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\begin{thebibliography}{85}%
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\makeatletter
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10.1021/acs.jctc.9b00176} {\bibfield {journal} {\bibinfo {journal} {J.
Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {15}},\ \bibinfo {pages}
{3591} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Lie}\ and\ \citenamefont
{Clementi}(1974)}]{LieCle-JCP-74a}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~C.}\ \bibnamefont
{Lie}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Clementi}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
}\textbf {\bibinfo {volume} {60}},\ \bibinfo {pages} {1275} (\bibinfo {year}
{1974})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Motta}\ \emph {et~al.}(2017)\citenamefont {Motta},
\citenamefont {Ceperley}, \citenamefont {Chan}, \citenamefont {Gomez},
\citenamefont {Gull}, \citenamefont {Guo}, \citenamefont {Jim\'enez-Hoyos},

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\BOOKMARK [1][-]{section*.18}{Results}{section*.2}% 18
\BOOKMARK [2][-]{section*.19}{Computational details}{section*.18}% 19
\BOOKMARK [2][-]{section*.20}{Dissociation of equally distant H10 chains}{section*.18}% 20
\BOOKMARK [2][-]{section*.21}{Dissociation of F2, N2}{section*.18}% 21
\BOOKMARK [2][-]{section*.21}{Dissociation of N2, O2 and F2}{section*.18}% 21
\BOOKMARK [1][-]{section*.22}{Conclusion}{section*.2}% 22

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@ -492,10 +492,10 @@ Such a property is guaranteed independently by i) the definition of the effectiv
\subsection{Requirements for the approximated functionals in the strong correlation regime}
\subsubsection{Requirements: separability of the energies and $S_z$ invariance}
An important requirement for any electronic structure method is the extensivity of the energy, \textit{i. e.} the additivity of the energies in the case of non interacting fragments, which is particularly important to avoid any ambiguity in computing interaction energies.
When two subsystems $A$ and $B$ dissociate in closed shell systems, as in the case of weak interactions for instance, a simple HF wave function leads to extensive energies.
When the two subsystems dissociate in open shell systems, such as in covalent bond breaking, it is well known that the HF approach fail and an alternative is to use a CASSCF wave function which, provided that the active space has been properly chosen, leads to additives energies.
Another important requirement is the independence of the energy with respect to the $S_z$ component of a given spin state.
An important requirement for any electronic structure method is the extensivity of the energy, \textit{i. e.} the additivity of the energies in the case of non interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies.
When two subsystems $A$ and $B$ dissociate in closed shell systems, as in the case of weak interactions for instance, a simple RHF wave function leads to extensive energies.
When the two subsystems dissociate in open shell systems, such as in covalent bond breaking, it is well known that the RHF approach fail and an alternative is to use a CASSCF wave function which, provided that the active space has been properly chosen, leads to additives energies.
Another important requirement is the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function.
Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components.
\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance}
@ -579,17 +579,17 @@ iii) and the PBE-ot-$0{\zeta}$ where no spin polarization is used and which ther
\section{Results}
\label{sec:results}
\subsection{Computational details}
The purpose of the present paper being the study of the basis set correction in the regime of strong correlation, we propose to study the potential energy surfaces (PES) until dissociation of an equally distant H$_{10}$ chain, F$_2$ and N$_2$.
The purpose of the present paper being the study of the basis set correction in the regime of strong correlation, we propose to study the potential energy surfaces (PES) until dissociation of an equally distant H$_{10}$ chain, together with the N$_2$, O$_2$ and F$_2$ molecules.
In a given basis set, to compute the approximation of the exact ground state energy using equation \eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the complementary basis set energy functional $\efuncbasisFCI$.
In the case of the F$_2$ and N$_2$ molecules, the approximation to the FCI energies are obtained using converged frozen-core (1s orbitals are kept frozen) CIPSI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.}
(see Refs \onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the Quantum Package software\cite{QP2}.
For all geometry and basis sets, the error with respect to actual FCI energy are estimated to be below 0.5 mH.
In the case of H$_{10}$, the approximation to $\efci$ are obtained from the data from of Ref. \onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space).
In the case of N$_2$, O$_2$ and F$_2$, the approximation to the FCI energies are obtained using converged frozen-core (1s orbitals are kept frozen) CIPSI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.}
(see Refs \onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the Quantum Package software\cite{QP2}. The estimated exact PES are obtained from Ref. \onlinecite{LieCle-JCP-74a}.
For all geometry and basis sets, the error with respect to actual FCI energies are estimated to be below 0.5 mH.
In the case of H$_{10}$, the approximation to $\efci$ together with the estimated exact curves are obtained from the data from of Ref. \onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space).
Regarding the complementary basis set energy functional, we use minimal valence CASSCF wave functions computed with the GAMESS-US software\cite{gamess} to obtain the wave functions $\psibasis$. Therefore, all all density related quantities (such as the total densities, different flavors of spin polarizations and on-top pair densities) together with the $\murpsi$ of equation \eqref{eq:def_mur} are obtained at minimal valence CASSCF level.
These CASSCF wave functions correspond to active spaces containing two electrons in the bonding and anti-bonding $\sigma$ orbitals and six electrons in the bonding and anti-bonding $\sigma$, $\pi_x$ and $\pi_y$ orbitals for F$_2$ and N$_2$, respectively.
Regarding the H$_{10}$ linear chains, the CASSCF wave functions contain ten electrons in the 10 orbitals needed to correctly dissociate into 10 hydrogen atoms in their 1s state.
Regarding the complementary basis set energy functional, we use CASSCF wave functions computed with the GAMESS-US software\cite{gamess} to obtain the wave functions $\psibasis$. Therefore, all density related quantities (such as the total densities, different flavors of spin polarizations and on-top pair densities) together with the $\murpsi$ of equation \eqref{eq:def_mur} are obtained at minimal valence CASSCF level.
These CASSCF wave functions correspond to the following active spaces: ten electrons in ten orbitals for H$_{10}$, six electrons in six orbitals for N$_2$, eight electrons in six orbitals for O$_2$ and two electrons in two orbitals for F$_2$.
In all cases, the orbitals involved in the active spaces are such that the dissociation limit corresponds to atomic.
\subsection{Dissociation of equally distant H$_{10}$ chains}
The study of equally distant H$_{10}$ chains is a good prototype for the study of strong correlation regime as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations can be performed at near CBS values can be obtained (see Ref. \onlinecite{h10_prx} for detailed study of that problem).
@ -604,10 +604,10 @@ These observations bring two important clues on the role of the different physic
i) the explicit use of the on-top pair density coming from the CASSCF wave function (see equation \eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see equation \eqref{eq:def_n2ueg}),
ii) removing the dependence on any kind of spin polarizations does not lead to significant loss of accuracy once that a minimal description of the on-top pair density of the system is used.
\subsection{Dissociation of F$_2$, N$_2$}
The study of N$_2$ and F$_2$ are complementary to the H$_{10}$ system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of H$_{10}$, but also these systems exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of $D_0$, while the shape of the curve far from the equilibrium geometry is governed by dispersion forces which are medium to long-range weak correlation effects.
\subsection{Dissociation of N$_2$, O$_2$ and F$_2$}
The study of N$_2$, O$_2$ and F$_2$ molecules are complementary to the H$_{10}$ system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of H$_{10}$, but also these systems exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of $D_0$, while the shape of the curve far from the equilibrium geometry is governed by dispersion forces which are medium to long-range weak correlation effects. Also, O$_2$ exhibit a triplet ground state and therefore is good check for the performance of the various types of functionals and their dependence on the spin polarization.
We report in figures \ref{fig:N2_avdz} and \ref{fig:F2_avdz} (\ref{fig:N2_avtz} and \ref{fig:F2_avtz}) the potential energy curves computed using the aug-cc-pVDZ (aug-cc-pVTZ) basis sets of F$_2$ and N$_2$, respectively, for different levels of computations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in table \ref{tab:d0}.
We report in figures \ref{fig:N2_avdz}, \ref{fig:O2_avdz} and \ref{fig:F2_avdz} (\ref{fig:N2_avtz}, \ref{fig:O2_avtz} and \ref{fig:F2_avtz}) the potential energy curves computed using the aug-cc-pVDZ (aug-cc-pVTZ) basis sets of N$_2$, O$_2$ and N$_2$, respectively, for different levels of computations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in table \ref{tab:d0}.
Just as the case of H$_{10}$, the quality of $D_0$ are globally improved and the chemical accuracy is reached at the aug-cc-pVTZ using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, which also give very similar results.
The latter observation confirms that even in the presence of higher electron density, the dependence on the on-top pair density allows to remove the dependence of any kind of spin polarizations.
@ -618,7 +618,8 @@ Interestingly, the complementary basis set functional fail provide a noticeable
\includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat.eps}\\
\includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat_zoom.eps}
\caption{
H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
H$_{10}$, cc-pvdz: Comparison between MRCI+Q and corrected MRCI+Q energies and the estimated exact one.
The MRCI+Q and estimated exact values are obtained from Ref. \onlinecite{h10_prx}.
\label{fig:H10_vdz}}
\end{figure}
@ -628,7 +629,8 @@ Interestingly, the complementary basis set functional fail provide a noticeable
\includegraphics[width=\linewidth]{data/H10/DFT_vtzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{fig2c}
\caption{
H$_{10}$, cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
H$_{10}$, cc-pvtz: Comparison between MRCI+Q and corrected MRCI+Q energies and the estimated exact one.
The MRCI+Q and estimated exact values are obtained from Ref. \onlinecite{h10_prx}.
\label{fig:H10_vtz}}
\end{figure}
@ -638,7 +640,8 @@ Interestingly, the complementary basis set functional fail provide a noticeable
\includegraphics[width=\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{fig2c}
\caption{
H$_{10}$, cc-pvqz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
H$_{10}$, cc-pvqz: Comparison between MRCI+Q and corrected MRCI+Q energies and the estimated exact one.
The MRCI+Q and estimated exact values are obtained from Ref. \onlinecite{h10_prx}.
\label{fig:H10_vqz}}
\end{figure}
@ -667,17 +670,20 @@ Interestingly, the complementary basis set functional fail provide a noticeable
\begin{table*}
\label{tab:d0}
\caption{Dissociation energy ($D_0$) computed at different levels in various basis sets. }
\caption{$D_0$ in mH and associated error with respect to the estimated exact values computed at different levels in various basis sets.
$^a$: The MRCI+Q and estimated exact curves are obtained from Ref. \onlinecite{h10_prx}.
$^b$: The estimated exact $D_0$ are obtained from Ref. \onlinecite{LieCle-JCP-74a}.
}
\begin{ruledtabular}
\begin{tabular}{lcccc}
System/basis & MRCI+Q & (MRCI+Q)+$\pbeuegXi$ & (MRCI+Q)+$\pbeontXi$ & (MRCI+Q)+$\pbeontns$ \\
System/basis & MRCI+Q$^a$ & (MRCI+Q)+$\pbeuegXi$ & (MRCI+Q)+$\pbeontXi$ & (MRCI+Q)+$\pbeontns$ \\
\hline
H$_{10}$, cc-pvdz & 622.1$/$43.3 & 642.6$/$22.8 & 649.2$/$16.2 & 649.5$/$15.9 \\
H$_{10}$, cc-pvtz & 655.2$/$10.2 & 661.9$/$3.5 & 666.0$/$-0.6 & 666.0$/$-0.6 \\
H$_{10}$, cc-pvqz & 661.2$/$4.2 & 664.1$/$1.3 & 666.4$/$-1.0 & 666.5$/$-1.1 \\
\hline
& \multicolumn{4}{c}{Estimated exact} \\
& \multicolumn{4}{c}{Estimated exact$^a$} \\
& \multicolumn{4}{c}{665.4 } \\
\hline
System/basis & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$ \\
@ -685,13 +691,21 @@ H$_{10}$, cc-pvqz & 661.2$/$4.2 & 664.1$/$1.3
N$_2$, aug-cc-pvdz & 321.9$/ $42.3 & 356.0$/$8.2 & 355.5$/$8.7 & 354.5$/$ 9.7 \\
N$_2$, aug-cc-pvtz & 348.5$/$15.7 & 361.8$/$2.4 & 363.5$/$0.7 & 363.2$/$1.0 \\
\hline
& \multicolumn{4}{c}{Estimated exact} \\
& \multicolumn{4}{c}{Estimated exact$^b$} \\
& \multicolumn{4}{c}{364.2 } \\
\hline
F$_2$, aug-cc-pvdz & 49.6$/$11.1 & 54.5$/$6.2 & 54.7$/$6.0 & 54.5$/$6.3 \\
F$_2$, aug-cc-pvtz & 59.3$/$1.4 & 61.2$/$-0.5 & 61.6$/$-0.9 & 61.5$/$-0.8 \\
System/basis & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$ \\
\hline
O$_2$, aug-cc-pvdz & 171.4$/$20.1 & 187.6$/$3.9 & 187.6$/$3.9 & 187.1$/$4.4 \\
O$_2$, aug-cc-pvtz & 184.5$/$7.0 & 190.3$/$1.2 & 191.2$/$0.3 & 191.0$/$0.5 \\
\hline
& \multicolumn{4}{c}{Estimated exact} \\
& \multicolumn{4}{c}{Estimated exact$^b$} \\
& \multicolumn{4}{c}{191.5 } \\
\hline
F$_2$, aug-cc-pvdz & 49.6$/$11.1 & 54.5$/$6.2 & 54.7$/$6.0 & 54.5$/$6.3 \\
F$_2$, aug-cc-pvtz & 59.3$/$1.4 & 61.2$/$-0.5 & 61.6$/$-0.9 & 61.5$/$-0.8 \\
\hline
& \multicolumn{4}{c}{Estimated exact$^b$} \\
& \multicolumn{4}{c}{60.7 } \\
\end{tabular}
\end{ruledtabular}