Removed F2

Manuscript/rsdft-cipsi-qmc.tex

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