From e5fd32c5e3a16cad1667d31ed063d6b349d61ce8 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 8 Aug 2020 22:41:39 +0200 Subject: [PATCH] saving work --- Manuscript/rsdft-cipsi-qmc.tex | 23 +++++++++++------------ 1 file changed, 11 insertions(+), 12 deletions(-) diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index b4832b0..0d84984 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -514,12 +514,12 @@ The take-home message here is that RS-DFT trial wave functions yield a lower fix \subsection{Link between RS-DFT and Jastrow factors } \label{sec:rsdft-j} %====================================================== -The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients +The data obtained in Sec.~\ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients giving trial wave functions with better nodes than FCI wave functions. Such behaviour can be compared to the common practice of re-optimizing the Slater part of a trial wave function in the presence of the Jastrow factor. In the present paragraph, we would like therefore to elaborate some more on the link between RS-DFT -and wave function optimization within the presence of a Jastrow factor. +and wave function optimization with the presence of a Jastrow factor. Let us assume a fixed Jastrow factor $J({\bf{r}_1}, \hdots , {\bf{r}_N})$, and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$, @@ -553,7 +553,7 @@ a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$. Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed on the same Slater determinant basis. In Fig.~\ref{fig:overlap}, we plot the overlaps -$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer, +$\braket*{\Psi^J}{\Psi^\mu}$ obtained for water, and in Fig.~\ref{dmc_small} the FN-DMC energy of the wave functions $\Psi^\mu$ together with that of $\Psi^J$. @@ -617,7 +617,7 @@ an impact on the CI coefficients similar to the Jastrow factor. \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, + density $n_2({\bf r},{\bf r})$ along the \ce{O-H} axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. } \label{fig:n2} \end{figure} @@ -628,7 +628,7 @@ an impact on the CI coefficients similar to the Jastrow factor. \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 $\mu$. } + the \ce{O-H} axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. } \label{fig:n1} \end{figure} %%% %%% %%% %%% @@ -637,20 +637,19 @@ 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 -$\expval{ n_2({\bf r},{\bf r}) }$ +report in Table~\ref{table_on_top} the integrated on-top pair density \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) +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 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. +$n_2({\bf r},{\bf r})$ along one \ce{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 +First, from Table~\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 $\mu$ @@ -658,7 +657,7 @@ 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 -In the high-density region of the O---H bond, the value of the on-top +In the high-density region of the \ce{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 @@ -668,7 +667,7 @@ and the overlap curve. 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 (\ie, $H^\mu[n]$ 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 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.