From b22dcbede273130e272db96ed99377383788d6c5 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 17 Aug 2020 22:39:16 +0200 Subject: [PATCH] 1st screening of Sec IV done --- Manuscript/rsdft-cipsi-qmc.tex | 73 ++++++++++++++++++---------------- 1 file changed, 38 insertions(+), 35 deletions(-) diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index 56d4ba8..58abc01 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -540,6 +540,7 @@ functional give very similar FN-DMC energies with respect to those obtained with srPBE, even if the RS-DFT energies obtained with these two functionals differ by several tens of m\hartree{}. +Accordingly, all the RS-DFT calculations are performed with the srPBE functional in the remaining of this paper. Another important aspect here is the compactness of the trial wave functions $\Psi^\mu$: at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$. @@ -558,7 +559,7 @@ and wave function optimization in the presence of a Jastrow factor. For the sake of simplicity, the molecular orbitals and the Jastrow factor are kept fixed; only the CI coefficients are varied. -Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_N)$ (where $\br_i$ is the position of the $i$th electron), +Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_\Nelec)$ (where $\br_i$ is the position of the $i$th electron), and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$, where \begin{equation} @@ -591,7 +592,7 @@ To do so, we have made the following numerical experiment. First, we extract the 200 determinants with the largest weights in the FCI wave function out of a large CIPSI calculation obtained with the VDZ-BFD basis. Within this set of determinants, we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}] -with different values of $\mu$. This gives the CI expansions $\Psi^\mu$. +for different values of $\mu$ \titou{using the srPBE functional}. This gives the CI expansions $\Psi^\mu$. Then, within the same set of determinants we optimize the CI coefficients $c_I$ [see Eq.~\eqref{eq:Slater}] in the presence of a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and \begin{subequations} @@ -604,8 +605,8 @@ a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ \end{subequations} The one-body Jastrow factor $J_\text{eN}$ contains the electron-nucleus terms (where $\Nat$ is the number of nuclei) with a single parameter $\alpha_A$ per nucleus. -The two-body Jastrow factor $J_\text{ee}$ contains the electron-electron terms -where the sum over $i < j$ loops over all electron pairs. +The two-body Jastrow factor $J_\text{ee}$ gathers the electron-electron terms +where the sum over $i < j$ loops over all unique electron pairs. In Eqs.~\eqref{eq:jast-eN} and \eqref{eq:jast-ee}, $r_{iA}$ is the distance between the $i$th electron and the $A$th nucleus while $r_{ij}$ is the interlectronic distance between electrons $i$ and $j$. The parameters $a=1/2$ and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$ @@ -625,7 +626,7 @@ We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed on the same set of Slater determinants. In Fig.~\ref{fig:overlap}, we plot the overlaps $\braket*{\Psi^J}{\Psi^\mu}$ obtained for water, -and in Fig.~\ref{dmc_small} the FN-DMC energy of the wave functions +and in Fig.~\ref{fig:dmc_small} the FN-DMC energy of the wave functions $\Psi^\mu$ together with that of $\Psi^J$. %%% FIG 3 %%% @@ -641,21 +642,21 @@ $\Psi^\mu$ together with that of $\Psi^J$. %%% FIG 4 %%% \begin{figure} - \centering \includegraphics[width=\columnwidth]{h2o-200-dmc.pdf} - \caption{\ce{H2O}, double-zeta basis set, 200 most important - 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.} - \label{dmc_small} + \caption{ + FN-DMC energies of $\Psi^\mu$ (red curve) as a function of $\mu$, together with + the FN-DMC energy of $\Psi^J$ (blue line) for \ce{H2O}. The width of the lines + represent the statistical error bars. + For these two trial wave functions, the CI expansion consists of the 200 most important + determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).} + \label{fig:dmc_small} \end{figure} %%% %%% %%% %%% There is a clear maximum of overlap at $\mu=1$~bohr$^{-1}$, which coincides with the minimum of the FN-DMC energy of $\Psi^\mu$. Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is compatible -with that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that +with that of $\Psi^\mu$ for $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that introducing short-range correlation with DFT has an impact on the CI coefficients similar to the Jastrow factor. @@ -664,7 +665,7 @@ an impact on the CI coefficients similar to the Jastrow factor. \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 $\mu$. \titou{Please remove table and merge data in the Fig. 5.}} - \label{table_on_top} + \label{tab:table_on_top} \begin{ruledtabular} \begin{tabular}{cc} $\mu$ & $\expval{ n_2(\br,\br) }$ \\ @@ -685,10 +686,11 @@ an impact on the CI coefficients similar to the Jastrow factor. %%% FIG 5 %%% \begin{figure} - \centering \includegraphics[width=\columnwidth]{density-mu.pdf} - \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 $\mu$. } + \caption{One-electron density $n(\br)$ along + the \ce{O-H} axis of \ce{H2O} as a function of $\mu$ for $\Psi^J$ (dashed curve) and $\Psi^\mu$. + For these two trial wave functions, the CI expansion consists of the 200 most important + determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).} \label{fig:n1} \end{figure} %%% %%% %%% %%% @@ -696,31 +698,32 @@ an impact on the CI coefficients similar to the Jastrow factor. %%% FIG 6 %%% \begin{figure} - \centering \includegraphics[width=\columnwidth]{on-top-mu.pdf} - \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 $\mu$. } + \caption{On-top pair + density $n_2(\br,\br)$ along the \ce{O-H} axis of \ce{H2O} as a function of $\mu$ + for $\Psi^J$ (dashed curve) and $\Psi^\mu$. + For these two trial wave functions, the CI expansion consists of the 200 most important + determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).} \label{fig:n2} \end{figure} %%% %%% %%% %%% 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 +report several quantities related to the one- and two-body densities 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 \begin{equation} \expval{ n_2(\br,\br) } = \int d\br \,\,n_2(\br,\br) \end{equation} -where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $N(N-1)$ where $N$ is the number of electrons] +where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $\Nelec(\Nelec-1)$] 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 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. +$n_2(\br,\br)$ along one of these \ce{O-H} axis of the water molecule. From these data, one can clearly notice several trends. -First, from Table~\ref{table_on_top}, we can observe that the overall +First, from Table~\ref{tab:table_on_top}, we can observe that the overall on-top pair density decreases when $\mu$ increases, which is expected as the two-electron interaction increases in $H^\mu[n]$. Second, the relative variations of the on-top pair density with $\mu$ @@ -734,7 +737,7 @@ $\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is the closest to $\mu=\infty$. The integrated on-top pair density obtained with $\Psi^J$ lies between the values obtained with $\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies -and the overlap curve. +and the overlap curve depicted in Figs.~\ref{fig:overlap} and \ref{fig:dmc_small} 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. @@ -743,24 +746,24 @@ 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. The roles of these two terms are therefore very different: with respect -to the exact ground state wave function $\Psi$, the non divergent two body interaction +to the exact ground-state wave function $\Psi$, the non-divergent two-body interaction 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 +This is clearly what has been observed from 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} +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. -Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Jastrow-Slater optimization: +Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Slater-Jastrow optimization: they both deal with an effective non-divergent interaction but still produce a reasonable one-body density. As a conclusion of the first part of this study, we can notice that: \begin{itemize} \item with respect to the nodes of a KS determinant or a FCI wave function, - one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller - fixed node error by properly choosing an optimal value of $\mu$ + 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, \item the optimal value of $\mu$ depends on the system and the basis set, and the larger the basis set, the larger the optimal value @@ -770,7 +773,7 @@ As a conclusion of the first part of this study, we can notice that: that the RS-DFT scheme essentially plays the role of a simple Jastrow factor, \ie, mimicking short-range correlation effects. The latter statement can be qualitatively understood by noticing that both RS-DFT - and transcorrelated approaches deal with an effective non-divergent + and the trans-correlated approach deal with an effective non-divergent electron-electron interaction, while keeping the density constant. \end{itemize} @@ -780,12 +783,12 @@ As a conclusion of the first part of this study, we can notice that: \label{sec:atomization} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -Atomization energies are challenging for post-Hartree-Fock methods +Atomization energies are challenging for post-HF methods because their calculation requires a perfect balance in the description of atoms and molecules. Basis sets used in molecular calculations are atom-centered, so they are always better adapted to atoms than molecules and atomization energies usually tend to be -underestimated with variational methods. +underestimated by variational methods. In the context of FN-DMC calculations, the nodal surface is imposed by the trial wavefunction which is expanded on an atom-centered basis set, so we expect the fixed-node error to be also tightly related to