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Manuscript/srDFT_SC.bbl
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@ -896,15 +896,13 @@
{Ruedenberg}},\ }\href {\doibase 10.1063/1.1869493} {\bibfield {journal}
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {122}},\
\bibinfo {pages} {154110} (\bibinfo {year} {2005})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Booth}\ \emph {et~al.}(2011)\citenamefont {Booth},
\citenamefont {Cleland}, \citenamefont {Thom},\ and\ \citenamefont
{Alavi}}]{BooCleThoAla-JCP-11}%
\bibitem [{\citenamefont {{P. C. Hiberty S. Humbel, C. P. Byrman and J. H. van
Lenthe}}()}]{HibHumByrLen-JCP-94}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~H.}\ \bibnamefont
{Booth}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Cleland}},
\bibinfo {author} {\bibfnamefont {A.~J.~W.}\ \bibnamefont {Thom}}, \ and\
\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Alavi}},\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf
{\bibinfo {volume} {135}},\ \bibinfo {pages} {084104} (\bibinfo {year}
{2011})}\BibitemShut {NoStop}%
\bibfield {author} {\bibinfo {author} {\bibnamefont {{P. C. Hiberty S.
Humbel, C. P. Byrman and J. H. van Lenthe}}},\ }\href {\doibase
10.1063/1.468459} {\bibfield {journal} {\bibinfo {journal} {J. Chem.
Phys.}\ }\textbf {\bibinfo {volume} {101}},\ \bibinfo {pages} {5969}},\
\Eprint {http://arxiv.org/abs/https://doi.org/10.1063/1.468459}
{https://doi.org/10.1063/1.468459} \BibitemShut {NoStop}%
\end{thebibliography}%

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Manuscript/srDFT_SC.bib
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@ -5248,6 +5248,9 @@ eprint = {https://doi.org/10.1063/1.2800017}
Journal = {J. Chem. Phys.},
Pages = {5969},
Volume = {101},
doi = {10.1063/1.468459},
URL = {https://doi.org/10.1063/1.468459},
eprint = {https://doi.org/10.1063/1.468459}
Year = {1994}}
@article{HibHum-JCP-94,

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Manuscript/srDFT_SC.tex
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@ -646,15 +646,16 @@ DFT: BLACK BOX and not CASSCF
\subsection{Computational details}
The purpose of the present paper being the study of the basis-set correction in the regime of strong correlation, we study the potential energy curves up to the dissociation limit of a \ce{H10} chain with equally-spaced atoms and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics. In a given basis set, in order to compute the approximation of the exact ground-state energy using Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$.
The purpose of the present paper being the study of the basis-set correction in regimes of both weak and/or strong correlation, we study the potential energy curves up to the dissociation limit of a \ce{H10} chain with equally-spaced atoms and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics. In a given basis set, in order to compute the approximation of the exact ground-state energy using Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$.
In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVXZ (X=D,T,Q), approximations to the FCI energies are obtained using converged frozen-core ($1s$ orbitals are kept frozen) selected CI 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 latest version of Quantum Package\cite{QP2} , and the correlation energy extrapolation by intrinsic scaling\cite{BytNagGorRue-JCP-07} (CEEIS) in the case of \ce{F2} for the cc-pVXZ (X=D,T,Q) basis set. The estimated exact potential energy curves are obtained from experimental data in Ref.~\onlinecite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from extrapolated 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 below $0.5$ mHa. 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).
In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVXZ (X=D,T), approximations to the FCI energies are obtained using converged frozen-core ($1s$ orbitals are kept frozen) selected CI 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 latest version of Quantum Package\cite{QP2} (exFCI), and the correlation energy extrapolation by intrinsic scaling\cite{BytNagGorRue-JCP-07} (CEEIS) in the case of \ce{F2} for the cc-pVXZ (X=D,T,Q) basis set. The estimated exact potential energy curves are obtained from experimental data in Ref.~\onlinecite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from extrapolated 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 on the order of $0.5$ mHa. For the \ce{N2}, \ce{O2}, and \ce{F2} molecules, we also performed single-point exFCI calculations in the aug-cc-pVQZ basis set at the equilibrium geometry to obtain estimations of the FCI dissociation energies in these basis sets.
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).
Regarding the \titou{complementary density functional}, we first perform full-valence complete-active-space self-consistent-field (CASSCF) calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-like quantities involved in the functional [density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. The CASSCF calculations have been 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}.
Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only \titou{complementary functionals}. Therefore, all density-like quantities exclude any contribution from the $1s$ core orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}.
Regarding the computational cost of the present approach, it should be stressed (see supplementary information) that the basis set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any the selected CI calculations. We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems.
Regarding the computational cost of the present approach, it should be stressed (see supplementary information) that the basis set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any of the selected CI calculations. We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*}
@ -696,13 +697,13 @@ Regarding the computational cost of the present approach, it should be stressed
\hline
\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]
& aug-cc-pVTZ & 60.1 [ ] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm]
& aug-cc-pVTZ & 60.1 [ ] & 61.0 [1.2] & 61.3 [0.9] & 61.3 [0.9] \\[0.1cm]
\hline
& & \tabc{CEEIS\fnm[3]} & \tabc{CEEIS\fnm[3]+$\pbeuegXi$} & \tabc{CEEIS\fnm[3]+$\pbeontXi$} & \tabc{CEEIS\fnm[3]+$\pbeontns$}\\
\hline
\ce{F2} & cc-pVDZ & 43.7 [18.5] & 51.0 [11.2] & 51.0 [11.2] & 50.7 [11.5] \\
& cc-pVTZ & 56.3 [5.9] & 59.2 [3.0] & 59.6 [2.6] & 59.5 [2.7] \\
& cc-pVQZ & [ ] & [ ] & [ ] & [ ] \\[0.1cm]
& cc-pVQZ & 59.9 [2.3] & 61.3 [0.9] & 61.6 [0.6] & 61.6 [0.6] \\[0.1cm]
& & \multicolumn{4}{c}{Estimated exact:\fnm[2] 62.2} \\
\end{tabular}
\end{ruledtabular}
@ -718,26 +719,26 @@ Regarding the computational cost of the present approach, it should be stressed
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 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. The computation of the atomization energies $D_0$ for each level of theory 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, independently of the approximation level of \titou{$\efuncbasis$}. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure and their functionals are robust when reaching the strong-correlation regime.
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. The computation of the atomization energies $D_0$ for each level of theory 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, independently of the approximation level of \titou{$\efuncbasis$}. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\textit{i.e.} the determination of the range-separation parameter and the definition of the functionals) is robust when reaching the strong-correlation regime.
In other words, smooth potential energy surfaces are obtained with the present basis-set correction.
More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, an error below $1.4$ mHa) from the cc-pVTZ basis set when using the $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not even reached at the standard MRCI+Q/cc-pVQZ level of theory.
Analyzing more carefully the performance of the different types of approximate density functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on $D_0$ being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provides two important clues on the role of the different physical ingredients used in these functionals: i) the explicit use of the on-top pair density coming from the \titou{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 somehow understandable, and ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it shows that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density.
Analyzing more carefully the performance of the different types of approximate density functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on $D_0$ 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 used in these functionals: i) the explicit use of the on-top pair density coming from the \titou{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 somehow understandable, and ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it shows 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.
%\PFL{Why can't we see the effect of dispersion in that system?}
\subsection{Dissociation of diatomics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}
\includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat.eps}
\includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat_zoom.eps}
\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 exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:C2}}
\end{figure*}
%\begin{figure*}
% \includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat.eps}
% \includegraphics[width=0.45\linewidth]{data/C2/DFT_avdzE_relat_zoom.eps}
% \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 exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
% \label{fig:C2}}
%\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -779,11 +780,12 @@ $^1$: CEEIS calculations obtained from non-relativistic calculations of Ref.~\on
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The \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 in these diatomics also increases while stretching the bond similarly to the case of \ce{H10}. 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. The dispersion forces in \ce{H10} play a much minor role in the PES due to the much smaller number of near-neighboring electron pairs compared to \ce{C2}, \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.
The \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 in these diatomics also increases while stretching the bond similarly to the case of \ce{H10}. 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. The dispersion forces in \ce{H10} play a much minor role in the PES due to the much smaller number of near-neighboring 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 computation of the atomization energies $D_0$ at each level of theory is reported in Table \ref{tab:d0}.
We report in Figs~\ref{fig:N2}, \ref{fig:O2} the potential energy curves of \ce{N2}, \ce{O2}, and computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets, and in Fig~\ref{fig:F2} the potential energy surface of \ce{F2} using the cc-pVXZ (X=D,T,Q) basis set. The computation of the atomization energies $D_0$ at each level of theory is reported in Table \ref{tab:d0}.
Just as in \ce{H10}, the quality of $D_0$ 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 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 \titou{open-shell} systems like \ce{O2}. More quantitatively, an error below 1.0 mHa 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 $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. BLABLA F2 diffuse
Just as in \ce{H10}, the quality of $D_0$ 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 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 \titou{open-shell} systems like \ce{O2}. More quantitatively, an error below 1.0 mHa 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 $\pbeontns$ 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 presence of diffuse function strongly improves the results, which is somehow understandable due to the strong breathing-orbital effect in this molecule induced by the ionic valence bond forms\cite{HibHumByrLen-JCP-94}.
It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the quality of $D_0$ slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI $D_0$ and very close the to chemical accuracy.
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 \titou{complementary basis functionals} can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is designed thanks to the universal condition provided by the electron-electron cusp 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 theoretically expected and predictable.
\titou{We hope to report further on this in the near future.}
@ -791,14 +793,14 @@ 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-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.
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{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 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{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 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.
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 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 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.
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 whereas the uncorrected near-FCI results are far from this accuracy within the same basis set.
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.