theory OK

Manuscript/srDFT_SC.bib

@@ -1,13 +1,25 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2019-12-13 10:34:29 +0100 
+%% Created for Pierre-Francois Loos at 2020-01-06 14:18:41 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{IrmGru-JCP-2019,
+	Author = {A. Irmler and A. Gruneis},
+	Date-Added = {2020-01-06 14:14:58 +0100},
+	Date-Modified = {2020-01-06 14:18:41 +0100},
+	Doi = {10.1063/1.5110885},
+	Journal = {J. Chem. Phys.},
+	Pages = {104107},
+	Title = {Particle-particle ladder based basis-set corrections applied to atoms and molecules using coupled-cluster theory},
+	Volume = {151},
+	Year = {2019},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.5110885}}
+
 @article{BooCleThoAla-JCP-11,
 	Author = {G. H. Booth and D. Cleland and A. J. W. Thom and A. Alavi},
 	Date-Added = {2019-12-13 10:13:33 +0100},
@@ -28,12 +40,14 @@
 	Volume = {340},
 	Year = {2001}}
 
-@article{LooPraSceGinTou-ARX-19,
+@article{LooPraSceGinTou-JCTC-20,
 	Author = {P.-F. Loos and B. Pradines and A. Scemama and E. Giner and J. Toulouse},
 	Date-Added = {2019-12-12 03:48:12 +0100},
-	Date-Modified = {2019-12-12 03:48:12 +0100},
-	Journal = {arXiv:1910.12238},
-	Title = {A density-based basis-set incompleteness correction for GW methods}}
+	Date-Modified = {2020-01-06 14:17:24 +0100},
+	Journal = {J. Chem. Theory Comput.},
+	Pages = {in press},
+	Title = {A density-based basis-set incompleteness correction for GW methods},
+	Year = {2020}}
 
 @article{KalMusTou-JCP-19,
 	Author = {C. Kalai and B. Mussard and J. Toulouse},
@@ -1772,19 +1786,17 @@
 	Note = {Note that this estimate of the exact well depth differs from the one used in Ref.~\onlinecite{UmrTouFilSorHeg-JJJ-XX} where we used instead the scalar-relativistic, valence-corrected estimate of Ref.~\onlinecite{BytRue-JCP-05} since calculations were performed with a relativistic pseudopotential.}}
 
 @article{BytNagGorRue-JCP-07,
-author = {Bytautas,Laimutis  and Nagata,Takeshi  and Gordon,Mark S.  and Ruedenberg,Klaus },
-title = {Accurate ab initio potential energy curve of F2. I. Nonrelativistic full valence configuration interaction energies using the correlation energy extrapolation by intrinsic scaling method},
-journal = {The Journal of Chemical Physics},
-volume = {127},
-number = {16},
-pages = {164317},
-year = {2007},
-doi = {10.1063/1.2800017},
-URL = {https://doi.org/10.1063/1.2800017},
-eprint = {https://doi.org/10.1063/1.2800017}
-}
-
-
+	Author = {Bytautas,Laimutis and Nagata,Takeshi and Gordon,Mark S. and Ruedenberg,Klaus},
+	Doi = {10.1063/1.2800017},
+	Eprint = {https://doi.org/10.1063/1.2800017},
+	Journal = {The Journal of Chemical Physics},
+	Number = {16},
+	Pages = {164317},
+	Title = {Accurate ab initio potential energy curve of F2. I. Nonrelativistic full valence configuration interaction energies using the correlation energy extrapolation by intrinsic scaling method},
+	Url = {https://doi.org/10.1063/1.2800017},
+	Volume = {127},
+	Year = {2007},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.2800017}}
 
 @article{CadWah-ADNDT-74,
 	Author = {P. E. Cade and A. C. Wahl},
@@ -5245,12 +5257,13 @@ eprint = {https://doi.org/10.1063/1.2800017}
 
 @article{HibHumByrLen-JCP-94,
 	Author = {{P. C. Hiberty S. Humbel, C. P. Byrman and J. H. van Lenthe}},
+	Doi = {10.1063/1.468459},
 	Journal = {J. Chem. Phys.},
 	Pages = {5969},
+	Url = {https://doi.org/10.1063/1.468459},
 	Volume = {101},
-        doi = {10.1063/1.468459},
-        URL = {https://doi.org/10.1063/1.468459},
-	Year = {1994}}
+	Year = {1994},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.468459}}
 
 @article{HibHum-JCP-94,
 	Author = {P. C. Hiberty and S. Humbel and P. Archirel},
@@ -12609,13 +12622,16 @@ eprint = {https://doi.org/10.1063/1.2800017}
 	Volume = {94},
 	Year = {1991}}
 
-@misc{IrmHulGru-arxiv-19,
-	Archiveprefix = {arXiv},
+@article{IrmHulGru-PRL-19,
 	Author = {Andreas Irmler and Felix Hummel and Andreas Gr{\"u}neis},
-	Eprint = {1903.05458},
-	Primaryclass = {cond-mat.mtrl-sci},
+	Date-Modified = {2020-01-06 14:18:23 +0100},
+	Doi = {10.1103/PhysRevLett.123.156401},
+	Journal = {Phys. Rev. Lett.},
+	Pages = {156401},
 	Title = {On the duality of ring and ladder diagrams and its importance for many-electron perturbation theories},
-	Year = {2019}}
+	Volume = {123},
+	Year = {2019},
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.123.156401}}
 
 @article{GruHirOhnTen-JCP-17,
 	Author = {A. Gr\"uneis and S. Hirata and Y.-Y. Ohnishi and S. Ten-no},

Manuscript/srDFT_SC.tex

@@ -309,12 +309,12 @@ The difficulty of obtaining a reliable theoretical description of a given chemic
 
 In practice, WFT uses a finite one-particle basis set (here denoted as $\basis$). 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 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, invariance with respect to the spin operator $S_z$) 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 electronic energy (and related 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, \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-arxiv-19} 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 distances. 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- and four-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, 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}
+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 electronic energy (and related 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, \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 distances. 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- and four-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, 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 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 $\mu$.
 % which can be seen as an empirical parameter. 
 
-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 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-ARX-19} The goal of the present work is to further develop this approach for the description of strongly correlated systems. 
+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 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.
 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}.