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@ -72,8 +72,10 @@ As the WFT method is relieved from describing the short-range part of the correl
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Solving the Schr\"odinger equation for atoms and molecules is a complex task that has kept theoretical and computational chemists busy for almost hundred years now. \cite{Schrodinger_1926}
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In order to achieve this formidable endeavour, various strategies have been carefully designed and implemented in quantum chemistry software packages.
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@ -176,8 +178,10 @@ still an active field of research. The present paper falls
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within this context.
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\section{Combining CIPSI with range-separated DFT}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Combining CIPSI with RS-DFT}
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\label{sec:rsdft-cipsi}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In single-determinant DMC calculations, the degrees of freedom used to
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reduce the fixed-node error are the molecular orbitals on which the
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@ -214,8 +218,9 @@ system.
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If one wants to have to exact CBS limit, a multi-determinant parameterization
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of the wave functions is required.
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%====================
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\subsection{CIPSI}
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%====================
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Beyond the single-determinant representation, the best
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multi-determinant wave function one can obtain is the FCI. FCI is
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a \emph{post-Hartree-Fock} method, and there exists several systematic
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@ -250,8 +255,10 @@ accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
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%=================================
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\subsection{Range-separated DFT}
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\label{sec:rsdft}
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%=================================
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Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004}
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the Coulomb operator entering the interelectronic repulsion is split into two pieces:
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@ -330,6 +337,7 @@ $w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and
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thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is
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the FCI wave function.
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%%% FIG 1 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=0.7\linewidth]{algorithm.pdf}
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@ -340,6 +348,7 @@ the FCI wave function.
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DIIS extrapolations of the one-electron density are introduced in both the outer and inner loops in order to speed up convergence of the self-consistent process.}
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\label{fig:algo}
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\end{figure*}
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%%% %%% %%% %%%
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Hence, range separation creates a continuous path connecting smoothly the KS determinant to the
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FCI wave function. Because the KS nodes are of higher quality than the
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@ -376,8 +385,10 @@ Note that, thanks to the self-consistent nature of the algorithm,
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the final trial wave function $\Psi^{\mu}$ is independent of the starting wave function $\Psi^{(0)}$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:comp-details}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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For all the systems considered here, experimental geometries have been used and they have been extracted from the NIST website \titou{[REF]}.
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Geometries for each system are reported in the {\SI}.
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@ -413,13 +424,14 @@ algorithm developed by Assaraf \textit{et al.}, \cite{Assaraf_2000}
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with a time step of $5 \times 10^{-4}$ a.u.
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\titou{All-electron move DMC?}
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\section{Influence of the range-separation parameter on the fixed-node
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error}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Influence of the range-separation parameter on the fixed-node error}
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\label{sec:mu-dmc}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% TABLE I %%%
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\begin{table}
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\caption{Fixed-node energies $\EDMC$ (in hartree) and number of determinants $\Ndet$ in \ce{H2O} with various trial wave functions.}
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\caption{Fixed-node energies $\EDMC$ (in hartree) and number of determinants $\Ndet$ in \ce{H2O} with various trial wave functions $\Psi^{\mu}$.}
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\label{tab:h2o-dmc}
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\centering
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\begin{ruledtabular}
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@ -445,6 +457,7 @@ with a time step of $5 \times 10^{-4}$ a.u.
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\end{table}
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%%% %%% %%% %%%
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%%% FIG 2 %%%
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
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@ -453,23 +466,26 @@ with a time step of $5 \times 10^{-4}$ a.u.
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functionals to build the trial wave function.}
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\label{fig:h2o-dmc}
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\end{figure}
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%%% %%% %%% %%%
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The first question we would like to address is the quality of the
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nodes of the wave functions $\Psi^{\mu}$ obtained with an intermediate
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nodes of the wave function $\Psi^{\mu}$ obtained with an intermediate
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range separation parameter (\ie, $0 < \mu < +\infty$).
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For this purpose, we consider a weakly correlated molecular system: the water
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molecule near its equilibrium geometry.\cite{Caffarel_2016}
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For this purpose, we consider a weakly correlated molecular system, namely the water
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molecule \titou{near its equilibrium geometry.} \cite{Caffarel_2016}
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We then generate trial wave functions $\Psi^\mu$ for multiple values of
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$\mu$, and compute the associated fixed-node energy keeping all the
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parameters having an impact on the nodal surface fixed (\titou{such as ??}).
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%======================================================
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\subsection{Fixed-node energy of $\Psi^\mu$}
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\label{sec:fndmc_mu}
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%======================================================
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From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc},
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one can clearly observe that using FCI trial
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one can clearly observe that relying on FCI trial
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wave functions ($\mu \to \infty$) give FN-DMC energies lower
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than the energies obtained with a single KS determinant ($\mu=0$):
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a gain of $3.2 \pm 0.6$~m\hartree{} at the double-zeta level and $7.2 \pm
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a gain of $3.2 \pm 0.6$~m\hartree{} at the double-$\zeta$ level and $7.2 \pm
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0.3$~m\hartree{} at the triple-zeta level are obtained.
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Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with
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intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that
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@ -490,8 +506,10 @@ to those obtained with the short-range PBE functional, even if the
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RS-DFT energies obtained with these two functionals differ by several
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tens of m\hartree{}.
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%======================================================
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\subsection{Link between RS-DFT and Jastrow factors }
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\label{sec:rsdft-j}
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%======================================================
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The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients
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giving trial wave functions with better nodes than FCI wave functions.
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Such behaviour can be compared to the common practice of
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@ -535,6 +553,7 @@ $\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer,
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and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions
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$\Psi^\mu$ together with that of $\Psi^J$.
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%%% FIG 3 %%%
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{overlap.pdf}
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@ -544,7 +563,9 @@ $\Psi^\mu$ together with that of $\Psi^J$.
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expansion optimized in the presence of a Jastrow factor $\Psi^J$.}
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\label{fig:overlap}
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\end{figure}
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%%% %%% %%% %%%
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%%% FIG 4 %%%
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
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@ -555,8 +576,7 @@ $\Psi^\mu$ together with that of $\Psi^J$.
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represent the statistical error bars.}
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\label{dmc_small}
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\end{figure}
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%%% %%% %%% %%%
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There is a clear maximum of overlap at $\mu=1$~bohr$^{-1}$, which
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coincides with the minimum of the FN-DMC energy of $\Psi^\mu$.
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@ -565,6 +585,7 @@ with that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that
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introducing short-range correlation with DFT has
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an impact on the CI coefficients similar to the Jastrow factor.
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%%% TABLE II %%%
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\begin{table}
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\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$
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for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
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@ -585,7 +606,9 @@ an impact on the CI coefficients similar to the Jastrow factor.
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
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%%% FIG 5 %%%
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{on-top-mu.pdf}
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@ -594,7 +617,9 @@ an impact on the CI coefficients similar to the Jastrow factor.
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for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
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\label{fig:n2}
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\end{figure}
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%%% %%% %%% %%%
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%%% FIG 6 %%%
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{density-mu.pdf}
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@ -602,6 +627,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
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the O---H axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
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\label{fig:n1}
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\end{figure}
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%%% %%% %%% %%%
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In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we
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@ -675,8 +701,10 @@ As a conclusion of the first part of this study, we can notice that:
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Energy differences in FN-DMC: atomization energies}
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\label{sec:atomization}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Atomization energies are challenging for post-Hartree-Fock methods
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because their calculation requires a perfect balance in the
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@ -693,7 +721,9 @@ the density and of electron correlation, but also reduces the
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imbalance in the quality of the description of the atoms and the
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molecule, leading to more accurate atomization energies.
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%============================
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\subsection{Size consistency}
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%============================
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An extremely important feature required to get accurate
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atomization energies is size-consistency (or strict separability),
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@ -765,7 +795,7 @@ are computed analytically and the computational cost of the
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pseudo-potential is dramatically reduced (for more detail, see
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Ref.~\onlinecite{Scemama_2015}).
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%\begin{squeezetable}
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%%% TABLE III %%%
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\begin{table}
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\caption{FN-DMC energies (in hartree) using the VDZ-BFD basis set
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and pseudo-potential of the fluorine atom and the dissociated fluorine
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@ -785,6 +815,7 @@ Ref.~\onlinecite{Scemama_2015}).
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
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In this section, we make a numerical verification that the produced
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wave functions are size-consistent for a given range-separation
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@ -798,7 +829,9 @@ size-consistent FN-DMC energies for all values of $\mu$ (within
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$2\times$ statistical error bars).
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%============================
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\subsection{Spin invariance}
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%============================
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Closed-shell molecules often dissociate into open-shell
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fragments. To get reliable atomization energies, it is important to
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@ -822,6 +855,7 @@ of $m_s$ lead to different energies.
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So in the context of RS-DFT, the determinantal expansions will be
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impacted by this spurious effect, as opposed to FCI.
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%%% TABLE IV %%%
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\begin{table}
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\caption{FN-DMC energies (in hartree) of the triplet carbon atom (BFD-VDZ) with
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different values of $m_s$.}
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@ -840,6 +874,7 @@ impacted by this spurious effect, as opposed to FCI.
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
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In this section, we investigate the impact of the spin contamination
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due to the short-range density functional on the FN-DMC energy. We have
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@ -859,11 +894,11 @@ below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$
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there is no bias (within the error bars), and the bias is not
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noticeable with $\mu=5$~bohr$^{-1}$.
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%============================
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\subsection{Benchmark}
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%============================
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%%% FIG 6 %%%
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\begin{squeezetable}
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\begin{table*}
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\caption{Mean absolute errors (MAE), mean signed errors (MSE) and
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@ -903,6 +938,7 @@ DMC@ & 0 & 4.61(34) & -3.62(34) & 5.30(09) & 3.52(19) & -
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\end{ruledtabular}
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\end{table*}
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\end{squeezetable}
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%%% %%% %%% %%%
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The 55 molecules of the benchmark for the Gaussian-1
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theory\cite{Pople_1989,Curtiss_1990} were chosen to test the
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@ -971,6 +1007,7 @@ extrapolated FCI energies. The same comment applies to
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$\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
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%%% FIG 7 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=\textwidth]{g2-dmc.pdf}
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@ -982,6 +1019,7 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
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with a cross.}
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\label{fig:g2-dmc}
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\end{figure*}
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%%% %%% %%% %%%
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Searching for the optimal value of $\mu$ may be too costly, so we have
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computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
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@ -995,6 +1033,7 @@ $\mu$. Although the FN-DMC energies are higher, the numbers show that
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they are more consistent from one system to another, giving improved
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cancellations of errors.
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%%% FIG 8 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=\textwidth]{g2-ndet.pdf}
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@ -1005,7 +1044,7 @@ cancellations of errors.
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with a cross.}
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\label{fig:g2-ndet}
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\end{figure*}
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%%% %%% %%% %%%
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The number of determinants in the trial wave functions are shown in
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figure~\ref{fig:g2-ndet}. As expected, the number of determinants
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@ -1035,9 +1074,9 @@ these orbitals to get an even more compact expansion. In that case, we would
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have converged to the Kohn-Sham orbitals with $\mu=0$, and the
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solution would have been the PBE single determinant.
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%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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%%%%%%%%%%%%%%%%%%%%
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We have seen that introducing short-range correation via
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a range-separated Hamiltonian in a full CI expansion yields improved
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@ -1067,21 +1106,23 @@ value of $\mu$ can be further reduced to $0.25$~bohr$^{-1}$ to get
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extremely compact wave functions at the price of less efficient
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cancellations of errors.
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%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{acknowledgments}
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This work was performed using HPC resources from GENCI-TGCC (Grand
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Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
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2019-0510.
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\end{acknowledgments}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Data availability}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The data that support the findings of this study are available within the article and its {\SI}, and are openly available in [repository name] at \url{http://doi.org/[doi]}, reference number [reference number].
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{rsdft-cipsi-qmc}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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