Merge branch 'master' of git.irsamc.ups-tlse.fr:scemama/RSDFT-CIPSI-QMC
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46b8ce3657
@ -624,7 +624,7 @@ $\Psi^\mu$ together with that of $\Psi^J$.
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\centering
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\includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
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\caption{\ce{H2O}, double-zeta basis set, 200 most important
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determinants of the FCI expansion (see \ref{sec:rsdft-j}).
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determinants of the FCI expansion (see Sec.~\ref{sec:rsdft-j}).
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FN-DMC energies of $\Psi^\mu$ (red curve), together with
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the FN-DMC energy of $\Psi^J$ (blue line). The width of the lines
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represent the statistical error bars.}
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@ -642,7 +642,8 @@ an impact on the CI coefficients similar to the Jastrow factor.
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%%% TABLE II %%%
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\begin{table}
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\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2(\br,\br) }$
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for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
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for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
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\titou{Please remove table and merge data in the Fig. 5.}}
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\label{table_on_top}
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\begin{ruledtabular}
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\begin{tabular}{cc}
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@ -666,7 +667,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
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\begin{figure}
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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 $n(\br)$ along
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\caption{\ce{H2O}, \titou{srLDA?} double-zeta basis set. One-electron density $n(\br)$ along
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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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@ -677,7 +678,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
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\begin{figure}
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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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\caption{\ce{H2O}, \titou{srLDA?} double-zeta basis set. On-top pair
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density $n_2(\br,\br)$ 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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@ -694,7 +695,7 @@ report in Table~\ref{table_on_top} the integrated on-top pair density
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where $n_2(\br_1,\br_2)$ 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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pair density and the density, we report in Figs.~\ref{fig:n1} and \ref{fig:n2}
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the plots of the total density $n(\br)$ and on-top pair density
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$n_2(\br,\br)$ along one \ce{O-H} axis of the water molecule.
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@ -727,7 +728,7 @@ increases the probability to find electrons at short distances in $\Psi^\mu$,
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while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
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provided that it is exact, maintains the exact one-body density.
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This is clearly what has been observed from the plots in
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Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}.
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Figs.~\ref{fig:n1} and \ref{fig:n2}.
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Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No,\cite{Ten-no2000Nov}
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the effective two-body interaction induced by the presence of a Jastrow factor
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can be non-divergent when a proper Jastrow factor is chosen.
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@ -815,7 +816,7 @@ Another source of size-consistency error in QMC calculations originates
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from the Jastrow factor. Usually, the Jastrow factor contains
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one-electron, two-electron and one-nucleus-two-electron terms.
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The problematic part is the two-electron term, whose simplest form can
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be expressed as in Eq.\eqref{eq:jast-ee}.
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be expressed as in Eq.~\eqref{eq:jast-ee}.
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The parameter
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$a$ is determined by cusp conditions, and $b$ is obtained by energy
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or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
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@ -872,7 +873,7 @@ parameter.
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We have computed the FN-DMC energy of the dissociated fluorine dimer, where
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the two atoms are at a distance of 50~\AA. We expect that the energy
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of this system is equal to twice the energy of the fluorine atom.
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The data in table~\ref{tab:size-cons} shows that it is indeed the
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The data in Table~\ref{tab:size-cons} shows that it is indeed the
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case, so we can conclude that the proposed scheme provides
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size-consistent FN-DMC energies for all values of $\mu$ (within
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$2\times$ statistical error bars).
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@ -928,11 +929,11 @@ In this section, we investigate the impact of the spin contamination
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due to the short-range density functional on the FN-DMC energy. We have
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computed the energies of the carbon atom in its triplet state
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with BFD pseudopotentials and the corresponding double-zeta basis
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set. The calculation was done with $m_s=1$ (3 $\uparrow$ electrons
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and 1 $\downarrow$ electrons) and with $m_s=0$ (2 $\uparrow$ and 2
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$\downarrow$ electrons).
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set. The calculation was done with $m_s=1$ (3 spin-up electrons
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and 1 spin-down electrons) and with $m_s=0$ (2 spin-up and 2
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spin-down electrons).
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The results are presented in table~\ref{tab:spin}.
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The results are presented in Table~\ref{tab:spin}.
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Although using $m_s=0$ the energy is higher than with $m_s=1$, the
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bias is relatively small, more than one order of magnitude smaller
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than the energy gained by reducing the fixed-node error going from the single
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@ -1071,7 +1072,7 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
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Searching for the optimal value of $\mu$ may be too costly, so we have
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computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
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are illustrated in figure~\ref{fig:g2-dmc}. As seen on the figure and
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are illustrated in Fig.~\ref{fig:g2-dmc}. As seen on the figure and
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in Table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is
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0.5~bohr$^{-1}$ for all three basis sets. It is the value for which
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the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23),
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@ -1095,7 +1096,7 @@ cancellations of errors.
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%%% %%% %%% %%%
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The number of determinants in the trial wave functions are shown in
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figure~\ref{fig:g2-ndet}. As expected, the number of determinants
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Fig.~\ref{fig:g2-ndet}. As expected, the number of determinants
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is smaller when $\mu$ is small and larger when $\mu$ is large.
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It is important to remark that the median of the number of
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determinants when $\mu=0.5$~bohr$^{-1}$ is below 100~000 determinants
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