Nice plots for on-top
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Data/H2O.density
2002
Data/H2O.density
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Data/H2O_0.5.density
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Data/H2O_0.5.density
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Data/H2O_1.0.density
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Data/H2O_1.0.density
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Data/H2O_1e6.density
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Data/H2O_1e6.density
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Data/H2O_2.0.density
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Data/H2O_2.0.density
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Data/H2O_5.0.density
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Data/H2O_5.0.density
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@ -6336,8 +6336,146 @@ to_kcal <- 627.502164882
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library(ggplot2)
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library(latex2exp)
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library(extrafont)
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library(RColorBrewer)
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loadfonts()
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#+end_src
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#+RESULTS:
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#+begin_example
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#+end_example
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** Read csv
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#+begin_src R :results output :session *R* :exports both
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mad = function(x) { mean(abs(x)) }
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@ -6652,27 +6790,119 @@ dev.off()
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#+end_src
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** On-top pair density
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#+begin_src gnuplot :file output.png
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#set terminal pdf
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#set output "on-top-mu.pdf"
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set xlabel "distance from O"
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set ylabel "on-top pair density"
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plot "H2O_1.e-6.density" w l title "n2(r,r), {/Symbol m}=0", \
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"H2O_0.25.density" w l title "n2(r,r), {/Symbol m}=0.25", \
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"H2O_0.5.density" w l title "n2(r,r), {/Symbol m}=0.5", \
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"H2O_1.0.density" w l title "n2(r,r), {/Symbol m}=1.0", \
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"H2O_1e6.density" w l title "n2(r,r), {/Symbol m}=10^6", \
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"H2O.density" w l title "n2(r,r), Jastrow"
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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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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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tmp_data <- read.csv("H2O_0.25.density")
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data.0.25 <- data.frame(mu=breaks[2], x=tmp_data$X..distance, n=tmp_data$on.top)
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tmp_data <- read.csv("H2O_0.5.density")
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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_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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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 <- rbind(data.0, data.0.25, data.0.5, data.1.0, data.inf)
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labels= TeX(breaks)
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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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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 = "Integrated 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
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#+end_src
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#+RESULTS:
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[[file:output.png]]
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[[file:/tmp/babel-eZHQur/figurey8Ummd.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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p
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dev.off()
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#+end_src
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#+RESULTS:
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:
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: png
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: 2
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** One-body density
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#+begin_src gnuplot :file output.png
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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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tmp_data <- read.csv("H2O_1.e-6.density")
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data.0 <- data.frame(mu=breaks[1], x=tmp_data$X..distance, n=tmp_data$density)
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tmp_data <- read.csv("H2O_0.25.density")
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data.0.25 <- data.frame(mu=breaks[2], x=tmp_data$X..distance, n=tmp_data$density)
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tmp_data <- read.csv("H2O_0.5.density")
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data.0.5 <- data.frame(mu=breaks[3], x=tmp_data$X..distance, n=tmp_data$density)
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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$X..distance, n=tmp_data$density)
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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$X..distance, n=tmp_data$density)
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tmp_data <- read.csv("H2O.density")
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data.J <- data.frame(mu=breaks[6], x=tmp_data$X..distance, n=tmp_data$density)
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data <- rbind(data.0, data.0.25, data.0.5, data.1.0, data.inf)
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labels= TeX(breaks)
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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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labels = labels)
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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 = "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
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#+end_src
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#+RESULTS:
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[[file:/tmp/babel-eZHQur/figureJZB5vg.png]]
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#+begin_src R :results output :session *R* :exports both
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pdf("../Manuscript/density-mu.pdf", family="Times", width=8, height=5)
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p
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dev.off()
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#+end_src
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#+RESULTS:
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:
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: png
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: 2
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#+begin_src gnuplot :file output2.png
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#set terminal pdf
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#set output "density.pdf"
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reset
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set xlabel "distance from O"
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set ylabel "Density"
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plot "H2O_1.e-6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0", \
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@ -6684,7 +6914,6 @@ plot "H2O_1.e-6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0", \
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#+end_src
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#+RESULTS:
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[[file:output.png]]
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** Optimal mu
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*** Table
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@ -512,7 +512,7 @@ but also the non-hermitian transcorrelated eigenvalue problem\cite{many_things}
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which is much easier to handle despite its non-hermicity.
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Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$.
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In a finite basis set and with a quite accurate Jastrow factor, it is known that the nodes
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of $\Psi^J$ are better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$.
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of $\Psi^J$ may be better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$.
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To do so, we have made the following numerical experiment.
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First, we extract the 200 determinants with the largest weights in the FCI wave
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@ -558,10 +558,50 @@ 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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In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$,
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we report several quantities related to the one- and two-body density
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of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
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First, we report in table~\ref{table_on_top} the integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$
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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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\label{table_on_top}
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\begin{ruledtabular}
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\begin{tabular}{cc}
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$\mu$ & $\expval{ n_2({\bf r},{\bf r}) }$ \\
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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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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{on-top-mu.pdf}
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\caption{\ce{H2O}, double-zeta basis set. On-top pair
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density $n_2({\bf r},{\bf r})$ along the O---H axis,
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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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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{density-mu.pdf}
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\caption{\ce{H2O}, double-zeta basis set. Density $n({\bf r})$ along
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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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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 density 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{table_on_top} the integrated on-top pair density
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$\expval{ n_2({\bf r},{\bf r}) }$
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\begin{equation}
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\expval{ n_2({\bf r},{\bf r}) } = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})
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\end{equation}
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@ -581,11 +621,15 @@ 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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The values of the on-top pair density in $\Psi^\mu$ are closer for
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certain values of $\mu$ to those of $\Psi^J$ than the FCI wave
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function.
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In the high-density region of the 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 is
|
||||
the closest to $\mu=\infty$. The integrated on-top pair density
|
||||
obtained with $\Psi^J$ lies between the values obtained with
|
||||
$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies
|
||||
and the overlap curve.
|
||||
|
||||
These data suggest that the wave functions $\Psi^{\mu=0.5}$ and $\Psi^J$ are similar,
|
||||
These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
|
||||
and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
|
||||
Considering the form of $\hat{H}^\mu[n]$ (see Eq.~\eqref{H_mu}),
|
||||
one can notice that the differences with respect to the usual Hamiltonian come
|
||||
|
|
@ -605,53 +649,23 @@ Therefore, one can understand the similarity between the eigenfunctions of $H^\m
|
|||
they both deal with an effective non-divergent interaction but still
|
||||
produce a reasonable one-body density.
|
||||
|
||||
\begin{table}
|
||||
\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$
|
||||
for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
|
||||
\label{table_on_top}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{cc}
|
||||
$\mu$ & $\expval{ n_2({\bf r},{\bf r}) }$ \\
|
||||
\hline
|
||||
0.00 & 1.443 \\
|
||||
0.25 & 1.438 \\
|
||||
0.50 & 1.423 \\
|
||||
1.00 & 1.378 \\
|
||||
2.00 & 1.325 \\
|
||||
5.00 & 1.288 \\
|
||||
$\infty$ & 1.277 \\
|
||||
\hline
|
||||
$\Psi^J$ & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\columnwidth]{on-top-mu.pdf}
|
||||
\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$ along the O---H axis, for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
|
||||
\label{fig:n2}
|
||||
\end{figure}
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\columnwidth]{density-mu.pdf}
|
||||
\caption{\ce{H2O}, double-zeta basis set. Density $n({\bf r})$ along the O---H axis, for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
|
||||
\label{fig:n1}
|
||||
\end{figure}
|
||||
|
||||
|
||||
As a conclusion of the first part of this study, we can notice that:
|
||||
i) with respect to the nodes of a KS determinant or a FCI wave function,
|
||||
one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller
|
||||
fixed node error by properly choosing an optimal value of $\mu$
|
||||
in RS-DFT calculations, ii) the value of the optimal $\mu$ depends
|
||||
on the system and the basis set, and the larger the basis set, the larger the optimal value of $\mu$,
|
||||
iii) numerical experiments (such as computation of overlap, FN-DMC energies) indicates
|
||||
that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
|
||||
\textit{i.e.} mimicking short-range correlation effects.
|
||||
The latter statement can be qualitatively understood by noticing that both RS-DFT
|
||||
and transcorrelated approach deal with an effective non-divergent electron-electron interaction, while keeping the density constant.
|
||||
|
||||
\begin{itemize}
|
||||
\item with respect to the nodes of a KS determinant or a FCI wave function,
|
||||
one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller
|
||||
fixed node error by properly choosing an optimal value of $\mu$
|
||||
in RS-DFT calculations,
|
||||
\item the optimal value of $\mu$ depends on the system and the
|
||||
basis set, and the larger the basis set, the larger the optimal value
|
||||
of $\mu$,
|
||||
\item numerical experiments (overlap $\braket{\Psi^\mu}{\Psi^J}$,
|
||||
one-body density, on-top pair density, and FN-DMC energy) indicate
|
||||
that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
|
||||
\textit{i.e.} mimicking short-range correlation effects. The latter
|
||||
statement can be qualitatively understood by noticing that both RS-DFT
|
||||
and transcorrelated approaches deal with an effective non-divergent
|
||||
electron-electron interaction, while keeping the density constant.
|
||||
\end{itemize}
|
||||
|
||||
|
||||
\section{Energy differences in FN-DMC: atomization energies}
|
||||
|
|
|
|||
Loading…
Reference in New Issue