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@ 309,3 +309,33 @@ densityfunctional theory: A systematically improvable approach},


doi = {10.1103/PhysRevLett.94.150201}


}




@article{Huron_1973,


author = {Huron, B. and Malrieu, J. P. and Rancurel, P.},


title = {{Iterative perturbation calculations of ground and excited state energies


from multiconfigurational zeroth{}order wavefunctions}},


journal = {J. Chem. Phys.},


volume = {58},


number = {12},


pages = {57455759},


year = {1973},


month = {Jun},


issn = {00219606},


publisher = {American Institute of Physics},


doi = {10.1063/1.1679199}


}






@article{Davidson_1975,


author = {Davidson, Ernest R.},


title = {{The iterative calculation of a few of the lowest eigenvalues and corresponding


eigenvectors of large realsymmetric matrices}},


journal = {J. Comput. Phys.},


volume = {17},


number = {1},


pages = {8794},


year = {1975},


month = {Jan},


issn = {00219991},


publisher = {Academic Press},


doi = {10.1016/00219991(75)900650}


}



@ 49,11 +49,36 @@


\section{Introduction}


\label{sec:intro}




\section{Combining rangeseparated DFT with CIPSI}


\label{sec:rsdftcipsi}




\section{Rangeseparated DFT in a nutshell}


\subsection{CIPSI}




\emph{Configuration interaction using a perturbative selection made


iteratively} (CIPSI)\cite{Huron_1973} belongs to the class of selected


CI methods, whose goal is to build compressed configuration interaction


(CI) wave functions with a controlled accuracy, together with an estimate


of the associated eigenvalue obtained with perturbation theory.




Using a conventional iterative diagonalization method such as


Davidson's,\cite{Davidson_1975} the size of the CI space is


predefined, and at each iteration the length of the vectors is equal


to the size of the CI space. With selected CI methods, at each


iteration the size of the determinant space is increased such that


one keeps only the most important determinants that will lead to an


accurate description of the state of interest. The lowest eigenpairs


are extracted from the CI matrix expressed in this determinant


subspace, and the CI energy can be estimated by computing a


secondorder perturbative correction (PT2) to the variational energy


computed in the complete CI space. The magnitude of the PT2 correction


is a measure of the distance to the exact eigenvalue, and is an


adjustable parameter to control the quality of the wave function.






\subsection{Rangeseparated DFT in a nutshell}


\label{sec:rsdft}




\subsection{Exact equations}


\subsubsection{Exact equations}


\label{sec:exact}




The exact groundstate energy of a $N$electron system with


@ 189,31 +214,13 @@ takes the form


\end{eqnarray}






\subsection{RSDFTCIPSI}




It is possible to use DFT for shortrange interactions and CIPSI for


the longrange. This scheme has been recently


implemented,\cite{GinPraFerAssSavTouJCP18}






\section{Computational details}


\label{sec:compdetails}




All the calculations were made using BFD


pseudopotentials\cite{Burkatzki_2008} with the associated double, triple


and quadruple zeta basis sets.


CCSD(T) and DFT calculations were made with


\emph{Gaussian09},\cite{g09} using an unrestricted HartreeFock


determinant as a reference for openshell systems. All the CIPSI


calculations and rangeseparated CIPSI calculations were made with


\emph{Quantum Package}.\cite{Garniron_2019,qp2_2020}


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


in the determinant localization approximation.\cite{Zen_2019}




In the determinant localization approximation, only the determinantal


component of the trial wave function is present in the expression of


the wave function on which the pseudopotential is localized. Hence,


the pseudopotential operator does not depend on the Jastrow factor, as


it is the case in allelectron calculations. This improves the


reproducibility of the results, as they depend only on parameters


optimized in a deterministic framework.






\section{Influence of the rangeseparation parameter on the fixednode


error}


@ 260,6 +267,49 @@ were used as trial wave functions for FNDMC calculations, and the


corresponding energies are shown in table~\ref{tab:h2odmc} and


figure~\ref{fig:h2odmc}.




Using FCI trial wave functions gives FNDMC energies which are lower


than the energies obtained with a single KohnSham determinant:


3~m$E_h$ at the doublezeta level and 7~m$E_h$ at the triplezeta


level. Interestingly, with the doublezeta basis one can obtain a


FNDMC energy 2.5~m$E_h$ 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 $\mu=\infty$.






\section{Computational details}


\label{sec:compdetails}




All the calculations were made using BFD


pseudopotentials\cite{Burkatzki_2008} with the associated double, triple


and quadruple zeta basis sets.


CCSD(T) and DFT calculations were made with


\emph{Gaussian09},\cite{g09} using an unrestricted HartreeFock


determinant as a reference for openshell systems. All the CIPSI


calculations and rangeseparated CIPSI calculations were made with


\emph{Quantum Package}.\cite{Garniron_2019,qp2_2020}


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


in the determinant localization approximation.\cite{Zen_2019}




In the determinant localization approximation, only the determinantal


component of the trial wave function is present in the expression of


the wave function on which the pseudopotential is localized. Hence,


the pseudopotential operator does not depend on the Jastrow factor, as


it is the case in allelectron calculations. This improves the


reproducibility of the results, as they depend only on parameters


optimized in a deterministic framework.






\section{Atomization energy benchmarks}


\label{sec:atomization}


@ 354,6 +404,27 @@ of the RSDFTCIPSI trial wave functions for energy differences.








\begin{figure}[h]


\centering


\includegraphics[width=\columnwidth]{g2dmc_dz.pdf}


\caption{Histogram of the errors in atomization energies with the doublezeta basis set.}


\label{fig:g2dmc_dz}


\end{figure}




\begin{figure}[h]


\centering


\includegraphics[width=\columnwidth]{g2dmc_tz.pdf}


\caption{Histogram of the errors in atomization energies with the triplezeta basis set.}


\label{fig:g2dmc_tz}


\end{figure}




\begin{figure}[h]


\centering


\includegraphics[width=\columnwidth]{g2dmc_qz.pdf}


\caption{Histogram of the errors in atomization energies with the quadruplezeta basis set.}


\label{fig:g2dmc_qz}


\end{figure}











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