Merge branch 'master' of https://git.irsamc.ups-tlse.fr/scemama/RSDFT-CIPSI-QMC

Manuscript/rsdft-cipsi-qmc.tex

@@ -624,7 +624,7 @@ $\Psi^\mu$ together with that of $\Psi^J$.
   \centering
   \includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
   \caption{\ce{H2O}, double-zeta basis set, 200 most important
-    determinants of the FCI expansion (see \ref{sec:rsdft-j}).
+    determinants of the FCI expansion (see Sec.~\ref{sec:rsdft-j}).
     FN-DMC energies of $\Psi^\mu$ (red curve), together with
     the FN-DMC energy of $\Psi^J$ (blue line). The width of the lines
     represent the statistical error bars.}
@@ -642,7 +642,8 @@ an impact on the CI coefficients similar to the Jastrow factor.
 %%% TABLE II %%%
 \begin{table}
   \caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2(\br,\br) }$
-           for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
+           for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. 
+           \titou{Please remove table and merge data in the Fig. 5.}}
   \label{table_on_top}
   \begin{ruledtabular}
     \begin{tabular}{cc}
@@ -666,7 +667,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
 \begin{figure}
   \centering
   \includegraphics[width=\columnwidth]{density-mu.pdf}
-  \caption{\ce{H2O}, double-zeta basis set. Density $n(\br)$ along
+  \caption{\ce{H2O}, \titou{srLDA?} double-zeta basis set. One-electron density $n(\br)$ along
     the \ce{O-H} axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
   \label{fig:n1}
 \end{figure}
@@ -677,7 +678,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
 \begin{figure}
   \centering
   \includegraphics[width=\columnwidth]{on-top-mu.pdf}
-  \caption{\ce{H2O}, double-zeta basis set. On-top pair
+  \caption{\ce{H2O}, \titou{srLDA?} double-zeta basis set. On-top pair
     density $n_2(\br,\br)$ along the \ce{O-H} axis,
     for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
   \label{fig:n2}
@@ -694,7 +695,7 @@ report in Table~\ref{table_on_top} the integrated on-top pair density
 where $n_2(\br_1,\br_2)$ 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}
+pair density and the density, we report in Figs.~\ref{fig:n1} and \ref{fig:n2}
 the plots of the total density $n(\br)$ and on-top pair density
 $n_2(\br,\br)$ along one \ce{O-H} axis of the water molecule.
 
@@ -727,7 +728,7 @@ increases the probability to find electrons at short distances in $\Psi^\mu$,
 while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
 provided that it is exact, maintains the exact one-body density.
 This is clearly what has been observed from the plots in
-Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}.
+Figs.~\ref{fig:n1} and \ref{fig:n2}.
 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
 can be non-divergent when a proper Jastrow factor is chosen.
@@ -815,7 +816,7 @@ Another source of size-consistency error in QMC calculations originates
 from the Jastrow factor. Usually, the Jastrow factor contains
 one-electron, two-electron and one-nucleus-two-electron terms.
 The problematic part is the two-electron term, whose simplest form can
-be expressed as in Eq.\eqref{eq:jast-ee}.
+be expressed as in Eq.~\eqref{eq:jast-ee}.
 The parameter
 $a$ is determined by cusp conditions, and $b$ is obtained by energy
 or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
@@ -872,7 +873,7 @@ parameter.
 We have computed the FN-DMC energy of the dissociated fluorine dimer, where
 the two atoms are at a distance of 50~\AA.  We expect that the energy
 of this system is equal to twice the energy of the fluorine atom.
-The data in table~\ref{tab:size-cons} shows that it is indeed the
+The data in Table~\ref{tab:size-cons} shows that it is indeed the
 case, so we can conclude that the proposed scheme provides
 size-consistent FN-DMC energies for all values of $\mu$ (within
 $2\times$ statistical error bars).
@@ -928,11 +929,11 @@ In this section, we investigate the impact of the spin contamination
 due to the short-range density functional on the FN-DMC energy. We have
 computed the energies of the carbon atom in its triplet state
 with BFD pseudopotentials and the corresponding double-zeta basis
-set. The calculation was done with $m_s=1$ (3 $\uparrow$ electrons
-and 1 $\downarrow$ electrons) and with $m_s=0$ (2 $\uparrow$ and 2
-$\downarrow$ electrons).
+set. The calculation was done with $m_s=1$ (3 spin-up electrons
+and 1 spin-down electrons) and with $m_s=0$ (2 spin-up and 2
+spin-down electrons).
 
-The results are presented in table~\ref{tab:spin}.
+The results are presented in Table~\ref{tab:spin}.
 Although using $m_s=0$ the energy is higher than with $m_s=1$, the
 bias is relatively small, more than one order of magnitude smaller
 than the energy gained by reducing the fixed-node error going from the single
@@ -1071,7 +1072,7 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
 
 Searching for the optimal value of $\mu$ may be too costly, so we have
 computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
-are illustrated in figure~\ref{fig:g2-dmc}. As seen on the figure and
+are illustrated in Fig.~\ref{fig:g2-dmc}. As seen on the figure and
 in Table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is
 0.5~bohr$^{-1}$ for all three basis sets. It is the value for which
 the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23),
@@ -1095,7 +1096,7 @@ cancellations of errors.
 %%% %%% %%% %%%
 
 The number of determinants in the trial wave functions are shown in
-figure~\ref{fig:g2-ndet}. As expected, the number of determinants
+Fig.~\ref{fig:g2-ndet}. As expected, the number of determinants
 is smaller when $\mu$ is small and larger when $\mu$ is large.
 It is important to remark that the median of the number of
 determinants when $\mu=0.5$~bohr$^{-1}$ is below 100~000 determinants