Nice plots for on-top
This commit is contained in:
parent
d2c52787d3
commit
3ebf2a6fb7
2002
Data/H2O.density
2002
Data/H2O.density
File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
2002
Data/H2O_0.5.density
2002
Data/H2O_0.5.density
File diff suppressed because it is too large
Load Diff
2002
Data/H2O_1.0.density
2002
Data/H2O_1.0.density
File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
2002
Data/H2O_1e6.density
2002
Data/H2O_1e6.density
File diff suppressed because it is too large
Load Diff
2002
Data/H2O_2.0.density
2002
Data/H2O_2.0.density
File diff suppressed because it is too large
Load Diff
2002
Data/H2O_5.0.density
2002
Data/H2O_5.0.density
File diff suppressed because it is too large
Load Diff
@ -6336,8 +6336,146 @@ to_kcal <- 627.502164882
|
||||
library(ggplot2)
|
||||
library(latex2exp)
|
||||
library(extrafont)
|
||||
library(RColorBrewer)
|
||||
loadfonts()
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
#+begin_example
|
||||
|
||||
More than one version of regular/bold/italic found for Cannam Codec Fixed. Skipping setup for this font.
|
||||
More than one version of regular/bold/italic found for Cannam Codec Proportional. Skipping setup for this font.
|
||||
aakar already registered with pdfFonts().
|
||||
Abyssinica SIL already registered with pdfFonts().
|
||||
Ani already registered with pdfFonts().
|
||||
AnjaliOldLipi already registered with pdfFonts().
|
||||
Chandas already registered with pdfFonts().
|
||||
Chilanka already registered with pdfFonts().
|
||||
DejaVu Math TeX Gyre already registered with pdfFonts().
|
||||
DejaVu Sans already registered with pdfFonts().
|
||||
DejaVu Sans Light already registered with pdfFonts().
|
||||
DejaVu Sans Condensed already registered with pdfFonts().
|
||||
DejaVu Sans Mono already registered with pdfFonts().
|
||||
DejaVu Serif already registered with pdfFonts().
|
||||
DejaVu Serif Condensed already registered with pdfFonts().
|
||||
Droid Sans Fallback already registered with pdfFonts().
|
||||
Dyuthi already registered with pdfFonts().
|
||||
FontAwesome already registered with pdfFonts().
|
||||
FreeMono already registered with pdfFonts().
|
||||
FreeSans already registered with pdfFonts().
|
||||
FreeSerif already registered with pdfFonts().
|
||||
Gargi already registered with pdfFonts().
|
||||
Garuda already registered with pdfFonts().
|
||||
Gubbi already registered with pdfFonts().
|
||||
Jamrul already registered with pdfFonts().
|
||||
JetBrains Mono already registered with pdfFonts().
|
||||
JetBrains Mono ExtraBold already registered with pdfFonts().
|
||||
JetBrains Mono Medium already registered with pdfFonts().
|
||||
JetBrains Mono NL already registered with pdfFonts().
|
||||
JetBrains Mono NL ExtraBold already registered with pdfFonts().
|
||||
JetBrains Mono NL Medium already registered with pdfFonts().
|
||||
KacstArt already registered with pdfFonts().
|
||||
KacstBook already registered with pdfFonts().
|
||||
KacstDecorative already registered with pdfFonts().
|
||||
KacstDigital already registered with pdfFonts().
|
||||
KacstFarsi already registered with pdfFonts().
|
||||
KacstLetter already registered with pdfFonts().
|
||||
KacstNaskh already registered with pdfFonts().
|
||||
KacstOffice already registered with pdfFonts().
|
||||
KacstOne already registered with pdfFonts().
|
||||
KacstPen already registered with pdfFonts().
|
||||
KacstPoster already registered with pdfFonts().
|
||||
KacstQurn already registered with pdfFonts().
|
||||
KacstScreen already registered with pdfFonts().
|
||||
KacstTitle already registered with pdfFonts().
|
||||
KacstTitleL already registered with pdfFonts().
|
||||
Kalapi already registered with pdfFonts().
|
||||
Kalimati already registered with pdfFonts().
|
||||
Karumbi already registered with pdfFonts().
|
||||
No regular (non-bold, non-italic) version of Keraleeyam. Skipping setup for this font.
|
||||
Khmer OS already registered with pdfFonts().
|
||||
Khmer OS System already registered with pdfFonts().
|
||||
More than one version of regular/bold/italic found for Kinnari. Skipping setup for this font.
|
||||
Laksaman already registered with pdfFonts().
|
||||
Lato Black already registered with pdfFonts().
|
||||
Lato already registered with pdfFonts().
|
||||
Lato Hairline already registered with pdfFonts().
|
||||
Lato Heavy already registered with pdfFonts().
|
||||
Lato Light already registered with pdfFonts().
|
||||
Lato Medium already registered with pdfFonts().
|
||||
Lato Semibold already registered with pdfFonts().
|
||||
Lato Thin already registered with pdfFonts().
|
||||
More than one version of regular/bold/italic found for Liberation Mono. Skipping setup for this font.
|
||||
More than one version of regular/bold/italic found for Liberation Sans. Skipping setup for this font.
|
||||
Liberation Sans Narrow already registered with pdfFonts().
|
||||
More than one version of regular/bold/italic found for Liberation Serif. Skipping setup for this font.
|
||||
Likhan already registered with pdfFonts().
|
||||
LKLUG already registered with pdfFonts().
|
||||
Lohit Assamese already registered with pdfFonts().
|
||||
Lohit Bengali already registered with pdfFonts().
|
||||
Lohit Devanagari already registered with pdfFonts().
|
||||
Lohit Gujarati already registered with pdfFonts().
|
||||
Lohit Gurmukhi already registered with pdfFonts().
|
||||
Lohit Kannada already registered with pdfFonts().
|
||||
Lohit Malayalam already registered with pdfFonts().
|
||||
Lohit Odia already registered with pdfFonts().
|
||||
Lohit Tamil already registered with pdfFonts().
|
||||
Lohit Tamil Classical already registered with pdfFonts().
|
||||
Lohit Telugu already registered with pdfFonts().
|
||||
Loma already registered with pdfFonts().
|
||||
Material Design Icons already registered with pdfFonts().
|
||||
Meera already registered with pdfFonts().
|
||||
Mitra Mono already registered with pdfFonts().
|
||||
mry_KacstQurn already registered with pdfFonts().
|
||||
Mukti Narrow already registered with pdfFonts().
|
||||
Nakula already registered with pdfFonts().
|
||||
Navilu already registered with pdfFonts().
|
||||
More than one version of regular/bold/italic found for Norasi. Skipping setup for this font.
|
||||
Noto Mono already registered with pdfFonts().
|
||||
OpenSymbol already registered with pdfFonts().
|
||||
Padauk already registered with pdfFonts().
|
||||
Padauk Book already registered with pdfFonts().
|
||||
More than one version of regular/bold/italic found for padmaa. Skipping setup for this font.
|
||||
No regular (non-bold, non-italic) version of padmaa-Bold.1.1. Skipping setup for this font.
|
||||
Pagul already registered with pdfFonts().
|
||||
Phetsarath OT already registered with pdfFonts().
|
||||
Pothana2000 already registered with pdfFonts().
|
||||
Purisa already registered with pdfFonts().
|
||||
Rachana already registered with pdfFonts().
|
||||
RaghuMalayalam already registered with pdfFonts().
|
||||
Rekha already registered with pdfFonts().
|
||||
Saab already registered with pdfFonts().
|
||||
Sahadeva already registered with pdfFonts().
|
||||
Samanata already registered with pdfFonts().
|
||||
Samyak Devanagari already registered with pdfFonts().
|
||||
Samyak Gujarati already registered with pdfFonts().
|
||||
Samyak Malayalam already registered with pdfFonts().
|
||||
Samyak Tamil already registered with pdfFonts().
|
||||
Sarai already registered with pdfFonts().
|
||||
Sawasdee already registered with pdfFonts().
|
||||
Source Code Pro Black already registered with pdfFonts().
|
||||
Source Code Pro already registered with pdfFonts().
|
||||
Source Code Pro ExtraLight already registered with pdfFonts().
|
||||
Source Code Pro Light already registered with pdfFonts().
|
||||
Source Code Pro Medium already registered with pdfFonts().
|
||||
Source Code Pro Semibold already registered with pdfFonts().
|
||||
Suruma already registered with pdfFonts().
|
||||
Tibetan Machine Uni already registered with pdfFonts().
|
||||
TlwgMono already registered with pdfFonts().
|
||||
TlwgTypewriter already registered with pdfFonts().
|
||||
Tlwg Typist already registered with pdfFonts().
|
||||
Tlwg Typo already registered with pdfFonts().
|
||||
Ubuntu already registered with pdfFonts().
|
||||
Ubuntu Light already registered with pdfFonts().
|
||||
Ubuntu Condensed already registered with pdfFonts().
|
||||
Ubuntu Mono already registered with pdfFonts().
|
||||
More than one version of regular/bold/italic found for Umpush. Skipping setup for this font.
|
||||
No regular (non-bold, non-italic) version of Uroob. Skipping setup for this font.
|
||||
utkal already registered with pdfFonts().
|
||||
Vemana2000 already registered with pdfFonts().
|
||||
Waree already registered with pdfFonts().
|
||||
#+end_example
|
||||
|
||||
** Read csv
|
||||
#+begin_src R :results output :session *R* :exports both
|
||||
mad = function(x) { mean(abs(x)) }
|
||||
@ -6652,27 +6790,119 @@ dev.off()
|
||||
#+end_src
|
||||
|
||||
** On-top pair density
|
||||
#+begin_src gnuplot :file output.png
|
||||
#set terminal pdf
|
||||
#set output "on-top-mu.pdf"
|
||||
set xlabel "distance from O"
|
||||
set ylabel "on-top pair density"
|
||||
plot "H2O_1.e-6.density" w l title "n2(r,r), {/Symbol m}=0", \
|
||||
"H2O_0.25.density" w l title "n2(r,r), {/Symbol m}=0.25", \
|
||||
"H2O_0.5.density" w l title "n2(r,r), {/Symbol m}=0.5", \
|
||||
"H2O_1.0.density" w l title "n2(r,r), {/Symbol m}=1.0", \
|
||||
"H2O_1e6.density" w l title "n2(r,r), {/Symbol m}=10^6", \
|
||||
"H2O.density" w l title "n2(r,r), Jastrow"
|
||||
|
||||
#+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")
|
||||
|
||||
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)
|
||||
|
||||
tmp_data <- read.csv("H2O_0.25.density")
|
||||
data.0.25 <- data.frame(mu=breaks[2], x=tmp_data$X..distance, n=tmp_data$on.top)
|
||||
|
||||
tmp_data <- read.csv("H2O_0.5.density")
|
||||
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_1e6.density")
|
||||
data.inf <- data.frame(mu=breaks[5], 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 <- rbind(data.0, data.0.25, data.0.5, data.1.0, data.inf)
|
||||
|
||||
labels= TeX(breaks)
|
||||
|
||||
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,
|
||||
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 = "Integrated 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
|
||||
|
||||
#+end_src
|
||||
|
||||
|
||||
#+RESULTS:
|
||||
[[file:output.png]]
|
||||
[[file:/tmp/babel-eZHQur/figurey8Ummd.png]]
|
||||
|
||||
#+begin_src R :results output :session *R* :exports both
|
||||
pdf("../Manuscript/on-top-mu.pdf", family="Times", width=8, height=5)
|
||||
p
|
||||
dev.off()
|
||||
#+end_src
|
||||
#+RESULTS:
|
||||
:
|
||||
: png
|
||||
: 2
|
||||
|
||||
** One-body density
|
||||
|
||||
#+begin_src gnuplot :file output.png
|
||||
#+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")
|
||||
|
||||
tmp_data <- read.csv("H2O_1.e-6.density")
|
||||
data.0 <- data.frame(mu=breaks[1], x=tmp_data$X..distance, n=tmp_data$density)
|
||||
|
||||
tmp_data <- read.csv("H2O_0.25.density")
|
||||
data.0.25 <- data.frame(mu=breaks[2], x=tmp_data$X..distance, n=tmp_data$density)
|
||||
|
||||
tmp_data <- read.csv("H2O_0.5.density")
|
||||
data.0.5 <- data.frame(mu=breaks[3], x=tmp_data$X..distance, n=tmp_data$density)
|
||||
|
||||
tmp_data <- read.csv("H2O_1.0.density")
|
||||
data.1.0 <- data.frame(mu=breaks[4], x=tmp_data$X..distance, n=tmp_data$density)
|
||||
|
||||
tmp_data <- read.csv("H2O_1e6.density")
|
||||
data.inf <- data.frame(mu=breaks[5], x=tmp_data$X..distance, n=tmp_data$density)
|
||||
|
||||
tmp_data <- read.csv("H2O.density")
|
||||
data.J <- data.frame(mu=breaks[6], x=tmp_data$X..distance, n=tmp_data$density)
|
||||
|
||||
|
||||
data <- rbind(data.0, data.0.25, data.0.5, data.1.0, data.inf)
|
||||
|
||||
labels= TeX(breaks)
|
||||
|
||||
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,
|
||||
labels = labels)
|
||||
p <- p + scale_x_continuous(name=TeX("$r_{O-H}$ (bohr)"))
|
||||
p <- p + scale_y_continuous(name = "Density (a.u.)")
|
||||
p <- p + theme(text = element_text(size = 20, family="Times"), legend.position=c(.85,.75), legend.text.align = 0)
|
||||
p
|
||||
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:/tmp/babel-eZHQur/figureJZB5vg.png]]
|
||||
|
||||
#+begin_src R :results output :session *R* :exports both
|
||||
pdf("../Manuscript/density-mu.pdf", family="Times", width=8, height=5)
|
||||
p
|
||||
dev.off()
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
:
|
||||
: png
|
||||
: 2
|
||||
|
||||
|
||||
|
||||
#+begin_src gnuplot :file output2.png
|
||||
#set terminal pdf
|
||||
#set output "density.pdf"
|
||||
reset
|
||||
set xlabel "distance from O"
|
||||
set ylabel "Density"
|
||||
plot "H2O_1.e-6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0", \
|
||||
@ -6684,7 +6914,6 @@ plot "H2O_1.e-6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0", \
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:output.png]]
|
||||
|
||||
** Optimal mu
|
||||
*** Table
|
||||
|
||||
Binary file not shown.
BIN
Manuscript/h2o-200-dmc.pdf
Normal file
BIN
Manuscript/h2o-200-dmc.pdf
Normal file
Binary file not shown.
Binary file not shown.
@ -512,7 +512,7 @@ but also the non-hermitian transcorrelated eigenvalue problem\cite{many_things}
|
||||
which is much easier to handle despite its non-hermicity.
|
||||
Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$.
|
||||
In a finite basis set and with a quite accurate Jastrow factor, it is known that the nodes
|
||||
of $\Psi^J$ are better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$.
|
||||
of $\Psi^J$ may be better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$.
|
||||
|
||||
To do so, we have made the following numerical experiment.
|
||||
First, we extract the 200 determinants with the largest weights in the FCI wave
|
||||
@ -558,10 +558,50 @@ with that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that
|
||||
introducing short-range correlation with DFT has
|
||||
an impact on the CI coefficients similar to the Jastrow factor.
|
||||
|
||||
In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$,
|
||||
we report several quantities related to the one- and two-body density
|
||||
of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
|
||||
First, we report in table~\ref{table_on_top} the integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$
|
||||
\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
|
||||
$\Psi^J$ & 1.404 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\columnwidth]{on-top-mu.pdf}
|
||||
\caption{\ce{H2O}, double-zeta basis set. On-top pair
|
||||
density $n_2({\bf r},{\bf r})$ along the O---H axis,
|
||||
for $\Psi^J$ (dashed curve) 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 $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
|
||||
\label{fig:n1}
|
||||
\end{figure}
|
||||
|
||||
|
||||
In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we
|
||||
report several quantities related to the one- and two-body density of
|
||||
$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we
|
||||
report in table~\ref{table_on_top} the integrated on-top pair density
|
||||
$\expval{ n_2({\bf r},{\bf r}) }$
|
||||
\begin{equation}
|
||||
\expval{ n_2({\bf r},{\bf r}) } = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})
|
||||
\end{equation}
|
||||
@ -581,11 +621,15 @@ 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
|
||||
The values of the on-top pair density in $\Psi^\mu$ are closer for
|
||||
certain values of $\mu$ to those of $\Psi^J$ than the FCI wave
|
||||
function.
|
||||
In the high-density region of the 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 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
Block a user