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Manuscript/srDFT_SC.aux
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@ -39,8 +39,7 @@
\citation{exicted}
\citation{GinPraFerAssSavTou-JCP-18}
\@writefile{toc}{\contentsline {section}{\numberline {II}Theory}{2}{section*.4}}
\@writefile{toc}{\contentsline {subsection}{\numberline {A}Basic formal equations}{2}{section*.5}}
\newlabel{sec:basic}{{II\tmspace +\thinmuskip {.1667em}A}{2}{}{section*.5}{}}
\newlabel{sec:theory}{{II}{2}{}{section*.4}{}}
\citation{GinPraFerAssSavTou-JCP-18}
\citation{GinPraFerAssSavTou-JCP-18}
\citation{GinPraFerAssSavTou-JCP-18}
@ -48,7 +47,8 @@
\citation{kato}
\citation{GinPraFerAssSavTou-JCP-18}
\citation{GinPraFerAssSavTou-JCP-18}
\citation{GinPraFerAssSavTou-JCP-18}
\@writefile{toc}{\contentsline {subsection}{\numberline {A}Basic formal equations}{3}{section*.5}}
\newlabel{sec:basic}{{II\tmspace +\thinmuskip {.1667em}A}{3}{}{section*.5}{}}
\newlabel{eq:levy}{{1}{3}{}{equation.2.1}{}}
\newlabel{eq:levy_func}{{2}{3}{}{equation.2.2}{}}
\newlabel{eq:e0approx}{{5}{3}{}{equation.2.5}{}}
@ -57,9 +57,7 @@
\newlabel{eq:wbasis}{{6}{3}{}{equation.2.6}{}}
\newlabel{eq:fbasis}{{8}{3}{}{equation.2.8}{}}
\newlabel{eq:cbs_wbasis}{{10}{3}{}{equation.2.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of an range-separation parameter varying in real space}{3}{section*.7}}
\newlabel{sec:mur}{{II\tmspace +\thinmuskip {.1667em}C}{3}{}{section*.7}{}}
\newlabel{eq:weelr}{{11}{3}{}{equation.2.11}{}}
\citation{GinPraFerAssSavTou-JCP-18}
\bibdata{srDFT_SCNotes,srDFT_SC}
\bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}}
\bibcite{ScoTho-JCP-17}{{2}{2017}{{Scott\ and\ Thom}}{{}}}
@ -88,6 +86,20 @@
\bibcite{CafGinScemRam-JCTC-14}{{25}{2014}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Giner, Scemama,\ and\ Ram{\'\i }rez-Sol{\'\i }s}}}
\bibcite{GinSceCaf-JCP-15}{{26}{2015}{{Giner, Scemama,\ and\ Caffarel}}{{}}}
\bibcite{CafAplGinScem-arxiv-16}{{27}{2016{}}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of an range-separation parameter varying in real space}{4}{section*.7}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {D}Approximation for $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ : link with RSDFT}{4}{section*.8}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Generic form of the approximated functionals}{4}{section*.9}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{4}{section*.10}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{4}{section*.11}}
\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{4}{section*.12}}
\newlabel{sec:results}{{III}{4}{}{section*.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{4}{section*.13}}
\newlabel{sec:conclusion}{{IV}{4}{}{section*.13}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{4}{figure.1}}
\newlabel{fig:N2_avdz}{{1}{4}{N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.1}{}}
\bibcite{CafAplGinSce-JCP-16}{{28}{2016{}}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Applencourt, Giner,\ and\ Scemama}}}
\bibcite{SchEva-JCP-16}{{29}{2016}{{Schriber\ and\ Evangelista}}{{}}}
\bibcite{LiuHofJCTC-16}{{30}{2016}{{Liu\ and\ Hoffmann}}{{}}}
@ -99,14 +111,6 @@
\bibcite{Zim-JCP-17}{{36}{2017}{{Zimmerman}}{{}}}
\bibcite{LiOttHolShaUmr-JCP-2018}{{37}{2018}{{Li\ \emph {et~al.}}}{{Li, Otten, Holmes, Sharma,\ and\ Umrigar}}}
\bibcite{ChiHolOttUmrShaZim-JPCA-18}{{38}{2018}{{Chien\ \emph {et~al.}}}{{Chien, Holmes, Otten, Umrigar, Sharma,\ and\ Zimmerman}}}
\newlabel{eq:cbs_mu}{{14}{4}{}{equation.2.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D}Approximation for $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ : link with RSDFT}{4}{section*.8}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Generic form of the approximated functionals}{4}{section*.9}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{4}{section*.10}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{4}{section*.11}}
\@writefile{toc}{\contentsline {section}{\numberline {III}Results}{4}{section*.12}}
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\newlabel{fig:N2_avdz}{{1}{4}{N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.1}{}}
\bibcite{SceBenJacCafLoo-JCP-18}{{39}{2018{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Benali, Jacquemin, Caffarel,\ and\ Loos}}}
\bibcite{LooSceBloGarCafJac-JCTC-18}{{40}{2018}{{Loos\ \emph {et~al.}}}{{Loos, Scemama, Blondel, Garniron, Caffarel,\ and\ Jacquemin}}}
\bibcite{GarSceGinCaffLoo-JCP-18}{{41}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}}
@ -118,6 +122,10 @@
\bibcite{KutKlo-JCP-91}{{47}{1991}{{Kutzelnigg\ and\ Klopper}}{{}}}
\bibcite{NogKut-JCP-94}{{48}{1994}{{Noga\ and\ Kutzelnigg}}{{}}}
\bibcite{HalHelJorKloKocOlsWil-CPL-98}{{49}{1998}{{Halkier\ \emph {et~al.}}}{{Halkier, Helgaker, J{\o }rgensen, Klopper, Koch, Olsen,\ and\ Wilson}}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{5}{figure.2}}
\newlabel{fig:N2_avtz}{{2}{5}{N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{5}{figure.3}}
\newlabel{fig:F2_avdz}{{3}{5}{F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.3}{}}
\bibcite{IrmHulGru-arxiv-19}{{50}{2019}{{Irmler, Hummel,\ and\ Grüneis}}{{}}}
\bibcite{Ten-TCA-12}{{51}{2012}{{Ten-no}}{{}}}
\bibcite{TenNog-WIREs-12}{{52}{2012}{{Ten-no\ and\ Noga}}{{}}}
@ -129,10 +137,12 @@
\bibcite{TouColSav-PRA-04}{{58}{2004}{{Toulouse, Colonna,\ and\ Savin}}{{}}}
\bibcite{FraMusLupTou-JCP-15}{{59}{2015}{{Franck\ \emph {et~al.}}}{{Franck, Mussard, Luppi,\ and\ Toulouse}}}
\bibcite{AngGerSavTou-PRA-05}{{60}{2005}{{\'Angy\'an\ \emph {et~al.}}}{{\'Angy\'an, Gerber, Savin,\ and\ Toulouse}}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{5}{figure.2}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.5}}
\newlabel{fig:H10_vdz}{{5}{6}{H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.6}}
\newlabel{fig:H10_vtz}{{6}{6}{H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.6}{}}
\bibcite{GolWerSto-PCCP-05}{{61}{2005}{{Goll, Werner,\ and\ Stoll}}{{}}}
\bibcite{TouGerJanSavAng-PRL-09}{{62}{2009}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Gerber, Jansen, Savin,\ and\ \'Angy\'an}}}
\bibcite{JanHenScu-JCP-09}{{63}{2009}{{Janesko, Henderson,\ and\ Scuseria}}{{}}}
@ -142,12 +152,6 @@
\bibcite{FroTouJen-JCP-07}{{67}{2007}{{Fromager, Toulouse,\ and\ Jensen}}{{}}}
\bibcite{FroCimJen-PRA-10}{{68}{2010}{{Fromager, Cimiraglia,\ and\ Jensen}}{{}}}
\bibcite{HedKneKieJenRei-JCP-15}{{69}{2015}{{Hedeg{\r a}rd\ \emph {et~al.}}}{{Hedeg{\r a}rd, Knecht, Kielberg, Jensen,\ and\ Reiher}}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.4}}
\newlabel{fig:F2_avtz}{{4}{6}{F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.5}}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. }}{6}{figure.6}}
\newlabel{fig:H10_vtz}{{6}{6}{H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.6}{}}
\bibcite{HedTouJen-JCP-18}{{70}{2018}{{Hedeg{\r a}rd, Toulouse,\ and\ Jensen}}{{}}}
\bibcite{FerGinTou-JCP-18}{{71}{2019}{{Fert{\'e}, Giner,\ and\ Toulouse}}{{}}}
\bibcite{GinPraFerAssSavTou-JCP-18}{{72}{2018}{{Giner\ \emph {et~al.}}}{{Giner, Pradines, Fert\'e, Assaraf, Savin,\ and\ Toulouse}}}
@ -157,5 +161,5 @@
\citation{aip41Control}
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@ -10,3 +10,4 @@
\BOOKMARK [3][-]{section*.10}{Introduction of the effective spin-density}{section*.8}% 10
\BOOKMARK [3][-]{section*.11}{Requirement for B for size extensivity}{section*.8}% 11
\BOOKMARK [1][-]{section*.12}{Results}{section*.2}% 12
\BOOKMARK [1][-]{section*.13}{Conclusion}{section*.2}% 13

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Manuscript/srDFT_SC.tex
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@ -269,7 +269,7 @@ In the MR approaches, the zeroth order wave function consists in a linear combin
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 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.
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.
@ -277,11 +277,13 @@ An alternative point of view is to leave the short-range correlation effects to
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 direction of strongly correlated systems.
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}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%
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}.
@ -387,6 +389,7 @@ which is fundamental to guarantee the good behaviour of the theory at the CBS li
\subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\label{sec:results}
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat.eps}
@ -459,6 +462,8 @@ which is fundamental to guarantee the good behaviour of the theory at the CBS li
\end{figure}
\section{Conclusion}
\label{sec:conclusion}