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Julien Toulouse 2019-12-11 21:48:10 +01:00
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@ -641,53 +641,49 @@ Also, as the frozen core approximation is used in all our CIPSI calculations, we
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*}
\label{tab:d0}
\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 the extrapolated valence-only non relativistic calculations of Ref. \onlinecite{BytLaiRuedenJCP05}.
}
\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.}
\begin{ruledtabular}
\begin{tabular}{lcccc}
\begin{tabular}{llcccc}
System/basis & MRCI+Q$^a$ & (MRCI+Q)+$\pbeuegXi$ & (MRCI+Q)+$\pbeontXi$ & (MRCI+Q)+$\pbeontns$ \\
System & Basis set & 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}$, cc-pvqz & 661.2$/$4.2 & 664.1$/$1.3 & 666.4$/$-1.0 & 666.5$/$-1.1 \\
\hline
& \multicolumn{4}{c}{Estimated exact$^a$} \\
& \multicolumn{4}{c}{665.4 } \\
H$_{10}$ & 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]
\hline
System & Basis set & 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] \\
& 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]
\hline
System/basis & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$ \\
System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
\hline
C$_2$, aug-cc-pvdz & 204.6$/$29.5 & 218.0$/$16.1 & 217.4$/$16.7 & 217.0$/$17.1 \\
C$_2$, aug-cc-pvtz & 223.4$/$10.9 & 228.1$/$6.0 & 228.6$/$5.5 & 226.5$/$5.6 \\
\hline
& \multicolumn{4}{c}{Estimated exact$^b$} \\
& \multicolumn{4}{c}{234.1 } \\
N$_2$ & 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]
\hline
System/basis & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$ \\
System & Basis set & 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 \\
N$_2$, aug-cc-pvtz & 348.5$/$14.2 & 361.5$/$1.2 & 363.5$/$-0.5 & 363.2$/$-0.3 \\
\hline
& \multicolumn{4}{c}{Estimated exact$^b$} \\
& \multicolumn{4}{c}{362.7 } \\
O$_2$ & 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]
\hline
System/basis & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$ \\
System & Basis set & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$\\
\hline
O$_2$, aug-cc-pvdz & 171.4$/$20.5 & 187.6$/$4.3 & 187.6$/$4.3 & 187.1$/$4.8 \\
O$_2$, aug-cc-pvtz & 184.5$/$7.4 & 190.3$/$1.6 & 191.2$/$0.7 & 191.0$/$0.9 \\
\hline
& \multicolumn{4}{c}{Estimated exact$^b$} \\
& \multicolumn{4}{c}{191.9 } \\
\hline
F$_2$, aug-cc-pvdz & 49.6$/$12.6 & 54.8$/$7.4 & 54.9$/$7.3 & 54.8$/$7.4 \\
F$_2$, aug-cc-pvtz & 59.3$/$2.9 & 61.2$/$1.0 & 61.5$/$0.7 & 61.5$/$0.7 \\
\hline
& \multicolumn{4}{c}{Estimated exact$^b$} \\
& \multicolumn{4}{c}{62.2 } \\
F$_2$ & 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} \\
\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}
\label{tab:extensiv_closed}
\end{table*}
@ -714,7 +710,7 @@ 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{
C$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
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}.
\label{fig:C2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -726,7 +722,7 @@ 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{
N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
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}.
\label{fig:N2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -738,7 +734,7 @@ 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{
O$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
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}.
\label{fig:O2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -750,7 +746,7 @@ 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{
F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
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}.
\label{fig:F2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -763,30 +759,20 @@ Just as in the case of H$_{10}$, the quality of $D_0$ are globally improved by a
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$_2$ linear molecules up to full dissociation limits at near FCI level in increasing basis sets, and investigated how the basis set correction affects the convergence toward the CBS limits of the PES 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 such effective interaction with a long-range interaction used in RS-DFT,
iii) the use of complementary correlation functional of RS-DFT.
In the present paper, we investigated points i) and iii) in the context of strong correlation and focussed on PES and atomization energies.
More precisely, we proposed a new scheme to design functionals fulfilling a) $S_z$ invariance, b) size extensivity. To achieve such requirements we proposed to use CASSCF wave functions leading to extensive energies, and to develop functionals using only $S_z$ invariant density-related quantities.
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.
The development of new $S_z$ invariant and size extensive 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 with multi-configurational wave function whose mathematical definition originally proposed by Perdrew and co-workers in Ref. \cite{PerSavBur-PRA-95} has only a clear mathematical ground for a single Slater determinant and can be become complex-valued in the case of multi-configurational wave functions. From a more fundamental aspect, this shows that the spin-polarization in DFT-related frameworks only mimic's the role of the on-top density.
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.
Regarding the results of the present approach, the basis set correction systematically improves the near FCI calculation in a given basis set. More quantitatively, it is shown that the atomization energy $D_0$ is within the chemical accuracy for all systems but C$_2$ within a triple zeta quality basis set, whereas the near FCI values 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.
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.
Also, it is shown that the basis set correction gives substantial differential contribution along the PES only close to the equilibrium geometry, meaning that it cannot recover the dispersion forces missing because the incompleteness of the basis set. Although it can be looked as a failure of the basis set correction, in our context such behaviour is actually preferable as the dispersion forces are long-range effects and the present approach was designed to recover electronic correlation effects near the electron coalescence.
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.
Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary materials) that it is minor with respect to WFT methods for all systems and basis set studied here. We believe that such approach is a significant step towards calculations near the CBS limit for strongly correlated systems.
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 interatomic 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.
Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary information) that it is minor with respect to WFT methods for all systems and basis sets studied here. We thus believe that this approach is a significant step towards calculations near the CBS limit for strongly correlated systems.
\bibliography{srDFT_SC,biblio}