scemama/RSDFT-CIPSI-QMC
2
1
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Removed F2 again

Browse Source
This commit is contained in:
Anthony Scemama 2020-08-08 01:06:13 +02:00
parent 56095fff4b
commit d2c52787d3
1 changed files with 37 additions and 33 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

70
Manuscript/rsdft-cipsi-qmc.tex
View File

@ -246,7 +246,7 @@ estimate of the FCI energy, using a fixed value of the PT2 correction
as a stopping criterion enforces a constant distance of all the as a stopping criterion enforces a constant distance of all the
calculations to the FCI energy. In this work, we target the chemical calculations to the FCI energy. In this work, we target the chemical
accuracy so all the CIPSI selections were made such that $|\EPT| < accuracy so all the CIPSI selections were made such that $|\EPT| <
1$~mE$_h$. 1$~m\hartree{}.
@ -531,7 +531,7 @@ $\Psi^\mu$ together with that of $\Psi^J$.
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=\columnwidth]{overlap.pdf} \includegraphics[width=\columnwidth]{overlap.pdf}
\caption{H$_2$O, double-zeta basis set, 200 most important \caption{\ce{H2O}, double-zeta basis set, 200 most important
determinants of the FCI expansion (see \ref{sec:rsdft-j}). determinants of the FCI expansion (see \ref{sec:rsdft-j}).
Overlap of the RS-DFT CI expansions $\Psi^\mu$ with the CI Overlap of the RS-DFT CI expansions $\Psi^\mu$ with the CI
expansion optimized in the presence of a Jastrow factor $\Psi^J$.} expansion optimized in the presence of a Jastrow factor $\Psi^J$.}
@ -541,7 +541,7 @@ $\Psi^\mu$ together with that of $\Psi^J$.
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=\columnwidth]{h2o-200-dmc.pdf} \includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
\caption{H$_2$O, double-zeta basis set, 200 most important \caption{\ce{H2O}, double-zeta basis set, 200 most important
determinants of the FCI expansion (see \ref{sec:rsdft-j}). determinants of the FCI expansion (see \ref{sec:rsdft-j}).
FN-DMC energies of $\Psi^\mu$ (red curve), together with FN-DMC energies of $\Psi^\mu$ (red curve), together with
the FN-DMC energy of $\Psi^J$ (blue line). The width of the lines the FN-DMC energy of $\Psi^J$ (blue line). The width of the lines
@ -551,58 +551,62 @@ $\Psi^\mu$ together with that of $\Psi^J$.
In the case of H$_2$O, there is a clear maximum of overlap at There is a clear maximum of overlap at $\mu=1$~bohr$^{-1}$, which
$\mu=1$~bohr$^{-1}$, which coincide with the minimum of the FN-DMC energy of $\Psi^\mu$. coincides with the minimum of the FN-DMC energy of $\Psi^\mu$.
Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is very Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is compatible
close to that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that with that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that
introducing short-range correlation with DFT has introducing short-range correlation with DFT has
an impact on the CI coefficients similar to the Jastrow factor. an impact on the CI coefficients similar to the Jastrow factor.
In the case of F$_2$, the Jastrow factor has
very little effect on the CI coefficients, as the overlap
$\braket*{\Psi^J}{\Psi^{\mu=\infty}}$ is very close to
$1$.
Nevertheless, a slight maximum is obtained for
$\mu=5$~bohr$^{-1}$.
In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$,
we report, in the case of the water molecule in the double-zeta basis set, we report several quantities related to the one- and two-body density
several quantities related to the one- and two-body density of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. 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 $\expval{ n_2({\bf r},{\bf r}) }$ First, we report in table~\ref{table_on_top} the integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$
\begin{equation} \begin{equation}
\expval{ n_2({\bf r},{\bf r}) } = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r}) \expval{ n_2({\bf r},{\bf r}) } = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})
\end{equation} \end{equation}
where $n_2({\bf r}_1,{\bf r}_2)$ is the two-body density (normalized to $N(N-1)$ where $N$ is the number of electrons) where $n_2({\bf r}_1,{\bf r}_2)$ is the two-body density (normalized to $N(N-1)$ where $N$ is the number of electrons)
obtained for both $\Psi^\mu$ and $\Psi^J$. obtained for both $\Psi^\mu$ and $\Psi^J$.
Then, in order to have a pictorial representation of both the on-top pair density and the density, we report in figures~\ref{fig:n1} and ~\ref{fig:n2} Then, in order to have a pictorial representation of both the on-top
the plots of the total density $n({\bf r})$ and on-top pair density $n_2({\bf r},{\bf r})$ along the OH axis of the water molecule. pair density and the density, we report in Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}
From these data, one can clearly observe several trends. the plots of the total density $n({\bf r})$ and on-top pair density
First, from table~\ref{table_on_top}, we can observe that the overall on-top pair density decreases $n_2({\bf r},{\bf r})$ along one O---H axis of the water molecule.
when one increases $\mu$, which is expected as the two-electron interaction increases in $H^\mu[n]$.
Second, the relative variation of the on-top pair density with $\mu$ 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. From these data, one can clearly notice several trends.
% TODO TOTO First, from Tab.~\ref{table_on_top}, we can observe that the overall
The value of the on-top pair density in $\Psi^\mu$ are closer for on-top pair density decreases when $\mu$ increases. This is expected
certain values of $\mu$ to that of $\Psi^J$ than the FCI wave as the two-electron interaction increases in $H^\mu[n]$.
Second, the relative variations of the on-top pair density with $\mu$
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
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. function.
These data suggest that the wave functions $\Psi^\mu$ and $\Psi^J$ are similar, These data suggest that the wave functions $\Psi^{\mu=0.5}$ and $\Psi^J$ are similar,
and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics. 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}), 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 one can notice that the differences with respect to the usual Hamiltonian come
from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional. and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional.
The role of these two terms are therefore very different: with respect The roles of these two terms are therefore very different: with respect
to the exact ground state wave function $\Psi$, the non divergent two body interaction to the exact ground state wave function $\Psi$, the non divergent two body interaction
increases the probability to find electrons at short distances in $\Psi^\mu$, increases the probability to find electrons at short distances in $\Psi^\mu$,
while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$, while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
provided that it is exact, maintains the exact one-body density. provided that it is exact, maintains the exact one-body density.
This is clearly what has been observed from the plots in figures ~\ref{fig:n1} and~\ref{fig:n2} in the case of the water molecule. This is clearly what has been observed from the plots in
Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No\cite{Ten-no2000Nov}, Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}.
the effective two-body interaction induced by the presence of a jastrow factor Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No,\cite{Ten-no2000Nov}
the effective two-body interaction induced by the presence of a Jastrow factor
can be non-divergent when a proper Jastrow factor is chosen. can be non-divergent when a proper Jastrow factor is chosen.
Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Jastrow-Slater optimization: Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Jastrow-Slater optimization:
they both deal with an effective non-divergent interaction but still produce reasonable one-body density. they both deal with an effective non-divergent interaction but still
produce a reasonable one-body density.
\begin{table} \begin{table}
\caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$ \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 $\mu$. } for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
\label{table_on_top} \label{table_on_top}
\begin{ruledtabular} \begin{ruledtabular}
@ -625,13 +629,13 @@ they both deal with an effective non-divergent interaction but still produce rea
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=\columnwidth]{on-top-mu.pdf} \includegraphics[width=\columnwidth]{on-top-mu.pdf}
\caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$ along the OH axis, for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. } \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} \label{fig:n2}
\end{figure} \end{figure}
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=\columnwidth]{density-mu.pdf} \includegraphics[width=\columnwidth]{density-mu.pdf}
\caption{H$_2$O, double-zeta basis set. Density $n({\bf r})$ along the OH axis, for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. } \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} \label{fig:n1}
\end{figure} \end{figure}