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@ -622,6 +622,52 @@ The performance of each of these functionals is tested in the following. Note th
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Results} \section{Results}
\label{sec:results} \label{sec:results}
%& \tabc{$\largemu$}
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\subsection{Computational details}
We present potential energy curves of small molecules up to the dissociation limit
to investigate the performance of the basis-set correction in regimes of both weak and strong correlation.
The considered systems are the \ce{H10} linear chain with equally-spaced atoms, and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics.
The computation of the ground-state energy in Eq.~\eqref{eq:e0approx} in a given basis set requires approximations to the FCI energy $\efci$ and to the basis-set correction $\efuncbasisFCI$.
For diatomics with the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-JCP-92} energies are obtained using frozen-core selected-CI calculations (using the CIPSI algorithm) followed by the extrapolation scheme proposed by Holmes \textit{et al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more detail). All these calculations are performed with the latest version of \textsc{Quantum Package}, \cite{QP2} and will be labeled as exFCI in the following. In the case of \ce{F2}, we also use the correlation energy extrapolated by intrinsic scaling (CEEIS) \cite{BytNagGorRue-JCP-07} as an estimate of the FCI correlation energy with the cc-pVXZ (X $=$ D, T, and Q) basis sets.~\cite{Dun-JCP-89} The estimated exact potential energy curves are obtained from experimental data \cite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be of the order of $0.5$~mHa.
For the three diatomics, we performed an additional exFCI calculation with the aug-cc-pVQZ basis set at the equilibrium geometry to obtain reliable estimates of the FCI/CBS dissociation energy.
In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data 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).
\alert{We note that, even though we use near-FCI energies in this work, the DFT-based basis-set correction could also be applied to any approximation to FCI such as multireference perturbation theory, similarly to what was done for weakly correlated systems for which the basis-set correction was applied to CCSD(T) calculations. \cite{LooPraSceTouGin-JCPL-19}}
Regarding the complementary functional, we first perform full-valence CASSCF calculations with the GAMESS-US software~\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, effective spin polarization $\tilde{\zeta}(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ are calculated with this full-valence CASSCF wave function. The CASSCF calculations are performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}. We note that, instead of using CASSCF wave functions for $\psibasis$, one could of course use the same selected-CI wave functions used for calculating the energy but the calculations of $n_2(\br{})$ and $\mu(\br{})$ would then be more costly.
Also, as the frozen-core approximation is used in all our selected-CI calculations, we use the corresponding valence-only complementary functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the 1s core orbitals, and the range-separation function follows the definition given in Eq.~\eqref{eq:def_mur_val}.
It should be stressed that the computational cost of the basis-set correction (see Appendix~\ref{app:computational}) is negligible compared to the cost the selected-CI calculations.
\subsection{H$_{10}$ chain}
The \ce{H10} chain with equally-spaced atoms is a prototype of strongly correlated systems as it consists in the simultaneous breaking of 10 interacting covalent $\sigma$ bonds.
As it is a relatively small system, benchmark calculations at near-CBS values are available (see Ref.~\onlinecite{h10_prx} for a detailed study of this system).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*} \begin{figure*}
\subfigure[cc-pVDZ]{ \subfigure[cc-pVDZ]{
\includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat.pdf} \includegraphics[width=0.45\linewidth]{data/H10/DFT_vdzE_relat.pdf}
@ -643,7 +689,6 @@ The performance of each of these functionals is tested in the following. Note th
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*} \begin{table*}
\caption{Atomization energies (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different levels of theory with various basis sets.} \caption{Atomization energies (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different levels of theory with various basis sets.}
@ -692,65 +737,19 @@ The performance of each of these functionals is tested in the following. Note th
\end{table*} \end{table*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation, and the corresponding atomization energies are reported in Table \ref{tab:d0}.
As a general trend, the addition of the basis-set correction globally improves
the quality of the potential energy curves, independently of the approximation level of $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\ie, the determination of the range-separation function and the definition of the functionals) is robust when reaching the strong-correlation regime.
In other words, smooth potential energy curves are obtained with the present basis-set correction.
More quantitatively, the values of the atomization energies are within chemical accuracy (\ie, an error below $1.4$ mHa) with the cc-pVTZ basis set when using the $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not yet reached at the standard MRCI+Q/cc-pVQZ level of theory.
\begin{table*} Analyzing more carefully the performance of the different types of approximate functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on the atomization energy being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provide two important clues on the role of the different physical ingredients included in these functionals: i) the explicit use of the on-top pair density originating from the CASSCF wave function [see Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [see Eq.~\eqref{eq:def_n2ueg}] which is somewhat understandable, and ii) removing the dependence on any kind of spin polarization does not lead to a significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it confirms that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density. This could have significant implications for the construction of more robust families of density-functional approximations within DFT.
{\color{red}
\caption{System-averaged on-top pair density $\langle n_2 \rangle$, extrapolated on-top pair density $\langle \mathring{n}_{2} \rangle$, and range-separation parameter $\langle \mu \rangle$ (all in atomic units) calculated with full-valence CASSCF and CIPSI wave functions (see text for details) for \ce{N2} and \ce{N} in the aug-cc-pVXZ basis sets (X $=$ D, T, and Q). All quantities were computed within the frozen-core approximation, \ie, excluding all contributions from the 1s orbitals.}
\begin{ruledtabular}
\begin{tabular}{lrccccccc}
%\begin{tabular}{lrccccccc}
System & \tabc{Basis set} &\tabc{$\ontopcas$}& \tabc{$\ontopextrap$}& \tabc{$\ontopcipsi$} & \tabc{$\ontopextrapcipsi$}& \tabc{$\muaverage$} & \tabc{$\muaveragecipsi$} \\
\hline
\ce{N2} & aug-cc-pVDZ & 1.17542 & 0.65966 & 1.02792 & 0.58228 & 0.946 & 0.962 \\
& aug-cc-pVTZ & 1.18324 & 0.77012 & 0.92276 & 0.61074 & 1.328 & 1.364 \\
& aug-cc-pVQZ & 1.18484 & 0.84012 & 0.83866 & 0.59982 & 1.706 & 1.746 \\[0.1cm]
\ce{N} & aug-cc-pVDZ & 0.34464 & 0.19622 & 0.25484 & 0.14686 & 0.910 & 0.922 \\
& aug-cc-pVTZ & 0.34604 & 0.22630 & 0.22344 & 0.14828 & 1.263 & 1.299 \\
& aug-cc-pVQZ & 0.34614 & 0.24666 & 0.21224 & 0.15164 & 1.601 & 1.653 \\
\end{tabular}
\end{ruledtabular}
\label{tab:d1}
}
\end{table*}
%& \tabc{$\largemu$} \subsection{Dissociation of diatomics}
%
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\subsection{Computational details} The \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study. The level of strong correlation in these diatomics also increases while stretching the bonds, similarly to the case of \ce{H10}, but in addition these molecules 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 the atomization energy at equilibrium, while the shape of the curve far from the equilibrium geometry is governed by dispersion interactions which are medium to long-range weak-correlation effects. \cite{AngDobJanGou-BOOK-20} The dispersion interactions in \ce{H10} play a minor role on the potential energy curve due to the much smaller number of near-neighbor electron pairs compared to \ce{N2}, \ce{O2} or \ce{F2}. Also, \ce{O2} has a triplet ground state and is therefore a good candidate for checking the spin-polarization dependence of the various functionals proposed here.
We present potential energy curves of small molecules up to the dissociation limit
to investigate the performance of the basis-set correction in regimes of both weak and strong correlation.
The considered systems are the \ce{H10} linear chain with equally-spaced atoms, and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics.
The computation of the ground-state energy in Eq.~\eqref{eq:e0approx} in a given basis set requires approximations to the FCI energy $\efci$ and to the basis-set correction $\efuncbasisFCI$.
For diatomics with the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-JCP-92} energies are obtained using frozen-core selected-CI calculations (using the CIPSI algorithm) followed by the extrapolation scheme proposed by Holmes \textit{et al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more detail). All these calculations are performed with the latest version of \textsc{Quantum Package}, \cite{QP2} and will be labeled as exFCI in the following. In the case of \ce{F2}, we also use the correlation energy extrapolated by intrinsic scaling (CEEIS) \cite{BytNagGorRue-JCP-07} as an estimate of the FCI correlation energy with the cc-pVXZ (X $=$ D, T, and Q) basis sets.~\cite{Dun-JCP-89} The estimated exact potential energy curves are obtained from experimental data \cite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be of the order of $0.5$~mHa.
For the three diatomics, we performed an additional exFCI calculation with the aug-cc-pVQZ basis set at the equilibrium geometry to obtain reliable estimates of the FCI/CBS dissociation energy.
In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data 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).
\alert{We note that, even though we use near-FCI energies in this work, the DFT-based basis-set correction could also be applied to any approximation to FCI such as multireference perturbation theory, similarly to what was done for weakly correlated systems for which the basis-set correction was applied to CCSD(T) calculations. \cite{LooPraSceTouGin-JCPL-19}}
Regarding the complementary functional, we first perform full-valence CASSCF calculations with the GAMESS-US software~\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, effective spin polarization $\tilde{\zeta}(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ are calculated with this full-valence CASSCF wave function. The CASSCF calculations are performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}. We note that, instead of using CASSCF wave functions for $\psibasis$, one could of course use the same selected-CI wave functions used for calculating the energy but the calculations of $n_2(\br{})$ and $\mu(\br{})$ would then be more costly.
Also, as the frozen-core approximation is used in all our selected-CI calculations, we use the corresponding valence-only complementary functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the 1s core orbitals, and the range-separation function follows the definition given in Eq.~\eqref{eq:def_mur_val}.
It should be stressed that the computational cost of the basis-set correction (see Appendix~\ref{app:computational}) is negligible compared to the cost the selected-CI calculations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*} \begin{figure*}
@ -768,22 +767,6 @@ It should be stressed that the computational cost of the basis-set correction (s
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{H$_{10}$ chain}
The \ce{H10} chain with equally-spaced atoms is a prototype of strongly correlated systems as it consists in the simultaneous breaking of 10 interacting covalent $\sigma$ bonds.
As it is a relatively small system, benchmark calculations at near-CBS values are available (see Ref.~\onlinecite{h10_prx} for a detailed study of this system).
We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation, and the corresponding atomization energies are reported in Table \ref{tab:d0}.
As a general trend, the addition of the basis-set correction globally improves
the quality of the potential energy curves, independently of the approximation level of $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\ie, the determination of the range-separation function and the definition of the functionals) is robust when reaching the strong-correlation regime.
In other words, smooth potential energy curves are obtained with the present basis-set correction.
More quantitatively, the values of the atomization energies are within chemical accuracy (\ie, an error below $1.4$ mHa) with the cc-pVTZ basis set when using the $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not yet reached at the standard MRCI+Q/cc-pVQZ level of theory.
Analyzing more carefully the performance of the different types of approximate functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on the atomization energy being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provide two important clues on the role of the different physical ingredients included in these functionals: i) the explicit use of the on-top pair density originating from the CASSCF wave function [see Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [see Eq.~\eqref{eq:def_n2ueg}] which is somewhat understandable, and ii) removing the dependence on any kind of spin polarization does not lead to a significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it confirms that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density. This could have significant implications for the construction of more robust families of density-functional approximations within DFT.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*} \begin{figure*}
\subfigure[aug-cc-pVDZ]{ \subfigure[aug-cc-pVDZ]{
@ -819,10 +802,6 @@ The estimated exact energies are based on a fit of experimental data and obtaine
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dissociation of diatomics}
The \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study. The level of strong correlation in these diatomics also increases while stretching the bonds, similarly to the case of \ce{H10}, but in addition these molecules 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 the atomization energy at equilibrium, while the shape of the curve far from the equilibrium geometry is governed by dispersion interactions which are medium to long-range weak-correlation effects. \cite{AngDobJanGou-BOOK-20} The dispersion interactions in \ce{H10} play a minor role on the potential energy curve due to the much smaller number of near-neighbor electron pairs compared to \ce{N2}, \ce{O2} or \ce{F2}. Also, \ce{O2} has a triplet ground state and is therefore a good candidate for checking the spin-polarization dependence of the various functionals proposed here.
We report in Figs.~\ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves of \ce{N2}, \ce{O2}, and \ce{F2} computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The atomization energies for each level of theory with different basis sets are reported in Table \ref{tab:d0}. We report in Figs.~\ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves of \ce{N2}, \ce{O2}, and \ce{F2} computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The atomization energies for each level of theory with different basis sets are reported in Table \ref{tab:d0}.
Just as in \ce{H10}, the accuracy of the atomization energies is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system like \ce{O2}. More quantitatively, an error below 1.0 mHa compared to the estimated exact valence-only atomization energy is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the accuracy of the results, which could have been anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule. \cite{HibHumByrLen-JCP-94} Just as in \ce{H10}, the accuracy of the atomization energies is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system like \ce{O2}. More quantitatively, an error below 1.0 mHa compared to the estimated exact valence-only atomization energy is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the accuracy of the results, which could have been anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule. \cite{HibHumByrLen-JCP-94}
@ -862,6 +841,28 @@ where $\mathring{n}_{2,\text{CISPI}}(\br{})=\ntwoextrap(n_{2,\text{CIPSI}}(\br{}
where $n_{\text{CASSCF}}(\br{})$ and $n_{\text{CIPSI}}(\br{})$ are the CASSCF and CIPSI densities, respectively. All the CIPSI quantities have been calculated with the largest variational wave function computed in the CIPSI calculation with a given basis, which contains here at least $10^7$ Slater determinants. In particular, $\murcipsi$ has been calculated from Eqs.~\eqref{eq:def_mur_val}--\eqref{eq:twordm_val} with the opposite-spin two-body density matrix $\Gam{pq}{rs}$ of the largest variational CIPSI wave function for a given basis. All quantities in Eqs.~\eqref{eq:ontopcas}--\eqref{eq:muaverage} were computed excluding all contributions from the 1s orbitals, \ie, they are ``valence-only'' quantities. where $n_{\text{CASSCF}}(\br{})$ and $n_{\text{CIPSI}}(\br{})$ are the CASSCF and CIPSI densities, respectively. All the CIPSI quantities have been calculated with the largest variational wave function computed in the CIPSI calculation with a given basis, which contains here at least $10^7$ Slater determinants. In particular, $\murcipsi$ has been calculated from Eqs.~\eqref{eq:def_mur_val}--\eqref{eq:twordm_val} with the opposite-spin two-body density matrix $\Gam{pq}{rs}$ of the largest variational CIPSI wave function for a given basis. All quantities in Eqs.~\eqref{eq:ontopcas}--\eqref{eq:muaverage} were computed excluding all contributions from the 1s orbitals, \ie, they are ``valence-only'' quantities.
} }
\begin{table*}
{\color{red}
\caption{System-averaged on-top pair density $\langle n_2 \rangle$, extrapolated on-top pair density $\langle \mathring{n}_{2} \rangle$, and range-separation parameter $\langle \mu \rangle$ (all in atomic units) calculated with full-valence CASSCF and CIPSI wave functions (see text for details) for \ce{N2} and \ce{N} in the aug-cc-pVXZ basis sets (X $=$ D, T, and Q). All quantities were computed within the frozen-core approximation, \ie, excluding all contributions from the 1s orbitals.}
\begin{ruledtabular}
\begin{tabular}{lrccccccc}
%\begin{tabular}{lrccccccc}
System & \tabc{Basis set} &\tabc{$\ontopcas$}& \tabc{$\ontopextrap$}& \tabc{$\ontopcipsi$} & \tabc{$\ontopextrapcipsi$}& \tabc{$\muaverage$} & \tabc{$\muaveragecipsi$} \\
\hline
\ce{N2} & aug-cc-pVDZ & 1.17542 & 0.65966 & 1.02792 & 0.58228 & 0.946 & 0.962 \\
& aug-cc-pVTZ & 1.18324 & 0.77012 & 0.92276 & 0.61074 & 1.328 & 1.364 \\
& aug-cc-pVQZ & 1.18484 & 0.84012 & 0.83866 & 0.59982 & 1.706 & 1.746 \\[0.1cm]
\ce{N} & aug-cc-pVDZ & 0.34464 & 0.19622 & 0.25484 & 0.14686 & 0.910 & 0.922 \\
& aug-cc-pVTZ & 0.34604 & 0.22630 & 0.22344 & 0.14828 & 1.263 & 1.299 \\
& aug-cc-pVQZ & 0.34614 & 0.24666 & 0.21224 & 0.15164 & 1.601 & 1.653 \\
\end{tabular}
\end{ruledtabular}
\label{tab:d1}
}
\end{table*}
\alert{ \alert{
We report in Table \ref{tab:d1} these quantities for \ce{N2} and \ce{N} for various basis sets. One notices that the system-averaged on-top pair density at the CIPSI level $\ontopcipsi$ is systematically lower than its CASSCF analogue $\ontopcas$, which is expected since short-range correlation, \ie, digging the correlation hole in a given basis set at near FCI level, is missing from the valence CASSCF wave function. We report in Table \ref{tab:d1} these quantities for \ce{N2} and \ce{N} for various basis sets. One notices that the system-averaged on-top pair density at the CIPSI level $\ontopcipsi$ is systematically lower than its CASSCF analogue $\ontopcas$, which is expected since short-range correlation, \ie, digging the correlation hole in a given basis set at near FCI level, is missing from the valence CASSCF wave function.
Also, $\ontopcipsi$ decreases in a monotonous way as the size of the basis set increases, leading to roughly a $20\%$ decrease from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas $\ontopcas$ is almost constant with respect to the basis set. Regarding the extrapolated on-top pair densities, $\ontopextrap$ and $\ontopextrapcipsi$, it is interesting to notice that they are substantially lower than their non-extrapolated counterparts, $\ontopcas$ and $\ontopcipsi$. Nevertheless, the behaviors of $\ontopextrap$ and $\ontopextrapcipsi$ are qualitatively different: $\ontopextrap$ clearly increases when enlarging the basis set whereas $\ontopextrapcipsi$ remains almost constant. More precisely, in the case of \ce{N2}, the value of $\ontopextrap$ increases by about 30$\%$ from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas the value of $\ontopextrapcipsi$ only fluctuates within 5$\%$ for the same basis sets. The behavior of $\ontopextrap$ can be understood by noticing that i) the value of $\murcas$ globally increases when enlarging the basis set (as evidenced by $\muaverage$), and ii) $\lim_{\mu \rightarrow \infty} \ntwoextrap(n_2,\mu) = n_2$ [see Eq.~\eqref{eq:def_n2extrap}]. Therefore, in the CBS limit, $\murcas \rightarrow \infty$ and one obtains Also, $\ontopcipsi$ decreases in a monotonous way as the size of the basis set increases, leading to roughly a $20\%$ decrease from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas $\ontopcas$ is almost constant with respect to the basis set. Regarding the extrapolated on-top pair densities, $\ontopextrap$ and $\ontopextrapcipsi$, it is interesting to notice that they are substantially lower than their non-extrapolated counterparts, $\ontopcas$ and $\ontopcipsi$. Nevertheless, the behaviors of $\ontopextrap$ and $\ontopextrapcipsi$ are qualitatively different: $\ontopextrap$ clearly increases when enlarging the basis set whereas $\ontopextrapcipsi$ remains almost constant. More precisely, in the case of \ce{N2}, the value of $\ontopextrap$ increases by about 30$\%$ from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas the value of $\ontopextrapcipsi$ only fluctuates within 5$\%$ for the same basis sets. The behavior of $\ontopextrap$ can be understood by noticing that i) the value of $\murcas$ globally increases when enlarging the basis set (as evidenced by $\muaverage$), and ii) $\lim_{\mu \rightarrow \infty} \ntwoextrap(n_2,\mu) = n_2$ [see Eq.~\eqref{eq:def_n2extrap}]. Therefore, in the CBS limit, $\murcas \rightarrow \infty$ and one obtains