pouet

Manuscript/rsdft-cipsi-qmc.tex

@@ -476,18 +476,10 @@ LDA exchange-correlation functional give very similar FN-DMC energy with respect
 to those obtained with the short-range PBE functional, even if the RS-DFT energies obtained 
 with these two functionals differ by several tens of m\hartree{}. 
 
-\begin{figure}
-  \centering
-  \includegraphics[width=\columnwidth]{overlap.pdf}
-  \caption{Overlap of the RS-DFT CI expansion with the
-    CI expansion optimized in the presence of a Jastrow factor.}
-  \label{fig:overlap}
-\end{figure}
-
-
 \subsection{Link between RS-DFT and jastrow factors }
-The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can give CI coefficients 
-which give trial wave functions with better nodes than FCI wave functions.
+\label{sec:rsdft-j}
+The data obtained in \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 
@@ -508,8 +500,9 @@ Such a wave function $\Psi^J$ satisfies the generalized hermitian eigenvalue equ
 but also the non-hermitian transcorrelated eigenvalue problem\cite{many_things}  
 \begin{equation}
  \label{eq:transcor}
- e^{-J}  H e^{J} \Psi^J = E \Psi^J.
+ e^{-J}  H e^{J} \Psi^J = E \Psi^J,
 \end{equation}
+which is much easier to handle despite its non-hermicity. 
 Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$. 
 In a finite basis set and with a quite accurate jastrow factor, it is known that the nodes 
 of $\Psi^J$ are better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$. 
@@ -524,12 +517,33 @@ 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 figure~\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 the fluorine dimer,
+and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions 
+$\Psi^\mu$ together with that of $\Psi^J$. 
+
+\begin{figure}
+  \centering
+  \includegraphics[width=\columnwidth]{overlap.pdf}
+  \caption{Overlap of the RS-DFT CI expansion with the
+    CI expansion optimized in the presence of a Jastrow factor.}
+  \label{fig:overlap}
+\end{figure}
+
+\begin{figure}
+  \centering
+  \includegraphics[width=\columnwidth]{x.png}
+  \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}).}
+  \label{dmc_small}
+\end{figure}
+
+
 
 In the case of H$_2$O, there is a clear maximum of overlap at
-$\mu=1$~bohr$^{-1}$. This confirms that introducing short-range
-correlation with DFT has the an impact on the CI coefficients similar to
-the Jastrow factor. In the case of F$_2$, the Jastrow factor has
+$\mu=1$~bohr$^{-1}$, which coincide with the minimum of the FN-DMC energy of $\Psi^\mu$. 
+???? 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$^{-1}$. 
+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
 $1$. 
@@ -537,18 +551,19 @@ 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 for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. 
-First, we report in table\ref{table_on_top} the integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$ 
+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 $\langle n_2({\bf r},{\bf r}) \rangle$ 
 \begin{equation}
- \langle n_2({\bf r},{\bf r}) \rangle = \int \text{d}{\bf r} n_2({\bf r},{\bf r})
+ \langle n_2({\bf r},{\bf r}) \rangle = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})
 \end{equation}
-where $n_2({\bf r},{\bf r})$ 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 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 table, one can clearly observe several trends. 
-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 $\mu=\infty$.  
-Second, the smaller the $\mu$, the larger the $\langle n_2({\bf r},{\bf r}) \rangle$. 
+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.  
 ??? 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 
 FCI wave function. 
 
@@ -565,10 +580,10 @@ while the effective one-body potential $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr
 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 
+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 while keeping the density constant. 
+they both deal with an effective non-divergent interaction but still produce reasonable one-body density. 
 
 \begin{table}
   \caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$ 
@@ -611,7 +626,7 @@ one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller
 fixed node error by properly choosing an optimal value of $\mu$ 
 in RS-DFT calculations, ii) the value of the optimal $\mu$ depends 
 on the system and the basis set, and the larger the basis set, the larger the optimal value of $\mu$, 
-iii) numerical experiments (such as computation of overlap) indicates 
+iii) numerical experiments (such as computation of overlap, FN-DMC energies) indicates 
 that the RS-DFT scheme essentially  plays the role of a simple Jastrow factor, 
 \textit{i.e.} mimicking short-range correlation effects. 
 The latter statement can be qualitatively understood by noticing that both RS-DFT 
@@ -619,7 +634,7 @@ and transcorrelated approach deal with an effective non-divergent electron-elect
 
 
 
-\section{Atomization energies}
+\section{Energy differences in FN-DMC: atomization energies}
 \label{sec:atomization}
 
 Atomization energies are challenging for post-Hartree-Fock methods