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Removed F2 again

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Anthony Scemama 2020-08-08 01:06:13 +02:00
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Manuscript/rsdft-cipsi-qmc.tex
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@ -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
calculations to the FCI energy. In this work, we target the chemical
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}
\centering
\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}).
Overlap of the RS-DFT CI expansions $\Psi^\mu$ with the CI
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}
\centering
\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}).
FN-DMC energies of $\Psi^\mu$ (red curve), together with
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
$\mu=1$~bohr$^{-1}$, which coincide 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
close to that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that
There is a clear maximum of overlap at $\mu=1$~bohr$^{-1}$, which
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 compatible
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 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 $\Psi^J$,
we report, in the case of the water molecule in the double-zeta basis set,
several quantities related to the one- and two-body density of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
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}
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 $\Psi^J$.
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}
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.
From these data, one can clearly observe several trends.
First, from table~\ref{table_on_top}, we can observe that the overall on-top pair density decreases
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.
% TODO TOTO
The value of the on-top pair density in $\Psi^\mu$ are closer for
certain values of $\mu$ to that of $\Psi^J$ than the FCI wave
Then, in order to have a pictorial representation of both the on-top
pair density and the density, we report in Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}
the plots of the total density $n({\bf r})$ and on-top pair density
$n_2({\bf r},{\bf r})$ along one O---H axis of the water molecule.
From these data, one can clearly notice several trends.
First, from Tab.~\ref{table_on_top}, we can observe that the overall
on-top pair density decreases when $\mu$ increases. This is expected
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.
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.
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
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.
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
increases the probability to find electrons at short distances in $\Psi^\mu$,
while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
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.
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
This is clearly what has been observed from the plots in
Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}.
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.
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}
\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 $\mu$. }
\label{table_on_top}
\begin{ruledtabular}
@ -625,13 +629,13 @@ they both deal with an effective non-divergent interaction but still produce rea
\begin{figure}
\centering
\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}
\end{figure}
\begin{figure}
\centering
\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}
\end{figure}