diff --git a/Data/RSDFT-CIPSI.org b/Data/RSDFT-CIPSI.org index ef981bf..a2a8c13 100644 --- a/Data/RSDFT-CIPSI.org +++ b/Data/RSDFT-CIPSI.org @@ -6792,7 +6792,7 @@ dev.off() ** On-top pair density #+begin_src R :results output graphics :file (org-babel-temp-file "figure" ".png") :exports both :width 600 :height 400 :session *R* -breaks <- c("0.00", "0.25", "0.50", "1.00", "$\\infty$", "Jastrow") +breaks <- c("0.00", "0.25", "0.50", "1.00", "2.00", "5.00", "$\\infty$", "Jastrow") tmp_data <- read.csv("H2O_1.e-6.density") data.0 <- data.frame(mu=breaks[1], x=tmp_data$X..distance, n=tmp_data$on.top) @@ -6806,32 +6806,49 @@ data.0.5 <- data.frame(mu=breaks[3], x=tmp_data$X..distance, n=tmp_data$on.top) tmp_data <- read.csv("H2O_1.0.density") data.1.0 <- data.frame(mu=breaks[4], x=tmp_data$X..distance, n=tmp_data$on.top) +tmp_data <- read.csv("H2O_2.0.density") +data.2.0 <- data.frame(mu=breaks[5], x=tmp_data$X..distance, n=tmp_data$on.top) + +tmp_data <- read.csv("H2O_5.0.density") +data.5.0 <- data.frame(mu=breaks[6], x=tmp_data$X..distance, n=tmp_data$on.top) + tmp_data <- read.csv("H2O_1e6.density") -data.inf <- data.frame(mu=breaks[5], x=tmp_data$X..distance, n=tmp_data$on.top) +data.inf <- data.frame(mu=breaks[7], x=tmp_data$X..distance, n=tmp_data$on.top) tmp_data <- read.csv("H2O.density") -data.J <- data.frame(mu=breaks[6], x=tmp_data$X..distance, n=tmp_data$on.top) +data.J <- data.frame(mu=breaks[8], x=tmp_data$X..distance, n=tmp_data$on.top) -data <- rbind(data.0, data.0.25, data.0.5, data.1.0, data.inf) +data <- rbind(data.0, data.0.25, data.0.5, data.1.0, data.2.0, data.5.0, data.inf) labels= TeX(breaks) +labels[1] <- TeX("\u00B5=$0.00$, $(1.443)$") +labels[2] <- TeX("\u00B5=$0.25$, $(1.438)$") +labels[3] <- TeX("\u00B5=$0.50$, $(1.423)$") +labels[4] <- TeX("\u00B5=$1.00$, $(1.378)$") +labels[5] <- TeX("\u00B5=$2.00$, $(1.325)$") +labels[6] <- TeX("\u00B5=$5.00$, $(1.288)$") +labels[7] <- TeX("\u00B5=\\infty, $(1.277)$") +labels[8] <- TeX("Jastrow $(1.404)$") + p <- ggplot(data, aes(x=x, y=n, col=mu)) p <- p + geom_line(lwd=1.5) p <- p + geom_line(data=data.J, lwd=1, col=1, linetype="dashed") -p <- p + scale_colour_discrete(name = TeX("$\\mu$"), breaks = breaks, +p <- p + scale_colour_discrete(name = "", breaks = breaks, labels = labels) #p <- p + scale_color_brewer(palette = "Paired") p <- p + scale_x_continuous(name=TeX("$r_{O-H}$ (bohr)")) p <- p + scale_y_continuous(name = "On-top pair density (a.u.)") -p <- p + theme(text = element_text(size = 20, family="Times"), legend.position=c(.85,.75), legend.text.align = 0) +p <- p + theme(text = element_text(size = 20, family="Times"), + legend.title=element_blank(), + legend.position=c(.81,.75), legend.text.align = 0) p #+end_src #+RESULTS: - [[file:/tmp/babel-eZHQur/figureWPqghB.png]] + [[file:/tmp/babel-eZHQur/figureKY394n.png]] #+begin_src R :results output :session *R* :exports both pdf("../Manuscript/on-top-mu.pdf", family="Times", width=8, height=5) @@ -7201,7 +7218,7 @@ p #+end_src #+RESULTS: - [[file:/tmp/babel-eZHQur/figureXcyXmu.png]] + [[file:/tmp/babel-eZHQur/figureB8MCqS.png]] #+begin_src R :results output :session *R* :exports both pdf("../Manuscript/g2-dmc.pdf", family="Times", width=16, height=5) @@ -7243,7 +7260,7 @@ p #+end_src #+RESULTS: - [[file:/tmp/babel-eZHQur/figureS69edW.png]] + [[file:/tmp/babel-eZHQur/figureFlQgjd.png]] #+begin_src R :results output :session *R* :exports both pdf("../Manuscript/g2-ndet.pdf", family="Times", width=16, height=5) diff --git a/Manuscript/on-top-mu.pdf b/Manuscript/on-top-mu.pdf index 987c887..76dcf30 100644 Binary files a/Manuscript/on-top-mu.pdf and b/Manuscript/on-top-mu.pdf differ diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index d67d205..b41dd3a 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -652,38 +652,18 @@ This confirms that introducing short-range correlation with DFT has an impact on the CI coefficients similar to a Jastrow factor. This is yet another key result of the present study. -%%% TABLE II %%% -\begin{table} - \caption{\ce{H2O}, double-$\zeta$ basis set. Integrated on-top pair density $\expval{ P }$ - for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. - \titou{Please remove table and merge data in Fig. 4.}} - \label{tab:table_on_top} - \begin{ruledtabular} - \begin{tabular}{cc} - $\mu$ & $\expval{ P }$ \\ - \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 - $\Psi^J$ & 1.404 \\ - \end{tabular} - \end{ruledtabular} -\end{table} -%%% %%% %%% %%% - %%% FIG 4 %%% \begin{figure*} \includegraphics[width=\columnwidth]{density-mu.pdf} \includegraphics[width=\columnwidth]{on-top-mu.pdf} - \caption{One-electron density $n(\br)$ (left) and on-top pair - 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 $\Psi^\mu$. - For these two trial wave functions, the CI expansion consists of the 200 most important - determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).} + \caption{\toto{One-electron density $n(\br)$ (left) and on-top pair + density $n_2(\br,\br)$ (right) along the \ce{O-H} axis of \ce{H2O} + as a function of $\mu$ for $\Psi^\mu$, and $\Psi^J$ (dashed + curve). + The integrated on-top pair density $\expval{P}$ is + given in the legend. + For all trial wave functions, the CI expansion consists of the 200 most important + determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}} \label{fig:densities} \end{figure*} %%% %%% %%% %%% @@ -691,7 +671,7 @@ This is yet another key result of the present study. In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we report several quantities related to the one- and two-body densities of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we -report in Table~\ref{tab:table_on_top} the integrated on-top pair density +report in the legend of Fig~\ref{fig:densities} the integrated on-top pair density \begin{equation} \expval{ P } = \int d\br \,\,n_2(\br,\br) \end{equation} @@ -703,21 +683,21 @@ the plots of the total density $n(\br)$ and on-top pair density $n_2(\br,\br)$ along one of the \ce{O-H} axis of the water molecule. From these data, one can clearly notice several trends. -First, from Table~\ref{tab:table_on_top}, we can observe that the overall -on-top pair density decreases when $\mu$ increases, which is expected -as the two-electron interaction increases in $H^\mu[n]$. +First, the overall on-top pair density decreases when $\mu$ increases, +which is expected as the two-electron interaction increases in +$H^\mu[n]$. Second, Fig.~\ref{fig:densities} shows that the relative variations of the on-top pair density with respect to $\mu$ are much more important than that of the one-body density, the latter being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the former can vary by about 10$\%$ in some regions. %TODO TOTO -In the high-density region of the \ce{O-H} bond, the value of the on-top +\toto{In the high-density region of the \ce{O-H} bond, the value of the on-top pair density obtained from $\Psi^J$ is superimposed with $\Psi^{\mu=0.5}$, and at a large distance the on-top pair density of $\Psi^J$ 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$^{-1}$ (see Table~\ref{tab:table_on_top}), consistently with the FN-DMC energies -and the overlap curve depicted in Fig.~\ref{fig:overlap}. +obtained with $\Psi^J$ is $\expval{P}=1.404$, between the values obtained with +$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies +and the overlap curve depicted in Fig.~\ref{fig:overlap}.} 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 (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.