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Manuscript/srDFT_SC.bbl
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10.1021/acs.jctc.9b00176} {\bibfield {journal} {\bibinfo {journal} {J. 10.1021/acs.jctc.9b00176} {\bibfield {journal} {\bibinfo {journal} {J.
Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {15}},\ \bibinfo {pages} Chem. Theory Comput.}\ }\textbf {\bibinfo {volume} {15}},\ \bibinfo {pages}
{3591} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% {3591} (\bibinfo {year} {2019})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Motta}\ \emph {et~al.}(2017)\citenamefont {Motta},
\citenamefont {Ceperley}, \citenamefont {Chan}, \citenamefont {Gomez},
\citenamefont {Gull}, \citenamefont {Guo}, \citenamefont {Jim\'enez-Hoyos},
\citenamefont {Lan}, \citenamefont {Li}, \citenamefont {Ma}, \citenamefont
{Millis}, \citenamefont {Prokof'ev}, \citenamefont {Ray}, \citenamefont
{Scuseria}, \citenamefont {Sorella}, \citenamefont {Stoudenmire},
\citenamefont {Sun}, \citenamefont {Tupitsyn}, \citenamefont {White},
\citenamefont {Zgid},\ and\ \citenamefont {Zhang}}]{h10_prx}%
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\bibinfo {author} {\bibfnamefont {G.~K.-L.}\ \bibnamefont {Chan}}, \bibinfo
{author} {\bibfnamefont {J.~A.}\ \bibnamefont {Gomez}}, \bibinfo {author}
{\bibfnamefont {E.}~\bibnamefont {Gull}}, \bibinfo {author} {\bibfnamefont
{S.}~\bibnamefont {Guo}}, \bibinfo {author} {\bibfnamefont {C.~A.}\
\bibnamefont {Jim\'enez-Hoyos}}, \bibinfo {author} {\bibfnamefont {T.~N.}\
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@ -12693,3 +12693,19 @@ eprint = {https://doi.org/10.1063/1.5122976}
year = {2005} year = {2005}
} }
@article{h10_prx,
title = {Towards the Solution of the Many-Electron Problem in Real Materials: Equation of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods},
author = {Motta, Mario and Ceperley, David M. and Chan, Garnet Kin-Lic and Gomez, John A. and Gull, Emanuel and Guo, Sheng and Jim\'enez-Hoyos, Carlos A. and Lan, Tran Nguyen and Li, Jia and Ma, Fengjie and Millis, Andrew J. and Prokof'ev, Nikolay V. and Ray, Ushnish and Scuseria, Gustavo E. and Sorella, Sandro and Stoudenmire, Edwin M. and Sun, Qiming and Tupitsyn, Igor S. and White, Steven R. and Zgid, Dominika and Zhang, Shiwei},
collaboration = {Simons Collaboration on the Many-Electron Problem},
journal = {Phys. Rev. X},
volume = {7},
issue = {3},
pages = {031059},
numpages = {28},
year = {2017},
month = {Sep},
publisher = {American Physical Society},
doi = {10.1103/PhysRevX.7.031059},
url = {https://link.aps.org/doi/10.1103/PhysRevX.7.031059}
}

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@ -13,7 +13,6 @@ Database file #1: srDFT_SCNotes.bib
Database file #2: srDFT_SC.bib Database file #2: srDFT_SC.bib
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\BOOKMARK [2][-]{section*.19}{Computational details}{section*.18}% 19 \BOOKMARK [2][-]{section*.19}{Computational details}{section*.18}% 19
\BOOKMARK [2][-]{section*.20}{Dissociation of equally distant H10 chains}{section*.18}% 20 \BOOKMARK [2][-]{section*.20}{Dissociation of equally distant H10 chains}{section*.18}% 20
\BOOKMARK [2][-]{section*.21}{Dissociation of F2, N2}{section*.18}% 21 \BOOKMARK [2][-]{section*.21}{Dissociation of F2, N2}{section*.18}% 21
\BOOKMARK [2][-]{section*.22}{Dissociation of H10}{section*.18}% 22 \BOOKMARK [1][-]{section*.22}{Conclusion}{section*.2}% 22
\BOOKMARK [1][-]{section*.23}{Conclusion}{section*.2}% 23

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@ -578,107 +578,44 @@ iii) and the PBE-ot-$0{\zeta}$ where no spin polarization is used and which ther
\section{Results} \section{Results}
\label{sec:results} \label{sec:results}
\subsection{Computational details} \subsection{Computational details}
In a given basis set, to compute the approximation of the exact ground state energy using equation \eqref{eq:e0approx}, one needs both an approximation to the FCI energy $\efci$ and an approximation to the basis set correction $\efuncbasisFCI$. The purpose of the present paper being the study of the basis set correction in the regime of strong correlation, we propose to study the potential energy surfaces (PES) until dissociation of an equally distant H$_{10}$ chain, F$_2$ and N$_2$.
In the case of the F$_2$ and N$_2$ molecules, the approximation to the FCI energies are obtained using converged frozen-core CIPSI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.} In a given basis set, to compute the approximation of the exact ground state energy using equation \eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the complementary basis set energy functional $\efuncbasisFCI$.
In the case of the F$_2$ and N$_2$ molecules, the approximation to the FCI energies are obtained using converged frozen-core (1s orbitals are kept frozen) CIPSI 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 Quantum Package software\cite{QP2}. (see Refs \onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the Quantum Package software\cite{QP2}.
For all geometry and basis sets, the error with respect to actual FCI energy are estimated to be below 0.5 mH. For all geometry and basis sets, the error with respect to actual FCI energy are estimated to be below 0.5 mH.
In the case of $H_{10}$, the approximation to $\efci$ are obtained from the data from 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 H$_{10}$, the approximation to $\efci$ are obtained from the data from 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 basis set correction, we use minimal valence CASSCF wave functions computed with the GAMESS-US software\cite{gamess} to obtain the wave functions $\psibasis$. Therefore, all all density related quantities (such as the total densities, different spin polarizations and on-top pair densities) together with $\murpsi$ are obtained at minimal valence CASSCF level. Regarding the complementary basis set energy functional, we use minimal valence CASSCF wave functions computed with the GAMESS-US software\cite{gamess} to obtain the wave functions $\psibasis$. Therefore, all all density related quantities (such as the total densities, different flavors of spin polarizations and on-top pair densities) together with the $\murpsi$ of equation \eqref{eq:def_mur} are obtained at minimal valence CASSCF level.
These CASSCF wave functions correspond to active spaces containing two electrons in the bonding and anti-bonding $\sigma$ orbitals and six electrons in the bonding and anti-bonding $\sigma$, $\pi_x$ and $\pi_y$ orbitals for F$_2$ and N$_2$, respectively. These CASSCF wave functions correspond to active spaces containing two electrons in the bonding and anti-bonding $\sigma$ orbitals and six electrons in the bonding and anti-bonding $\sigma$, $\pi_x$ and $\pi_y$ orbitals for F$_2$ and N$_2$, respectively.
Regarding the H$_{10}$ linear chains, the CASSCF contains ten electrons in the 10 orbitals needed to correctly dissociate into 10 hydrogen atoms in their 1s state. Regarding the H$_{10}$ linear chains, the CASSCF wave functions contain ten electrons in the 10 orbitals needed to correctly dissociate into 10 hydrogen atoms in their 1s state.
These energy contributions coming from the basis set correction correspond to approximations of $\efuncbasisFCI$ in equation \eqref{eq:e0approx}.
\subsection{Dissociation of equally distant H$_{10}$ chains} \subsection{Dissociation of equally distant H$_{10}$ chains}
We report in figures \ref{fig:H10_vdz} and \ref{fig:H10_vdz} (\ref{fig:H10_vtz} and \ref{fig:H10_vtz}) the potential energy curves computed using the cc-pVXZ (X=D,T,Q) basis sets of H$_{10}$, for different levels of computations. The study of equally distant H$_{10}$ chains is a good prototype for the study of strong correlation regime 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 can be performed at near CBS values can be obtained (see Ref. \onlinecite{h10_prx} for detailed study of that problem).
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 surfaces are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, from the
We report in figures \ref{fig:H10_vdz}, \ref{fig:H10_vtz}, \ref{fig:H10_vqz} the PES computed using the cc-pVXZ (X=D,T,Q) basis sets of H$_{10}$, 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 surfaces are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no divergence of bizarre behaviour are found when stretching the bonds, which show that the functionals are robust when reaching the strong correlation regime.
More quantitatively, the values of $D_0$ are within the chemical accuracy (\textit{i. e.} an error below 1.4 mH) from the cc-pVTZ basis set when using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, whereas such accuracy is not reached at the cc-pVQZ basis set using MRCI+Q.
Regarding in more details the performance of the different types of approximated functionals, the results show that the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference being 0.3 mH on $D_0$), and they give slightly more accurate than the PBE-UEG-$\tilde{\zeta}$.
These observations bring two important clues on the role of the different physical ingredients used in the functionals:
i) the explicit use of the on-top pair density coming from the CASSCF wave function (see equation \eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see equation \eqref{eq:def_n2ueg}),
ii) removing the dependence on any kind of spin polarizations does not lead to significant loss of accuracy once that a minimal description of the on-top pair density of the system is used.
\subsection{Dissociation of F$_2$, N$_2$} \subsection{Dissociation of F$_2$, N$_2$}
We report in figures \ref{fig:N2_avdz} and \ref{fig:N2_avdz} (\ref{fig:N2_avtz} and \ref{fig:N2_avtz}) the potential energy curves computed using the aug-cc-pVDZ (aug-cc-pVTZ) basis sets of F$_2$ and N$_2$, respectively, for different levels of computations. 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 surfaces are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, from the The study of N$_2$ and F$_2$ 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 also these systems 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 forces which are medium to long-range weak correlation effects.
\subsection{Dissociation of H$_{10}$}
%%%%%%%%%%%%%%%%%%%%%%%% We report in figures \ref{fig:N2_avdz} and \ref{fig:F2_avdz} (\ref{fig:N2_avtz} and \ref{fig:F2_avtz}) the potential energy curves computed using the aug-cc-pVDZ (aug-cc-pVTZ) basis sets of F$_2$ and N$_2$, respectively, for different levels of computations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in table \ref{tab:d0}.
\begin{figure} Just as the case of H$_{10}$, the quality of $D_0$ are globally improved and the chemical accuracy is reached at the aug-cc-pVTZ using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, which also give very similar results.
\includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat.eps} The latter observation confirms that even in the presence of higher electron density, the dependence on the on-top pair density allows to remove the dependence of any kind of spin polarizations.
\includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_error.eps}
\caption{
N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
\label{fig:N2_avdz}}
\end{figure}
\begin{figure} Interestingly, the complementary basis set functional fail provide a noticeable improvement of the PES near twice the equilibrium geometry, both for F$_2$ and N$_2$. Acknowledging that the weak correlation effects in these regions are dominated by dispersion forces which are long-range effects, the failure of the present approximations for the complementary basis set functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics near the electron-electron cusp: the $\murpsi$ is designed by looking at the 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 forces can be considered as a good behaviour.
\includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat.eps}
\includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{data/N2/DFT_avtzE_error.eps}\\
% \includegraphics[width=\linewidth]{fig2c}
\caption{
N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
\label{fig:N2_avtz}}
\end{figure}
\begin{table*}
\label{tab:d0}
\caption{Dissociation energy ($D_0$) computed at different levels in various basis sets. }
\begin{ruledtabular}
\begin{tabular}{lcccc}
%\hline
System/basis & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$ \\
\hline
N$_2$, aug-cc-pvdz & 321.9$/ $42.3 & 356.0$/$8.2 & 355.5$/$8.7 & 354.5$/$ 9.7 \\
N$_2$, aug-cc-pvtz & 348.5$/$15.7 & 361.8$/$2.4 & 363.5$/$0.7 & 363.2$/$1.0 \\
\hline
& \multicolumn{4}{c}{Estimated exact} \\
& \multicolumn{4}{c}{364.2 } \\
\hline
F$_2$, aug-cc-pvdz & 49.6$/$11.1 & 54.5$/$6.2 & 54.7$/$6.0 & 54.5$/$6.3 \\
F$_2$, aug-cc-pvtz & 59.3$/$1.4 & 61.2$/$-0.5 & 61.6$/$-0.9 & 61.5$/$-0.8 \\
\hline
& \multicolumn{4}{c}{Estimated exact} \\
& \multicolumn{4}{c}{60.7 } \\
\hline
System/basis & MRCI+Q & (MRCI+Q)+$\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}$, cc-pvqz & 661.2$/$4.2 & 664.1$/$1.3 & 666.4$/$-1.0 & 666.5$/$-1.1 \\
\hline
& \multicolumn{4}{c}{Estimated exact} \\
& \multicolumn{4}{c}{665.4 } \\
\end{tabular}
\end{ruledtabular}
\label{tab:extensiv_closed}
\end{table*}
\begin{figure}
\includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat.eps}
\includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{data/F2/DFT_avdzE_error.eps}\\
% \includegraphics[width=\linewidth]{fig2c}
\caption{
F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
\label{fig:F2_avdz}}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat.eps}
\includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{data/F2/DFT_avtzE_error.eps}\\
% \includegraphics[width=\linewidth]{fig2c}
\caption{
F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
\label{fig:F2_avtz}}
\end{figure}
\begin{figure} \begin{figure}
% \includegraphics[width=\linewidth]{data/H10/DFT_avdzE_relat.eps}
\includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat.eps}\\ \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat.eps}\\
\includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat_zoom.eps} \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{data/H10/DFT_vdzE_error.eps}\\
% \includegraphics[width=\linewidth]{fig2c}
\caption{ \caption{
H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one. H$_{10}$, cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
\label{fig:H10_vdz}} \label{fig:H10_vdz}}
@ -707,6 +644,80 @@ H$_{10}$, cc-pvqz & 661.2$/$4.2 & 664.1$/$1.3
\section{Conclusion} \section{Conclusion}
\label{sec:conclusion} \label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat.eps}
\includegraphics[width=\linewidth]{data/N2/DFT_avdzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{data/N2/DFT_avdzE_error.eps}
\caption{
N$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
\label{fig:N2_avdz}}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat.eps}
\includegraphics[width=\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{data/N2/DFT_avtzE_error.eps}\\
% \includegraphics[width=\linewidth]{fig2c}
\caption{
N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
\label{fig:N2_avtz}}
\end{figure}
\begin{table*}
\label{tab:d0}
\caption{Dissociation energy ($D_0$) computed at different levels in various basis sets. }
\begin{ruledtabular}
\begin{tabular}{lcccc}
System/basis & MRCI+Q & (MRCI+Q)+$\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}$, cc-pvqz & 661.2$/$4.2 & 664.1$/$1.3 & 666.4$/$-1.0 & 666.5$/$-1.1 \\
\hline
& \multicolumn{4}{c}{Estimated exact} \\
& \multicolumn{4}{c}{665.4 } \\
\hline
System/basis & FCI & FCI+$\pbeuegXi$ & FCI+$\pbeontXi$ & FCI+$\pbeontns$ \\
\hline
N$_2$, aug-cc-pvdz & 321.9$/ $42.3 & 356.0$/$8.2 & 355.5$/$8.7 & 354.5$/$ 9.7 \\
N$_2$, aug-cc-pvtz & 348.5$/$15.7 & 361.8$/$2.4 & 363.5$/$0.7 & 363.2$/$1.0 \\
\hline
& \multicolumn{4}{c}{Estimated exact} \\
& \multicolumn{4}{c}{364.2 } \\
\hline
F$_2$, aug-cc-pvdz & 49.6$/$11.1 & 54.5$/$6.2 & 54.7$/$6.0 & 54.5$/$6.3 \\
F$_2$, aug-cc-pvtz & 59.3$/$1.4 & 61.2$/$-0.5 & 61.6$/$-0.9 & 61.5$/$-0.8 \\
\hline
& \multicolumn{4}{c}{Estimated exact} \\
& \multicolumn{4}{c}{60.7 } \\
\end{tabular}
\end{ruledtabular}
\label{tab:extensiv_closed}
\end{table*}
\begin{figure}
\includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat.eps}
\includegraphics[width=\linewidth]{data/F2/DFT_avdzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{data/F2/DFT_avdzE_error.eps}\\
% \includegraphics[width=\linewidth]{fig2c}
\caption{
F$_2$, aug-cc-pvdz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
\label{fig:F2_avdz}}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat.eps}
\includegraphics[width=\linewidth]{data/F2/DFT_avtzE_relat_zoom.eps}
% \includegraphics[width=\linewidth]{data/F2/DFT_avtzE_error.eps}\\
% \includegraphics[width=\linewidth]{fig2c}
\caption{
F$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one.
\label{fig:F2_avtz}}
\end{figure}