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Pierre-Francois Loos 2019-12-12 23:37:52 +01:00
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Manuscript/srDFT_SC.tex
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@ -624,7 +624,7 @@ We then define \titou{four} functionals:
\end{equation}
\item[iv)] $\pbeontns$ which combines the on-top pair density of Eq.~\eqref{eq:def_n2extrap} and zero spin polarization:
\begin{equation}
\label{eq:def_pbeueg_iii}
\label{eq:def_pbeueg_iv}
\bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns).
\end{equation}
\end{itemize}
@ -727,7 +727,7 @@ 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 \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}.
Potential energy curves of the \ce{C2} molecule calculated with near-FCI and basis-set corrected near-FCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:C2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -739,7 +739,7 @@ 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 \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}.
Potential energy curves of the \ce{N2} molecule calculated with near-FCI and basis-set corrected near-FCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:N2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -751,7 +751,7 @@ 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 \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}.
Potential energy curves of the \ce{O2} molecule calculated with near-FCI and basis-set corrected near-FCI using the aug-cc-pVDZ (top) and \titou{aug-cc-pVTZ (bottom)} basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:O2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -763,7 +763,7 @@ 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{
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}.
Potential energy curves of the \ce{F2} molecule calculated with near-FCI and basis-set corrected near-FCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:F2}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -779,17 +779,19 @@ Regarding now the performance of the basis-set correction along the whole potent
\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 \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.
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-large 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 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 fit of this effective interaction with the long-range interaction used in RS-DFT, and iii) the use of a short-range, complementary correlation functional from RS-DFT. In the present paper, we investigated 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 i) spin-multiplet degeneracy, and ii) size consistency. To fulfil such requirements we proposed to use \titou{CAS} 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.
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 eschew its spin polarization dependency without loss of accuracy. This avoids the commonly used effective spin polarization \trashPFL{calculated from a multideterminant wave function} originally proposed in Ref.~\onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly becoming complex-valued in the multideterminant case. From a more fundamental aspect, this shows that, in a DFT framework, the spin polarization mimics the role of the on-top pair density.
\titou{Consequently, we believe that one could potentially develop new families of density functional approximations where the spin polarization is abondonned and replaced by 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 \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 mHa 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 this accuracy within the same basis set. In the case of \ce{C2}, an error of 5.5 mHa is obtained with respect to the estimated exact $D_0$, and we leave for future study the detailed investigation of the reasons behind 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.
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 energy missing because of the basis set incompleteness. This behaviour is actually expected as dispersion has a long-range correlation nature and the present approach is designed to recover only short-range correlation effects.
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.
\PFL{I think the paragraph below should be placed WAY earlier.}
Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary information) that it represents, for all systems and basis sets studied here, a minor computational overhead. We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems.
\bibliography{srDFT_SC}