Removed F2
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Data/RSDFT-CIPSI.org
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@ -411,37 +411,27 @@ All-electron move DMC.}
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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} and \ce{F2} 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.}
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\label{tab:h2o-dmc}
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\centering
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\begin{ruledtabular}
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\begin{tabular}{ccrlrl}
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& & \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
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\cline{3-4} \cline{5-6}
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System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
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\begin{tabular}{crlrl}
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& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
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\cline{2-3} \cline{4-5}
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$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
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\hline
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\ce{H2O}
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& $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
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& $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
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& $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
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& $0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
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& $0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
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& $1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
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& $1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
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& $2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
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& $3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
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& $5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
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& $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
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& $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
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\\
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\ce{F2}
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& $0.00$ & $23$ & $-48.419\,5(4)$ \\
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& $0.25$ & $8$ & $-48.421\,9(4)$ \\
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& $0.50$ & $1743$ & $-48.424\,8(8)$ \\
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& $1.00$ & $11952$ & $-48.432\,4(3)$ \\
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& $2.00$ & $829438$ & $-48.441\,0(7)$ \\
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& $5.00$ & $5326459$ & $-48.445(2)$ \\
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& $\infty$ & $8302442$ & $-48.437(3)$ \\
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$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
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$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
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$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
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$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
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$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
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$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
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$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
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$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
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$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
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$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
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$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
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$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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@ -456,46 +446,41 @@ System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\ED
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\label{fig:h2o-dmc}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{f2-dmc.pdf}
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\caption{Fixed-node energies of \ce{F2} for different
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values of $\mu$ using the srPBE density functional to build the trial wave function.}
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\label{fig:f2-dmc}
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\end{figure}
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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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range separation parameter (\textit{i.e.} $0 < \mu < +\infty$).
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We generated trial wave functions $\Psi^\mu$ with multiple values of
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$\mu$, and computed the associated fixed node energy keeping all the
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parameters having an impact on the nodal surface fixed.
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We considered two weakly correlated molecular systems: the water
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molecule and the fluorine dimer, near their equilibrium.
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geometry\cite{Caffarel_2016}
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We considered a weakly correlated molecular systems: the water
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molecule near its equilibrium geometry.\cite{Caffarel_2016}
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\subsection{Fixed-node energy of $\Psi^\mu$}
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\label{sec:fndmc_mu}
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From Table~\ref{tab:h2o-dmc} and Figs.~\ref{fig:h2o-dmc}
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and~\ref{fig:f2-dmc}, one can clearly observe that using FCI trial
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wave functions ($\mu = \infty$) gives FN-DMC energies which are lower
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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 a FCI trial
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wave functions ($\mu = \infty$) give an FN-DMC energies lower
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than the energies obtained with a single Kohn-Sham determinant ($\mu=0$):
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a gain of $3.2 \pm 0.6$~m\hartree{} at the double-zeta level and $7.2 \pm
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0.3$~m\hartree{} at the triple-zeta level are obtained for water, and
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a gain of $18 \pm 3$~m\hartree{} for F$_2$.
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Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with intermediate values of $\mu$,
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the figures~\ref{fig:h2o-dmc} show that a smooth behaviour is obtained:
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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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a smooth behaviour is obtained:
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starting from $\mu=0$ (\textit{i.e.} the KS determinant),
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the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$,
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and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function).
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For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set,
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with the optimal value of $\mu$, one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
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and $8 \pm 3$~m\hartree{} for the water and difluorine molecules, respectively.
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The optimal value of $\mu$ is $\mu=1.75$~bohr$^{-1}$ and $\mu=5$~bohr$^{-1}$ for the water and fluorine dimer, respectively.
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When the basis set is increased, the gain in FN-DMC energy with respect to the FCI trial wave function is reduced, and the optimal value of $\mu$ is shifted towards large $\mu$.
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Eventually, the nodes of the wave functions $\Psi^\mu$ obtained with short-range
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LDA exchange-correlation functional give very similar FN-DMC energy with respect
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to those obtained with the short-range PBE functional, even if the RS-DFT energies obtained
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with these two functionals differ by several tens of m\hartree{}.
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with the optimal value of $\mu$, one can obtain a lowering of the
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FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}, with an optimal value at
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$\mu=1.75$~bohr$^{-1}$.
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When the basis set is increased, the gain in FN-DMC energy with
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respect to the FCI trial wave function is reduced, and the optimal
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value of $\mu$ is slightly shifted towards large $\mu$. Eventually, the nodes
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of the wave functions $\Psi^\mu$ obtained with the short-range
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LDA exchange-correlation functional give very similar FN-DMC energies with respect
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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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\subsection{Link between RS-DFT and Jastrow factors }
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\label{sec:rsdft-j}
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@ -545,15 +530,21 @@ $\Psi^\mu$ together with that of $\Psi^J$.
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{overlap.pdf}
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\caption{Overlap of the RS-DFT CI expansion with the
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CI expansion optimized in the presence of a Jastrow factor.}
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\caption{H$_2$O, double-zeta basis set, 200 most important
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determinants of the FCI expansion (see \ref{sec:rsdft-j}).
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Overlap of the RS-DFT CI expansions $\Psi^\mu$ with the CI
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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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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{x.png}
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\caption{H$_2$O, double-zeta basis set. FN-DMC energy of $\Psi^\mu$ built with the 200 largest determinants of a large CIPSI expansion, together with the FN-DMC energy of $\Psi^J$ (see \ref{sec:rsdft-j}).}
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\includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
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\caption{H$_2$O, double-zeta basis set, 200 most important
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determinants of the FCI expansion (see \ref{sec:rsdft-j}).
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FN-DMC energies of $\Psi^\mu$ (red curve), together with
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the FN-DMC energy of $\Psi^J$ (blue line). The width of the lines
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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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