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6
Manuscript/srDFT_SC.tex
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@ -317,7 +317,7 @@ An alternative way to improve the convergence towards the complete basis set (CB
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-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 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. \trashPFL{These systems represent prototypes of strongly correlated systems.} Finally, we conclude in Sec.~\ref{sec:conclusion}. 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}.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory} \section{Theory}
@ -328,7 +328,7 @@ As the theory behind the present basis-set correction has been exposed in detail
\subsection{Basic equations} \subsection{Basic equations}
\label{sec:basic} \label{sec:basic}
The exact ground-state energy $E_0$ of a $N$-electron system can, in principle, be obtained in DFT by a minimization over \titou{$N$-electron density} $\denr$ The exact ground-state energy $E_0$ of a $N$-electron system can, in principle, be obtained in DFT by a minimization over \titou{one-electron densities} $\denr$
\begin{equation} \begin{equation}
\label{eq:levy} \label{eq:levy}
E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\}, E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\},
@ -534,7 +534,7 @@ Another important requirement is spin-multiplet degeneracy, \ie, the independenc
A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$. In the case of the functional $\ecmd(\argecmd)$, this means removing the dependency on the spin polarization $\zeta(\br{})$ originating from the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ [see Eq.~\eqref{eq:def_ecmdpbe}]. A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$. In the case of the functional $\ecmd(\argecmd)$, this means removing the dependency on the spin polarization $\zeta(\br{})$ originating from the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ [see Eq.~\eqref{eq:def_ecmdpbe}].
To do so, it has been proposed to substitute 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}) To do so, it has been proposed to substitute 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}) \manu{is zero if square roots is complex}
\begin{equation} \begin{equation}
\label{eq:def_effspin} \label{eq:def_effspin}
\tilde{\zeta}(n,n_{2}) = \tilde{\zeta}(n,n_{2}) =

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new/O2_avtz/data/data_DFT_avtz
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@ -1,10 +1,9 @@
# # FCI PBE PBEot
1.95 -150.07019460 -0.0514036781 -0.0580919429 -0.0581323214 1.95 -150.06963414
2.10 -150.12342824 -0.0498191601 -0.0566199123 -0.0566634475 2.10 -150.12303107
2.2816 -150.14178444 -0.0483549667 -0.0552424714 -0.0552906635 2.2816 -150.14208721
2.40 -150.13899522 -0.0476012736 -0.0545246645 -0.0545764508 2.40 -150.13989970
2.60 -150.11950071 -0.0465873654 -0.0535431346 -0.0536019959 2.60 -150.12002711
3.00 -150.06324132 -0.0451765580 -0.0521176567 -0.0521957399 3.00 -150.06398251
4.00 -149.88767371 -0.0435573953 -0.0499577563 -0.0501613581 4.00 -149.96272858
5.00 -149.88756789 -0.0432114786 -0.0495348356 -0.0497456437 10.00 -149.95859616
10.0 -149.95830994

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new/O2_avtz/data/exact-O2
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@ -6,8 +6,4 @@
3.00 -150.2314 3.00 -150.2314
4.00 -150.1476 4.00 -150.1476
5.00 -150.1274 5.00 -150.1274
<<<<<<< HEAD
=======
7.00 -150.1218
>>>>>>> 7d17cec3065dbb2c36d90f24f286b41985ef6529
10.0 -150.1214 10.0 -150.1214