effects and the present approach was designed to recover electronic correlation effects near the electron coalescence.  
 
+Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary materials) that it is completely negligible with respect to WFT methods for all systems and basis set studied here. We believe that it opens the road to a significant improvement of the results in WFT for strong correlation.  
+
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 \end{document}