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@ -476,18 +476,10 @@ LDA exchange-correlation functional give very similar FN-DMC energy with respect
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to those obtained with the short-range PBE functional, even if the RS-DFT energies obtained
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with these two functionals differ by several tens of m\hartree{}.
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{overlap.pdf}
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\caption{Overlap of the RS-DFT CI expansion with the
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CI expansion optimized in the presence of a Jastrow factor.}
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\label{fig:overlap}
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\end{figure}
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\subsection{Link between RS-DFT and jastrow factors }
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The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can give CI coefficients
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which give trial wave functions with better nodes than FCI wave functions.
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\label{sec:rsdft-j}
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The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients
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giving trial wave functions with better nodes than FCI wave functions.
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Such behaviour can be compared to the common practice of
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re-optimizing the Slater part of a trial wave function in the presence of the Jastrow factor.
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In the present paragraph, we would like therefore to elaborate some more on the link between RS-DFT
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@ -508,8 +500,9 @@ Such a wave function $\Psi^J$ satisfies the generalized hermitian eigenvalue equ
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but also the non-hermitian transcorrelated eigenvalue problem\cite{many_things}
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\begin{equation}
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\label{eq:transcor}
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e^{-J} H e^{J} \Psi^J = E \Psi^J.
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e^{-J} H e^{J} \Psi^J = E \Psi^J,
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\end{equation}
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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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@ -524,12 +517,33 @@ a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$.
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Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
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on the same Slater determinant basis.
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In figure~\ref{fig:overlap}, we plot the overlaps
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$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer.
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$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer,
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and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions
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$\Psi^\mu$ together with that of $\Psi^J$.
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{overlap.pdf}
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\caption{Overlap of the RS-DFT CI expansion with the
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CI expansion optimized in the presence of a Jastrow factor.}
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\label{fig:overlap}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{x.png}
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\caption{H$_2$O, double-zeta basis set. FN-DMC energy of $\Psi^\mu$ built with the 200 largest determinants of a large CIPSI expansion, together with the FN-DMC energy of $\Psi^J$ (see \ref{sec:rsdft-j}).}
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\label{dmc_small}
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\end{figure}
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In the case of H$_2$O, there is a clear maximum of overlap at
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$\mu=1$~bohr$^{-1}$. This confirms that introducing short-range
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correlation with DFT has the an impact on the CI coefficients similar to
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the Jastrow factor. In the case of F$_2$, the Jastrow factor has
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$\mu=1$~bohr$^{-1}$, which coincide with the minimum of the FN-DMC energy of $\Psi^\mu$.
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???? hypothetic: Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is very close to that of $\Psi^\mu$ with $\mu=1$~bohr$^{-1}$.
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This confirms that 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 the case of F$_2$, the Jastrow factor has
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very little effect on the CI coefficients, as the overlap
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$\braket{\Psi^J}{\Psi^{\mu=\infty}}$ is very close to
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$1$.
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@ -537,18 +551,19 @@ Nevertheless, a slight maximum is obtained for
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$\mu=5$~bohr$^{-1}$.
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In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$,
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we report, in the case of the water molecule in the double-zeta basis set,
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several quantities related to the one- and two-body density for $\Psi^J$ 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 $\langle n_2({\bf r},{\bf r}) \rangle$
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several quantities related to the one- and two-body density 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 $\langle n_2({\bf r},{\bf r}) \rangle$
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\begin{equation}
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\langle n_2({\bf r},{\bf r}) \rangle = \int \text{d}{\bf r} n_2({\bf r},{\bf r})
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\langle n_2({\bf r},{\bf r}) \rangle = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})
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\end{equation}
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where $n_2({\bf r},{\bf r})$ is the two-body density (normalized to $N(N-1)$ where $N$ is the number of electrons)
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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)
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obtained for both $\Psi^\mu$ and $\Psi^J$.
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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}
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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.
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From these data table, one can clearly observe several trends.
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First, 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 indistinguishable between $\mu=0$ and $\mu=\infty$.
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Second, the smaller the $\mu$, the larger the $\langle n_2({\bf r},{\bf r}) \rangle$.
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From these data, one can clearly observe several trends.
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First, from table~\ref{table_on_top}, we can observe that the overall on-top pair density decreases
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when one increases $\mu$, which is expected as the two-electron interaction increases in $H^\mu[n]$.
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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.
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??? Hypothetic: 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
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FCI wave function.
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@ -565,10 +580,10 @@ while the effective one-body potential $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr
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provided that it is exact, maintains the exact one-body density.
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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.
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Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No\cite{Ten-no2000Nov},
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the effective two-body interaction induced by the presence of a Jastrow factor
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the effective two-body interaction induced by the presence of a jastrow factor
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can be non-divergent when a proper Jastrow factor is chosen.
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Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the jastrow-Slater optimization:
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they both deal with an effective non-divergent interaction while keeping the density constant.
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they both deal with an effective non-divergent interaction but still produce reasonable one-body density.
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\begin{table}
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\caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$
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@ -611,7 +626,7 @@ one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller
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fixed node error by properly choosing an optimal value of $\mu$
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in RS-DFT calculations, ii) the value of the optimal $\mu$ depends
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on the system and the basis set, and the larger the basis set, the larger the optimal value of $\mu$,
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iii) numerical experiments (such as computation of overlap) indicates
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iii) numerical experiments (such as computation of overlap, FN-DMC energies) indicates
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that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
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\textit{i.e.} mimicking short-range correlation effects.
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The latter statement can be qualitatively understood by noticing that both RS-DFT
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@ -619,7 +634,7 @@ and transcorrelated approach deal with an effective non-divergent electron-elect
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\section{Atomization energies}
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\section{Energy differences in FN-DMC: atomization energies}
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\label{sec:atomization}
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Atomization energies are challenging for post-Hartree-Fock methods
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