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@ 175,20 +175,24 @@ Nevertheless, as the orbitals are oneelectron functions,


the procedure of orbital optimization in the presence of the


Jastrow factor can be interpreted as a selfconsistent field procedure


with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.


So \eg{KS}DFT can be viewed as a very cheap way of introducing the effect of


correlation in the \eg{orbital }parameters determining the nodal surface \eg{of a single Slater determinant}.


\sout{But in the general case, even} \eg{Nevertheless, even using the exact exchange correlation potential at } the CBS limit, a fixednode error \sout{will} \eg{necessary} remains because the singledeterminant ans\"atz does not


have enough \sout{freedom} \eg{flexibility} to describe the \sout{exact} nodal surface \eg{of the exact correlated wave function of a generic $N$electron system}.


If one wants to have to exact CBS limit, a multideterminant parameterization of the wave functions is required.


So KSDFT can be viewed as a very cheap way of introducing the effect of


correlation in the orbital parameters determining the nodal surface


of a single Slater determinant.


Nevertheless, even when using the exact exchange correlation potential at the


CBS limit, a fixednode error necessarily remains because the


singledeterminant ansätz does not have enough flexibility to describe the


nodal surface of the exact correlated wave function of a generic $N$electron


system.


If one wants to have to exact CBS limit, a multideterminant parameterization


of the wave functions is required.




\subsection{CIPSI}




Beyond the singledeterminant representation, the best


multideterminant wave function one can obtain is the FCI. FCI is


a \emph{postHartreeFock} method, and \sout{there is a continuous} \eg{there exists several systematic improvements}


\sout{connections} between the HartreeFock and FCI wave functions:


\sout{Multiple paths exist: one can for example use}


increasing the maximum degree of excitation of CI methods (CISD, CISDT,


a \emph{postHartreeFock} method, and there exists several systematic


improvements between the HartreeFock and FCI wave functions:


increasing the maximum degree of excitation of CI methods (CISD, CISDT,


CISDTQ, \emph{etc}), or increasing the complete active space


(CAS) wave functions until all the orbitals are in the active space.


Selected CI methods take a shorter path between the HartreeFock


@ 206,7 +210,8 @@ Within the \emph{Configuration interaction using a perturbative


selection made iteratively} (CIPSI)\cite{Huron_1973} method, the PT2


correction is computed along with the determinant selection. So the


magnitude of $\EPT$ can be made the only parameter of the algorithm,


and we choose this parameter as the convergence criterion.


and we choose this parameter as the convergence criterion of the CIPSI


algorithm.




Considering that the perturbatively corrected energy is a reliable


estimate of the FCI energy, using a fixed value of the PT2 correction


@ 237,7 +242,9 @@ The parameter $\mu$ controls the range of the separation, and allows


to go continuously from the KohnSham Hamiltonian ($\mu=0$) to


the FCI Hamiltinoan ($\mu = \infty$).




\eg{To rigorously connect wave function theory and DFT,} the universal \eg{LevyLieb} density functional\cite{LevPNAS79,LieIJQC83} is decomposed as


To rigorously connect wave function theory and DFT, the universal


LevyLieb density functional\cite{LevPNAS79,LieIJQC83} is


decomposed as


\begin{equation}


\mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n],


\label{Fdecomp}


@ 314,63 +321,44 @@ 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


\eg{convergence of the selfconsistent} calculation, in which the set of determinants is kept fixed, and only


the diagonalization of the RSHamiltonian is performed iteratively.


convergence of the selfconsistent calculation, in which the set of


determinants is kept fixed, and only the diagonalization of the


RSHamiltonian is performed iteratively.


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.


\sout{As always, } \eg{As mentioned above,} the convergence criterion for CIPSI was set to $\EPT <


1$~m$E_h$.


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


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




\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{g16} using an unrestricted HartreeFock


determinant as a reference for openshell systems.




\subsection{Approximations}


In this work, we use the shortrange version of the


PerdewBurkeErnzerhof (PBE)~\cite{PerBurErnPRL96} exchange and


correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSavCP06}


(see also Refs.~\onlinecite{TouColSavJCP05,GolWerStoPCCP05}).


All the CIPSI calculations were made with \emph{Quantum


Package}.\cite{Garniron_2019,qp2_2020} We used the shortrange version


of the PerdewBurkeErnzerhof (PBE)~\cite{PerBurErnPRL96} exchange


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$E_h \vee \Ndet > 10^7$.




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


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


where 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, in the DLA the fixednode


energy is independent of the Jatrow factor, as in allelectron


calculations.






\subsection{RSDFTCIPSI}






\begin{enumerate}


\item Total energies and nodal quality:


\begin{itemize}


\item Facts: KS occupied orbitals closer to NOs than HF


\item Even if exact functional, complete basis set, still approximated nodes for KS


\item KS > exponentially fast convergence (as HF) with basis because of non divergence of effective KS potential (citer le papier de Gill)


\item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS


\item With FCI, good limit at CBS ==> exact energy


\item But slow convergence with basis set because of divergence of the ee interaction not well represented in atom centered basis set


\item Exponential increase of number of Slater determinants


\item Cite papiers RSDFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté)


\item Question: does such a scheme provide better nodal quality ?


\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC


\item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI


\begin{itemize}


\item less determinants $\Rightarrow$ large systems


\item only one parameter to optimize $\Rightarrow$ deterministic


\item $\Rightarrow$ reproducible


\end{itemize}


\item with the optimal $\mu$:


\begin{itemize}


\item Direct optimization of FNDMC with one parameter


\item Do we improve energy differences ?


\item system dependent


\item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$


\item large wave functions


\end{itemize}


\begin{itemize}


\item plot $N_{det}$ en fonction de $\mu$


\end{itemize}


\end{itemize}


\end{enumerate}








% Overlap with reoptimized


% Plot Ndets as a function of mu






\section{Influence of the rangeseparation parameter on the fixednode


@ 416,26 +404,24 @@ correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSavCP06}


values of $\mu$.}


\label{fig:f2dmc}


\end{figure}


\eg{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.} $\mu > 0$ and $\mu < + \infty$).


Therefore, we computed the fixed node energy obtained with $\Psi^{\mu}$


without re optimizing any parameters having an impact on the nodes


(such as Slater determinant coefficients or orbitals),


and this for several values of $\mu$. We considered two weakly correlated molecular systems:


the water molecule and fluorine dimer, both studied near their equilibrium geometry\cite{Caffarel_2016}.


All RSDFTCIPSI wave functions were obtained calculations using BFD pseudopotentials


and the corresponding double and triplezeta basis sets for the water molecule,


and doublezeta quality for the fluorine dimer.}


\sout{The water molecule was taken at the equilibrium


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 considered two weakly correlated molecular systems: the water


molecule and fluorine dimer, near their equilibrium


geometry\cite{Caffarel_2016}.




geometry,\cite{Caffarel_2016} and


generated with BFD pseudopotentials and the corresponding doublezeta


basis set using multiple values of the rangeseparation parameter


$\mu$. The convergence criterion for stopping the CIPSI calculation


was set to 1~m$E_h$ on the PT2 correction.


$\mu$.


Then, these wave functions


were used as trial wave functions for FNDMC calculations,}


\eg{We report the values of the FNDMC energies of the water molecule in table~\ref{tab:h2odmc} and


figure~\ref{fig:h2odmc}.}


We report the values of 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:


@ 473,28 +459,6 @@ Nevertheless, one can notice that the value of such optimal $\mu$ is sensibly la


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


\label{fig:overlap}


\end{figure}


\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{g16} 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}


@ 668,7 +632,6 @@ been the KS single determinant.




We could have obtained


singledeterminant wave functions by using the natural orbitals of a first


% TODO : Get FCI calculations on Irene and redo figure with Ndet > 10^7








@ 691,4 +654,39 @@ Simulation of Functional Materials.




\bibliography{rsdftcipsiqmc}




\begin{enumerate}


\item Total energies and nodal quality:


\begin{itemize}


\item Facts: KS occupied orbitals closer to NOs than HF


\item Even if exact functional, complete basis set, still approximated nodes for KS


\item KS > exponentially fast convergence (as HF) with basis because of non divergence of effective KS potential (citer le papier de Gill)


\item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS


\item With FCI, good limit at CBS ==> exact energy


\item But slow convergence with basis set because of divergence of the ee interaction not well represented in atom centered basis set


\item Exponential increase of number of Slater determinants


\item Cite papiers RSDFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté)


\item Question: does such a scheme provide better nodal quality ?


\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC


\item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI


\begin{itemize}


\item less determinants $\Rightarrow$ large systems


\item only one parameter to optimize $\Rightarrow$ deterministic


\item $\Rightarrow$ reproducible


\end{itemize}


\item with the optimal $\mu$:


\begin{itemize}


\item Direct optimization of FNDMC with one parameter


\item Do we improve energy differences ?


\item system dependent


\item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$


\item large wave functions


\end{itemize}


\begin{itemize}


\item plot $N_{det}$ en fonction de $\mu$


\end{itemize}


\end{itemize}


\end{enumerate}








\end{document}



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