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\begin{abstract}


By combining densityfunctional theory (DFT) and wave function theory (WFT) via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixednode diffusion Monte Carlo (FNDMC) energies with compact multideterminant trial wave functions.


By combining densityfunctional theory (DFT) and wave function theory (WFT) via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixednode diffusion Monte Carlo (FNDMC) energies with compact multideterminant trial wave functions.


These compact trial wave functions are generated via the diagonalization of the RSDFT Hamiltonian.


In particular, we combine here shortrange correlation functionals with selected configuration interaction (SCI).


As the WFT method is relieved from describing the shortrange part of the correlation hole around the electronelectron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional SCI calculation.


@ 124,7 +124,8 @@ the FNDMC energy associated with a given trial wave function is an upper


bound to the exact energy, and the latter is recovered only when the


nodes of the trial wave function coincide with the nodes of the exact


wave function.


The trial wave function in FNDMC is then a key ingredient which dictates via the quality of its nodal surface the accuracy of the resulting energy and properties.


The trial wave function, which can be single or multideterminantal in nature depending on the type of correlation at play, is then a key ingredient dictating via the quality of its nodal surface the accuracy of the resulting energy and properties.




The polynomial scaling of its computational cost with respect to the number of electrons and with the size


of the trial wave function makes the FNDMC method particularly attractive.


This favorable scaling, its very low memory requirements and


@ 135,6 +136,19 @@ those obtained with the FCI method in computationally tractable basis


sets because the constraints imposed by the fixednode approximation


are less severe than the constraints imposed by the finitebasis


approximation.


However, because it is not possible to minimize directly the FNDMC energy with respect


to the linear and nonlinear parameters of the trial wave function, the


fixednode approximation is much more difficult to control than the


finitebasis approximation.


The conventional approach consists in multiplying the trial wave


function by a positive function, the \emph{Jastrow factor}, taking


account of the bulk of the dynamical correlation.


%electronelectron cusp and the shortrange correlation effects.


The trial wave function is then reoptimized within variational


Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal


surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}


Using this technique, it has been shown that the chemical accuracy could be reached within


FNDMC.\cite{Petruzielo_2012}






%However, it is usually harder to control the FN error in DMC, and this


@ 150,26 +164,14 @@ correlated systems, such as organic molecules near their equilibrium


geometry, is usually well represented with a single Slater


determinant. This feature is in part responsible for the success of


DFT and coupled cluster theory.


DMC with a singledeterminant trial wave function can be used as a


singlereference postHartreeFock method, with an accuracy comparable


Likewise, DMC with a singledeterminant trial wave function can be used as a


singlereference postHartreeFock method for weakly correlated systems, with an accuracy comparable


to coupled cluster.\cite{Dubecky_2014,Grossman_2002}




As it is not possible to minimize directly the FNDMC energy with respect


to the linear and nonlinear parameters of the trial wave function, the


fixednode approximation is much more difficult to control than the


finitebasis approximation.


The conventional approach consists in multiplying the trial wave


function by a positive function, the \emph{Jastrow factor}, taking


account of the electronelectron cusp and the shortrange correlation


effects. The wave function is then reoptimized within variational


Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal


surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}


Using this technique, it has been shown that the chemical accuracy could be reached within


FNDMC.\cite{Petruzielo_2012}




Another approach consists in considering the FNDMC method as a


``postFCI'' method. The trial wave function is obtained by


approaching the FCI with a SCI method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}


This approach obviously fails in the presence of strong correlation, like in


transition metal complexes, lowspin openshell systems, and covalent bond breaking situations which cannot be even qualitatively described by a single electronic configuration.


In such cases, a viable alternative is to consider the FNDMC method as a


``postFCI'' method. The multideterminant trial wave function is then produced by


approaching FCI with a SCI method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}


When the basis set is enlarged, the trial wave function gets closer to


the exact wave function, so we expect the nodal surface to be


improved.\cite{Caffarel_2016}


@ 194,12 +196,12 @@ applicable for large systems with a multiconfigurational character is


still an active field of research. The present paper falls


within this context.


The central idea of the present work, and the launchpad for the remainder of this study, is that one can combine the various strengths of these three philosophies in order to create a hybrid method with more attractive properties.


In particular, we show here that one can combine FBFCI and KSDFT via the range separation (RS) of the interelectronic Coulomb operator to obtain accurate FNDMC energies with compact multideterminant trial wave functions.


In particular, we show here that one can combine FBFCI and KSDFT via the range separation (RS) of the interelectronic Coulomb operator to obtain accurate FNDMC energies with compact multideterminant trial wave functions.








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Combining CIPSI with RSDFT}


\section{Combining WFT and DFT}


\label{sec:rsdftcipsi}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




@ 326,7 +328,7 @@ $\hat{W}_\text{ee}^{\text{lr},\mu}$ the longrange


electronelectron interaction,


$n_\Psi$ the oneelectron density associated with $\Psi$,


and $\hat{V}_{\text{ne}}$ the electronnucleus potential.


The minimizing multideterminant wave function $\Psi^\mu$


The minimizing multideterminant wave function $\Psi^\mu$


can be determined by the selfconsistent eigenvalue equation


\begin{equation}


\label{rsdfteigenequation}


@ 528,7 +530,7 @@ tens of m\hartree{}.




An other important aspect here regards 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/VDZBFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2odmc}). Even at the srPBE/VTZBFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.


The takehome message of this numerical study is that RSDFT trial wave functions can yield a lower fixednode energy with more compact multideterminant expansion as compared to FCI.


The takehome message of this numerical study is that RSDFT trial wave functions can yield a lower fixednode energy with more compact multideterminant expansion as compared to FCI.




%======================================================


\subsection{Link between RSDFT and Jastrow factors }


@ 537,7 +539,7 @@ The takehome message of this numerical study is that RSDFT trial wave function


The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RSDFT can provide


trial wave functions with better nodes than FCI wave function.


Such behaviour can be directly compared to the common practice of


reoptimizing the multideterminant part of a trial wave function $\Psi$ (the socalled Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}


reoptimizing the multideterminant part of a trial wave function $\Psi$ (the socalled Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}


Hence, in the present paragraph, we would like to elaborate further on the link between RSDFT


and wave function optimization in the presence of a Jastrow factor.


For simplicity in the comparison, the molecular orbitals and the Jastrow



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