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@ 175,23 +175,21 @@ 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 DFT can be viewed as a very cheap way of introducing the effect of


correlation in the parameters determining the nodal surface. But in


the general case, even at the complete basis set limit a fixednode


error will remain because the singledeterminant ans\"atz does not


have enough freedom to describe the exact nodal surface.


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


parameterization of the wave functions is required.


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.




\subsection{CIPSI}




Beyond the singledeterminant representation, the best


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


a \emph{postHartreeFock} method, and there is a continuous


connection between the HartreeFock and FCI wave functions.


Multiple paths exist: one can for example use


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


CISDTQ, \emph{etc}), or use increasingly large complete active space


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,


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


determinant and the FCI wave function by increasing iteratively the


@ 239,7 +237,7 @@ The parameter $\mu$ controls the range of the separation, and allows


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


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




The universal density functional is decomposed as


\eg{To rigorously connect wave function theory and DFT,} the universal \eg{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}


@ 248,7 +246,7 @@ where $n$ is a oneparticle density,


$\mathcal{F}^{\mathrm{lr},\mu}$ is a longrange universal density


functional and $\bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}$ is the


complementary shortrange Hartreeexchangecorrelation (Hxc) density


functional.


functional\cite{Savin_1996,Toulouse_2004}.


One obtains the following expression for the groundstate


electronic energy


\begin{equation}


@ 316,12 +314,12 @@ 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


calculation, in which the set of determinants is kept fixed, and only


\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.


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 always, the convergence criterion for CIPSI was set to $\EPT <


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


1$~m$E_h$.






@ 418,20 +416,31 @@ correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSavCP06}


values of $\mu$.}


\label{fig:f2dmc}


\end{figure}


The water molecule was taken at the equilibrium


geometry,\cite{Caffarel_2016} and RSDFTCIPSI wave functions were


\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


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. Then, these wave functions


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}.


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


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}.}




Using FCI trial wave functions gives FNDMC energies which are lower


\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:


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


\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


trial wave function, using the RSDFTCIPSI with a rangeseparation


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


@ 448,6 +457,14 @@ 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.


}






\begin{figure}


\centering



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