Removed table
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@ -6792,7 +6792,7 @@ dev.off()
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** On-top pair density
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#+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R*
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breaks <- c("0.00", "0.25", "0.50", "1.00", "$\\infty$", "Jastrow")
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breaks <- c("0.00", "0.25", "0.50", "1.00", "2.00", "5.00", "$\\infty$", "Jastrow")
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tmp_data <- read.csv("H2O_1.e-6.density")
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data.0 <- data.frame(mu=breaks[1], x=tmp_data$X..distance, n=tmp_data$on.top)
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@ -6806,32 +6806,49 @@ data.0.5 <- data.frame(mu=breaks[3], x=tmp_data$X..distance, n=tmp_data$on.top)
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tmp_data <- read.csv("H2O_1.0.density")
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data.1.0 <- data.frame(mu=breaks[4], x=tmp_data$X..distance, n=tmp_data$on.top)
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tmp_data <- read.csv("H2O_2.0.density")
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data.2.0 <- data.frame(mu=breaks[5], x=tmp_data$X..distance, n=tmp_data$on.top)
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tmp_data <- read.csv("H2O_5.0.density")
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data.5.0 <- data.frame(mu=breaks[6], x=tmp_data$X..distance, n=tmp_data$on.top)
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tmp_data <- read.csv("H2O_1e6.density")
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data.inf <- data.frame(mu=breaks[5], x=tmp_data$X..distance, n=tmp_data$on.top)
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data.inf <- data.frame(mu=breaks[7], x=tmp_data$X..distance, n=tmp_data$on.top)
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tmp_data <- read.csv("H2O.density")
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data.J <- data.frame(mu=breaks[6], x=tmp_data$X..distance, n=tmp_data$on.top)
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data.J <- data.frame(mu=breaks[8], x=tmp_data$X..distance, n=tmp_data$on.top)
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data <- rbind(data.0, data.0.25, data.0.5, data.1.0, data.inf)
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data <- rbind(data.0, data.0.25, data.0.5, data.1.0, data.2.0, data.5.0, data.inf)
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labels= TeX(breaks)
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labels[1] <- TeX("\u00B5=$0.00$, $(1.443)$")
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labels[2] <- TeX("\u00B5=$0.25$, $(1.438)$")
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labels[3] <- TeX("\u00B5=$0.50$, $(1.423)$")
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labels[4] <- TeX("\u00B5=$1.00$, $(1.378)$")
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labels[5] <- TeX("\u00B5=$2.00$, $(1.325)$")
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labels[6] <- TeX("\u00B5=$5.00$, $(1.288)$")
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labels[7] <- TeX("\u00B5=\\infty, $(1.277)$")
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labels[8] <- TeX("Jastrow $(1.404)$")
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p <- ggplot(data, aes(x=x, y=n, col=mu))
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p <- p + geom_line(lwd=1.5)
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p <- p + geom_line(data=data.J, lwd=1, col=1, linetype="dashed")
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p <- p + scale_colour_discrete(name = TeX("$\\mu$"), breaks = breaks,
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p <- p + scale_colour_discrete(name = "", breaks = breaks,
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labels = labels)
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#p <- p + scale_color_brewer(palette = "Paired")
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p <- p + scale_x_continuous(name=TeX("$r_{O-H}$ (bohr)"))
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p <- p + scale_y_continuous(name = "On-top pair density (a.u.)")
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p <- p + theme(text = element_text(size = 20, family="Times"), legend.position=c(.85,.75), legend.text.align = 0)
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p <- p + theme(text = element_text(size = 20, family="Times"),
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legend.title=element_blank(),
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legend.position=c(.81,.75), legend.text.align = 0)
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p
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#+end_src
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#+RESULTS:
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[[file:/tmp/babel-eZHQur/figureWPqghB.png]]
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[[file:/tmp/babel-eZHQur/figureKY394n.png]]
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#+begin_src R :results output :session *R* :exports both
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pdf("../Manuscript/on-top-mu.pdf", family="Times", width=8, height=5)
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@ -7201,7 +7218,7 @@ p
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#+end_src
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#+RESULTS:
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[[file:/tmp/babel-eZHQur/figureXcyXmu.png]]
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[[file:/tmp/babel-eZHQur/figureB8MCqS.png]]
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#+begin_src R :results output :session *R* :exports both
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pdf("../Manuscript/g2-dmc.pdf", family="Times", width=16, height=5)
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@ -7243,7 +7260,7 @@ p
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#+end_src
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#+RESULTS:
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[[file:/tmp/babel-eZHQur/figureS69edW.png]]
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[[file:/tmp/babel-eZHQur/figureFlQgjd.png]]
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#+begin_src R :results output :session *R* :exports both
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pdf("../Manuscript/g2-ndet.pdf", family="Times", width=16, height=5)
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Binary file not shown.
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@ -643,38 +643,18 @@ This confirms that introducing short-range correlation with DFT has
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an impact on the CI coefficients similar to a Jastrow factor.
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This is yet another key result of the present study.
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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 $\expval{ P }$
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for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
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\titou{Please remove table and merge data in Fig. 4.}}
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\label{tab:table_on_top}
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\begin{ruledtabular}
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\begin{tabular}{cc}
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$\mu$ & $\expval{ P }$ \\
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\hline
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0.00 & 1.443 \\
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0.25 & 1.438 \\
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0.50 & 1.423 \\
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1.00 & 1.378 \\
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2.00 & 1.325 \\
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5.00 & 1.288 \\
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$\infty$ & 1.277 \\
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\hline
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$\Psi^J$ & 1.404 \\
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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 4 %%%
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\begin{figure*}
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\includegraphics[width=\columnwidth]{density-mu.pdf}
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\includegraphics[width=\columnwidth]{on-top-mu.pdf}
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\caption{One-electron density $n(\br)$ (left) and on-top pair
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density $n_2(\br,\br)$ (right) along the \ce{O-H} axis of \ce{H2O} as a function of $\mu$ for $\Psi^J$ (dashed curve) and $\Psi^\mu$.
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For these two trial wave functions, the CI expansion consists of the 200 most important
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determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
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\caption{\toto{One-electron density $n(\br)$ (left) and on-top pair
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density $n_2(\br,\br)$ (right) along the \ce{O-H} axis of \ce{H2O}
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as a function of $\mu$ for $\Psi^\mu$, and $\Psi^J$ (dashed
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curve).
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The integrated on-top pair density $\expval{P}$ is
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given in the legend.
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For all trial wave functions, the CI expansion consists of the 200 most important
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determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}}
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\label{fig:densities}
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\end{figure*}
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%%% %%% %%% %%%
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@ -682,7 +662,7 @@ This is yet another key result of the present study.
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In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we
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report several quantities related to the one- and two-body densities of
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$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we
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report in Table~\ref{tab:table_on_top} the integrated on-top pair density
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report in the legend of Fig~\ref{fig:densities} the integrated on-top pair density
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\begin{equation}
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\expval{ P } = \int d\br \,\,n_2(\br,\br)
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\end{equation}
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@ -694,21 +674,21 @@ the plots of the total density $n(\br)$ and on-top pair density
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$n_2(\br,\br)$ along one of the \ce{O-H} axis of the water molecule.
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From these data, one can clearly notice several trends.
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First, from Table~\ref{tab:table_on_top}, we can observe that the overall
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on-top pair density decreases when $\mu$ increases, which is expected
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as the two-electron interaction increases in $H^\mu[n]$.
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First, the overall on-top pair density decreases when $\mu$ increases,
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which is expected as the two-electron interaction increases in
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$H^\mu[n]$.
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Second, Fig.~\ref{fig:densities} shows that the relative variations of the on-top pair density with respect to $\mu$
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are much more important than that of the one-body density, the latter
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being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the
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former can vary by about 10$\%$ in some regions.
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%TODO TOTO
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In the high-density region of the \ce{O-H} bond, the value of the on-top
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\toto{In the high-density region of the \ce{O-H} bond, the value of the on-top
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pair density obtained from $\Psi^J$ is superimposed with
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$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density of $\Psi^J$ is
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the closest to $\mu=\infty$. The integrated on-top pair density
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obtained with $\Psi^J$ lies between the values obtained with
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$\mu=0.5$ and $\mu=1$~bohr$^{-1}$ (see Table~\ref{tab:table_on_top}), consistently with the FN-DMC energies
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and the overlap curve depicted in Fig.~\ref{fig:overlap}.
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obtained with $\Psi^J$ is $\expval{P}=1.404$, between the values obtained with
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$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies
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and the overlap curve depicted in Fig.~\ref{fig:overlap}.}
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These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
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and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
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