saving work

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.