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@ -514,12 +514,12 @@ The take-home message here is that RS-DFT trial wave functions yield a lower fix
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\subsection{Link between RS-DFT and Jastrow factors }
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\label{sec:rsdft-j}
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%======================================================
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The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients
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The data obtained in Sec.~\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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and wave function optimization within the presence of a Jastrow factor.
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and wave function optimization with the presence of a Jastrow factor.
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Let us assume a fixed Jastrow factor $J({\bf{r}_1}, \hdots , {\bf{r}_N})$,
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and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$,
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@ -553,7 +553,7 @@ 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 Fig.~\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,
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and in Fig.~\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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@ -617,7 +617,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
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\centering
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\includegraphics[width=\columnwidth]{on-top-mu.pdf}
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\caption{\ce{H2O}, double-zeta basis set. On-top pair
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density $n_2({\bf r},{\bf r})$ along the O---H axis,
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density $n_2({\bf r},{\bf r})$ along the \ce{O-H} axis,
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for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
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\label{fig:n2}
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\end{figure}
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@ -628,7 +628,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
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\centering
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\includegraphics[width=\columnwidth]{density-mu.pdf}
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\caption{\ce{H2O}, double-zeta basis set. Density $n({\bf r})$ along
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the O---H axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
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the \ce{O-H} axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
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\label{fig:n1}
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\end{figure}
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%%% %%% %%% %%%
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@ -637,20 +637,19 @@ an impact on the CI coefficients similar to the Jastrow factor.
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In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we
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report several quantities related to the one- and two-body density of
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$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we
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report in table~\ref{table_on_top} the integrated on-top pair density
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$\expval{ n_2({\bf r},{\bf r}) }$
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report in Table~\ref{table_on_top} the integrated on-top pair density
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\begin{equation}
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\expval{ n_2({\bf r},{\bf r}) } = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})
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\end{equation}
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where $n_2({\bf r}_1,{\bf r}_2)$ is the two-body density (normalized to $N(N-1)$ where $N$ is the number of electrons)
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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
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pair density and the density, we report in Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}
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the plots of the total density $n({\bf r})$ and on-top pair density
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$n_2({\bf r},{\bf r})$ along one O---H axis of the water molecule.
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$n_2({\bf r},{\bf r})$ along one \ce{O-H} axis of the water molecule.
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From these data, one can clearly notice several trends.
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First, from Tab.~\ref{table_on_top}, we can observe that the overall
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First, from Table~\ref{table_on_top}, we can observe that the overall
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on-top pair density decreases when $\mu$ increases. This is expected
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as the two-electron interaction increases in $H^\mu[n]$.
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Second, the relative variations of the on-top pair density with $\mu$
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@ -658,7 +657,7 @@ are much more important than that of the one-body density, the latter
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being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the
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former can vary by about 10$\%$ in some regions.
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%TODO TOTO
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In the high-density region of the O---H bond, the value of the on-top
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In the high-density region of the \ce{O-H} bond, the value of the on-top
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pair density obtained from $\Psi^J$ is superimposed with
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$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is
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the closest to $\mu=\infty$. The integrated on-top pair density
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@ -668,7 +667,7 @@ and the overlap curve.
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These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
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and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
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Considering the form of $\hat{H}^\mu[n]$ (see Eq.~\eqref{H_mu}),
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Considering the form of $\hat{H}^\mu[n]$ [see Eq.~\eqref{H_mu}],
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one can notice that the differences with respect to the usual Hamiltonian come
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from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
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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.
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