Overlap
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\newcommand{\EPT}{E_{\text{PT2}}}


\newcommand{\EDMC}{E_{\text{FNDMC}}}


\newcommand{\Ndet}{N_{\text{det}}}


\newcommand{\hartree}{$E_h$}




\newcommand{\LCT}{Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France}


\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}


@ 318,18 +319,17 @@ determinants.


We can follow this path by performing FCI calculations using the


RSDFT Hamiltonian with different values of $\mu$. In this work, we


have used the CIPSI algorithm to peform approximate FCI calculations


with the RSDFT Hamiltonians,\cite{GinPraFerAssSavTouJCP18} as shown


in figure~\ref{fig:algo}. In the outer loop (red), a CIPSI selection


is performed with a RSHamiltonian parameterized using the current


density. An inner loop (blue) is introduced to accelerate the


with the RSDFT Hamiltonians,\cite{GinPraFerAssSavTouJCP18}


$\hat{H}^\mu$ as shown in figure~\ref{fig:algo}. In the outer loop


(red), a CIPSI selection is performed with a RSHamiltonian


parameterized using the current density.


An inner loop (blue) is introduced to accelerate the


convergence of the selfconsistent calculation, in which the set of


determinants is kept fixed, and only the diagonalization of the


RSHamiltonian is performed iteratively.


RSHamiltonian is performed iteratively with the updated density.


The convergence of the algorithm was further improved


by introducing a direct inversion in the iterative subspace (DIIS)


step to extrapolate the density both in the outer and inner loops.


As mentioned above, the convergence criterion for CIPSI was set to


$\EPT < 1$~m$E_h$.




\section{Computational details}


\label{sec:compdetails}


@ 349,7 +349,7 @@ and correlation functionals of


Ref.~\onlinecite{GolWerStoLeiGorSavCP06} (see also


Refs.~\onlinecite{TouColSavJCP05,GolWerStoPCCP05}).


The convergence criterion for stopping the CIPSI calculations


was $\EPT < 1$~m$E_h \vee \Ndet > 10^7$.


was $\EPT < 1$~m\hartree{} $\vee \Ndet > 10^7$.




Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}


in the determinant localization approximation (DLA),\cite{Zen_2019}


@ 418,52 +418,79 @@ $\infty$ & $8302442$ & $48.437(3)$ \\


The first question we would like to address is the quality of the


nodes of the wave functions $\Psi^{\mu}$ obtained with an intermediate


range separation parameter $\mu$ (\textit{i.e.} $0 < \mu < +\infty$).


We generated trial wave functions $\Psi^\mu$ with multiple


values of $\mu$, and computed the associated fixed node energy


keeping fixed all the parameters having an impact on the nodal surface.


We generated trial wave functions $\Psi^\mu$ with multiple values of


$\mu$, and computed the associated fixed node energy keeping all the


parameters having an impact on the nodal surface fixed.


We considered two weakly correlated molecular systems: the water


molecule and fluorine dimer, near their equilibrium


molecule and the fluorine dimer, near their equilibrium


geometry\cite{Caffarel_2016}.


We report the FNDMC energies of the water molecule in


table~\ref{tab:h2odmc} and figure~\ref{fig:h2odmc}.




\eg{From table~\ref{tab:h2odmc} and figure~\ref{fig:h2odmc} one can clearly observe that }using FCI trial wave functions gives FNDMC energies which are lower


than the energies obtained with a single KohnSham determinant:


\eg{a gain of} 3~m$E_h$ at the doublezeta level and 7~m$E_h$ at the triplezeta


level \eg{are obtained}. Interestingly, with the doublezeta basis one can obtain a


FNDMC energy 2.5~m$E_h$ lower than the energy obtained with the FCI


From table~\ref{tab:h2odmc} and figures~\ref{fig:h2odmc}


and~\ref{fig:f2dmc}, one can clearly observe that using FCI trial


wave functions gives FNDMC energies which are lower than the energies


obtained with a single KohnSham determinant:


a gain of $3.2 \pm 0.6$~m\hartree{} at the doublezeta level and $7.2 \pm


0.3$~m\hartree{} at the triplezeta level are obtained for water, and


a gain of $18 \pm 3$~m\hartree{} for F$_2$. Interestingly, with the


doublezeta basis one can obtain for water a FNDMC energy $2.6 \pm


0.7$~m\hartree{} lower than the energy obtained with the FCI


trial wave function, using the RSDFTCIPSI with a rangeseparation


parameter $\mu=1.75$. This can be explained by the inability of the


basis set to properly describe shortrange correlation, shifting


the nodes from their optimal position. Using DFT to take account of


shortrange correlation frees the determinant expansion from describing


shortrange effects, and enables a better placement of the nodes.


At the triplezeta level, the shortrange correlations can be better


described, and the improvement due to DFT is insignificant. However,


it is important to note that the same FNDMC energy can be


obtained with a CI expansion which is eight times smaller when srDFT


is introduced. One can also remark that the minimum has been


shifted towards the FCI, which is consistent with the fact that


in the CBS limit we expect the minimum of the FNDMC energy to be


obtained for the FCI wave function, at $\mu=\infty$.




\eg{To further study the behaviour of the FNDMC energy as a function of $\mu$,


we report in table ???? and figure~\ref{fig:f2dmc} the FNDMC energies obtained


with a similar procedure for the fluorine dimer.


The global behaviour and shape of the curves show a very similar behaviour


with respect to that obtained on the water molecule: there exist an "optimal" value of $\mu$ which provides a lower FNDMC energy than both the KS determinant (\textit{i.e.} $\mu = 0$) and the FCI wave function (\textit{i.e.} $\mu=\infty$).


Nevertheless, one can notice that the value of such optimal $\mu$ is sensibly larger in F$_2$ than H$_2$O: this is probably the signature of the fact that the average inter electronic distance in the valence is smaller in F$_2$ than in H$_2$O due to the larger nuclear charge and corresponding shrinking of the electronic density.


}




parameter $\mu=1.75$~bohr$^{1}$. This can be explained by the


inability of the basis set to properly describe the shortrange


correlation effects, shifting the nodes from their optimal


position. Using DFT to take account of shortrange correlation frees


the determinant expansion from describing shortrange effects, and


enables a placement of the nodes closer to the optimum. In the case


of F$_2$, a similar behavior with a gain of $8 \pm 4$ m\hartree{} is


observed for $\mu\sim 5$~bohr$^{1}$.


The optimal value of $\mu$ is larger than in the case of water. This


is probably the signature of the fact that the average


electronelectron distance in the valence is smaller in F$_2$ than in


H$_2$O due to the larger nuclear charge shrinking the electron


density. At the triplezeta level, the shortrange correlations can be


better described by the determinant expansion, and the improvement due


to DFT is insignificant. However, it is important to note that the


same FNDMC energy can be obtained with a CI expansion which is eight


times smaller when srDFT is introduced. One can also remark that the


minimum has been slightly shifted towards the FCI, which is consistent


with the fact that in the CBS limit we expect the minimum of the


FNDMC energy to be obtained for the FCI wave function, i.e. at


$\mu=\infty$.




\begin{figure}


\centering


\includegraphics[width=\columnwidth]{overlap.pdf}


\caption{Overlap of the RSDFTCIPSI wave functions with the


wave function reoptimized in the presence of a Jastrow factor.}


\caption{Overlap of the RSDFT CI expansion with the


CI expansion optimized in the presence of a Jastrow factor.}


\label{fig:overlap}


\end{figure}




This data confirms that RSDFT/CIPSI can give improved CI coefficients


with small basis sets, similarly to the common practice of


reoptimizing the wave function in the presence of the Jastrow


factor. To confirm that the introduction of RSDFT has the same impact


that the Jastrow factor on the CI coefficients, 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.


Within this set of determinants, we diagonalize selfconsistently the


RSDFT Hamiltonian with different values of $\mu$. This gives the CI


expansions $\Psi^\mu$. Then, within the same set of determinants we


optimize the CI coefficients in the presence of a simple one and


twobody Jastrow factor. This gives the CI expansion $\Psi^J$.


In figure~\ref{fig:overlap}, we plot the overlaps


$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer.




In the case of H$_2$O, there is a clear maximum of overlap at


$\mu=1$~bohr$^{1}$. This confirms that introducing shortrange


correlation with DFT has the same impact on the CI coefficients than


with the Jastrow factor. In the case of F$_2$, the Jastrow factor has


very little effect on the CI coefficients, as the overlap


$\braket{\Psi^J}{\Psi^{\mu=\infty}$ is very close to


$1$. Nevertheless, a slight maximum is obtained for


$\mu=5$~bohr$^{1}$.








\section{Atomization energy benchmarks}


\label{sec:atomization}




@ 621,7 +648,7 @@ DMC@RSDFTCIPSI & 0 & 4.61(34) & 3.62(\phantom{0.}34) & 5.30




The number of determinants in the wave functions are shown in


figure~\ref{fig:n2ndet}. For all the calculations, the stopping


criterion of the CIPSI algorithm was $\EPT < 1$~m$E_h$ or $\Ndet >


criterion of the CIPSI algorithm was $\EPT < 1$~m\hartree{} or $\Ndet >


10^7$.


For FCI, we have given extrapolated values at $\EPT\rightarrow 0$.


At $\mu=0$ the number of determinants is not equal to one because



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