From 9d0ebecc827b827c7e0269f6174e328d74ae5373 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 17 Aug 2020 10:47:27 +0200 Subject: [PATCH] minor corrections --- Manuscript/rsdft-cipsi-qmc.tex | 29 +++++++++++++++-------------- 1 file changed, 15 insertions(+), 14 deletions(-) diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index 019bbe8..2de0ea6 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -623,7 +623,7 @@ $\Psi^\mu$ 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.} @@ -641,7 +641,8 @@ 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} @@ -665,7 +666,7 @@ 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} @@ -676,7 +677,7 @@ 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} @@ -693,7 +694,7 @@ 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. @@ -726,7 +727,7 @@ 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. @@ -814,7 +815,7 @@ 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} @@ -871,7 +872,7 @@ 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). @@ -927,11 +928,11 @@ 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 @@ -1070,7 +1071,7 @@ $\mu=0.5$~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), @@ -1094,7 +1095,7 @@ 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