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@ -481,7 +481,7 @@ It is also noteworthy that, with the present definition, $\wbasisval$ still tend
\subsubsection{Generic form of the approximate functionals}
\label{sec:functional_form}
As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}~\cite{TouGorSav-TCA-05}. Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form
As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the \titou{complementary basis} functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}\cite{TouGorSav-TCA-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form
\begin{equation}
\begin{aligned}
\label{eq:def_ecmdpbebasis}
@ -634,7 +634,8 @@ iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair den
\includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat.eps}
\includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}
\caption{
Potential energy curves of the H$_{10}$ chain with equally distant atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the 1) cc-pVDZ, 2) cc-pVTZ, and 3) cc-pVQZ basis sets. The MRCI+Q energies and the estimated exact energies are from Ref.~\onlinecite{h10_prx}.
Potential energy curves of the H$_{10}$ chain with equally distant atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top), cc-pVTZ (middle), and cc-pVQZ (bottom) basis sets.
The MRCI+Q energies and the estimated exact energies have been extracted from Ref.~\onlinecite{h10_prx}.
\label{fig:H10}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -652,50 +653,46 @@ Also, as the frozen core approximation is used in all our CIPSI calculations, we
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*}
\label{tab:d0}
\caption{Atomization energies $D_0$ (in mH) and associated errors (in square brackets) with respect to the estimated exact values computed at different approximation levels with different basis sets.}
\caption{Atomization energies $D_0$ (in mH) and associated errors (in square brackets) with respect to the estimated exact values computed at different approximation levels with various basis sets.}
\begin{ruledtabular}
\begin{tabular}{llcccc}
System & Basis set & MRCI+Q$^a$ & (MRCI+Q)+$\pbeuegXi$ & (MRCI+Q)+$\pbeontXi$ & (MRCI+Q)+$\pbeontns$ \\
System & Basis set & MRCI+Q\fnm[1] & (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] \\
\ce{H10} & cc-pVDZ & 622.1 [43.3] & 642.6 [22.8] & 649.2 [16.2] & 649.5 [15.9] \\
& cc-pVTZ & 655.2 [10.2] & 661.9 [3.5] & 666.0 [-0.6] & 666.0 [-0.6] \\
& cc-pVQZ & 661.2 [4.2] & 664.1 [1.3] & 666.4 [-1.0] & 666.5 [-1.1] \\[0.1cm]
%\cline{2-6}
&\multicolumn{5}{c}{Estimated exact$^a$: 665.4} \\[0.2cm]
& &\multicolumn{4}{c}{Estimated exact:\fnm[1] 665.4} \\[0.2cm]
\hline
System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
& & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
\hline
C$_2$ & aug-cc-pVDZ & 204.6 [29.5] & 218.0 [16.1] & 217.4 [16.7] & 217.0 [17.1] \\
\ce{C2} & aug-cc-pVDZ & 204.6 [29.5] & 218.0 [16.1] & 217.4 [16.7] & 217.0 [17.1] \\
& aug-cc-pVTZ & 223.4 [10.9] & 228.1 [6.0] & 228.6 [5.5] & 226.5 [5.6] \\[0.1cm]
%\cline{2-6}
& \multicolumn{5}{c}{Estimated exact$^b$: 234.1} \\[0.2cm]
& & \multicolumn{4}{c}{Estimated exact:\fnm[2] 234.1} \\[0.2cm]
\hline
System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
& & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
\hline
N$_2$ & aug-cc-pVDZ & 321.9 [40.8] & 356.2 [6.5] & 355.5 [7.2] & 354.6 [8.1] \\
\ce{N2} & aug-cc-pVDZ & 321.9 [40.8] & 356.2 [6.5] & 355.5 [7.2] & 354.6 [8.1] \\
& aug-cc-pVTZ & 348.5 [14.2] & 361.5 [1.2] & 363.5 [-0.5] & 363.2 [-0.3] \\[0.1cm]
& \multicolumn{5}{c}{Estimated exact$^b$: 362.7} \\[0.2cm]
& & \multicolumn{4}{c}{Estimated exact:\fnm[2] 362.7} \\[0.2cm]
\hline
System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
& & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
\hline
O$_2$ & aug-cc-pVDZ & 171.4 [20.5] & 187.6 [4.3] & 187.6 [4.3] & 187.1 [4.8] \\
\ce{O2} & aug-cc-pVDZ & 171.4 [20.5] & 187.6 [4.3] & 187.6 [4.3] & 187.1 [4.8] \\
& aug-cc-pVTZ & 184.5 [7.4] & 190.3 [1.6] & 191.2 [0.7] & 191.0 [0.9] \\[0.1cm]
& \multicolumn{5}{c}{Estimated exact$^b$: 191.9} \\[0.2cm]
& & \multicolumn{4}{c}{Estimated exact:\fnm[2] 191.9} \\[0.2cm]
\hline
System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
& & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
\hline
F$_2$ & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\
\ce{F2} & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\
& aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm]
& \multicolumn{5}{c}{Estimated exact$^b$: 62.2} \\
& & \multicolumn{4}{c}{Estimated exact:\fnm[2] 62.2} \\
\end{tabular}
\end{ruledtabular}
\begin{flushleft}
\vspace{-0.2cm}
$^a$ From Ref.~\onlinecite{h10_prx}. \\
$^b$ From the extrapolated valence-only non-relativistic calculations of Ref.~\onlinecite{BytLaiRuedenJCP05}.
\end{flushleft}
\fnt[1]{From Ref.~\onlinecite{h10_prx}.}
\fnt[2]{From the extrapolated valence-only non-relativistic calculations of Ref.~\onlinecite{BytLaiRuedenJCP05}.}
\label{tab:extensiv_closed}
\end{table*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -704,7 +701,7 @@ $^b$ From the extrapolated valence-only non-relativistic calculations of Ref.~\o
\subsection{H$_{10}$ chain}
The study of the H$_{10}$ chain with equally distant atoms is a good prototype of strongly-correlated systems 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 at near CBS values can be obtained (see Ref.~\onlinecite{h10_prx} for a detailed study of this problem).
The study of the \ce{H10} chain with equally distant atoms is a good prototype of strongly-correlated systems 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 at near CBS values can be obtained (see Ref.~\onlinecite{h10_prx} for a detailed study of this problem).
We report in Figure \ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X=D,T,Q) basis sets for different levels of approximations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy curves are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no bizarre behaviors are found when stretching the bonds, which shows that the functionals are robust when reaching the strong correlation regime.
@ -721,7 +718,7 @@ Regarding in more details the performance of the different types of approximate
\includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat.eps}
\includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat_zoom.eps}
\caption{
Potential energy curves of the C$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
Potential energy curves of the \ce{C2} molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:C2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -733,7 +730,7 @@ Regarding in more details the performance of the different types of approximate
\includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.eps}
\includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps}
\caption{
Potential energy curves of the N$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
Potential energy curves of the \ce{N2} molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:N2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -745,7 +742,7 @@ Regarding in more details the performance of the different types of approximate
% \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.eps}
% \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.eps}
\caption{
Potential energy curves of the O$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
Potential energy curves of the \ce{O2} molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:O2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -757,29 +754,29 @@ Regarding in more details the performance of the different types of approximate
\includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat.eps}
\includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat_zoom.eps}
\caption{
Potential energy curves of the F$_2$ molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
Potential energy curves of the \ce{F2} molecule calculated with near-FCI and basis-set corrected near-FCI using the 1) aug-cc-pVDZ and 2) aug-cc-pVTZ basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:F2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The C$_2$, 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 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 $D_0$, 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. Also, O$_2$ has a triplet ground state and is therefore a good check for the performance of the dependence on the spin polarization of the different types of functionals proposed here.
The \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study as the level of strong correlation increases while stretching the bond 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 $D_0$, 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. Also, \ce{O2} has a triplet ground state and is therefore a good check for the performance of the dependence on the spin polarization of the different types of functionals proposed here.
We report in Figure \ref{fig:C2}, \ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves computed using the aug-cc-pVDZ and aug-cc-pVTZ basis sets of C$_2$, N$_2$, O$_2$, and N$_2$, respectively, for different approximation levels. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}.
Just as in the case of H$_{10}$, the quality of $D_0$ are globally improved by adding the basis-set correction and it is remarkable that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ give very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for a high-spin system like O$_2$. More quantitatively, an error below 1.0 mH on the estimated exact valence-only $D_0$ is found for N$_2$, O$_2$, and F$_2$ with the aug-cc-pVTZ basis set using the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at the near-FCI level. In the case of C$_2$ with the aug-cc-pVTZ basis set, an error of about 5.5 mH is found with respect to the estimated exact $D_0$. Such an error is remarkably large with respect to the other diatomic molecules studied here and might be associated to the level of strong correlation in the C$_2$ molecule.
Just as in the case of \ce{H10}, the quality of $D_0$ are globally improved by adding the basis-set correction and it is remarkable that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ give very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for a high-spin system like \ce{O2}. More quantitatively, an error below 1.0 mH on the estimated exact valence-only $D_0$ is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at the near-FCI level. In the case of \ce{C2} with the aug-cc-pVTZ basis set, an error of about 5.5 mH is found with respect to the estimated exact $D_0$. Such an error is remarkably large with respect to the other diatomic molecules studied here and might be associated to the level of strong correlation in the \ce{C2} molecule.
Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary basis functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron cusp: the local range-separation function $\mu(\br{})$ is designed by looking at the electron-electron coalescence point and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is expected.
\section{Conclusion}
\label{sec:conclusion}
In the present paper we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We studied the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit at near-FCI level in increasing basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems.
In the present paper we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We studied the \ce{H10}, \ce{C2}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit at near-FCI level in increasing basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems.
The DFT-based basis-set correction relies on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the Coulomb interaction projected into an incomplete basis set $\basis$; ii) the fitting of this effective interaction with the long-range interaction used in RS-DFT, iii) the use of a complementary correlation functional of RS-DFT. In the present paper, we investigated points i) and iii) in the context of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling a) spin-multiplet degeneracy, and b) size consistency. To fulfil such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and to develop functionals using only $S_z$-independent density-like quantities.
The development of new $S_z$-independent and size-consistent functionals has lead us to investigate the role of two related quantities: the spin-polarization and the on-top pair density. One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can avoid dependence to any form of spin polarization without loss of accuracy. This avoids the commonly used effective spin polarization calculated from a multideterminant wave function originally proposed in Ref.~\onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly become complex-valued for some multideterminant wave functions. From a more fundamental aspect, this shows that the spin polarization in DFT-related frameworks only mimics the role of the on-top pair density.
Regarding the results of the present approach, the basis-set correction systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple-zeta quality basis sets chemically accurate atomization energies $D_0$ are obtained for all systems but C$_2$, whereas the uncorrected near-FCI results are far from that accuracy within the same basis set. In the case of C$_2$, an error of 5.5 mH is obtained with respect to the estimated exact $D_0$, and we leave for further study the detailed investigation of the reasons of this relatively unusual poor performance of the basis-set correction.
Regarding the results of the present approach, the basis-set correction systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple-zeta quality basis sets chemically accurate atomization energies $D_0$ are obtained for all systems but \ce{C2}, whereas the uncorrected near-FCI results are far from that accuracy within the same basis set. In the case of \ce{C2}, an error of 5.5 mH is obtained with respect to the estimated exact $D_0$, and we leave for further study the detailed investigation of the reasons of this relatively unusual poor performance of the basis-set correction.
Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long internuclear distances it cannot recover the dispersion interactions missing because of the incompleteness of the basis set. This behaviour is actually expected as the dispersion interactions are long-range correlation effects and the present approach was designed to only recover electron correlation effects near the electron-electron coalescence.