done with Sec IV
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@ 556,7 +556,7 @@ This is a key result of the present study.


\label{sec:rsdftj}


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


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.


trial wave functions with better nodes than FCI wave functions.


As mentioned in Sec.~\ref{sec:SD}, such behavior 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_2006,Umrigar_2007,Toulouse_2007,Toulouse_2008}


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


@ 564,7 +564,7 @@ and wave function optimization in the presence of a Jastrow factor.


For the sake of simplicity, the molecular orbitals and the Jastrow


factor are kept fixed; only the CI coefficients are varied.




Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_\Nelec)$ (where $\br_i$ is the position of the $i$th electron and $\Nelec$ the total number of electrons),


Let us then assume a fixed Jastrow factor $J(\br_1, \ldots , \br_\Nelec)$ (where $\br_i$ is the position of the $i$th electron and $\Nelec$ the total number of electrons),


and a corresponding SlaterJastrow wave function $\Phi = e^J \Psi$,


where


\begin{equation}


@ 597,8 +597,8 @@ To do so, 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 obtained with the VDZBFD basis. Within this set of determinants,


we solve the selfconsistent equations of RSDFT [see Eq.~\eqref{rsdfteigenequation}]


for different values of $\mu$ using the srPBE functional. This gives the CI expansions $\Psi^\mu$.


Then, within the same set of determinants we optimize the CI coefficients $c_I$ [see Eq.~\eqref{eq:Slater}] in the presence of


for different values of $\mu$ using the srPBE functional. This gives the CI expansions of $\Psi^\mu$.


Then, within the same set of determinants we optimize the CI coefficients in the presence of


a simple one and twobody Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and


\begin{subequations}


\begin{gather}


@ 617,7 +617,7 @@ The parameters $a=1/2$


and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$


were obtained by energy minimization of a single determinant.


The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements


of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the


of the Hamiltonian ($\mathbf{H}$) and overlap ($\mathbf{S}$) matrices in the


basis of Jastrowcorrelated determinants $e^J D_i$:


\begin{subequations}


\begin{gather}


@ 678,17 +678,16 @@ report several quantities related to the one and twobody densities of


$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we


report in the legend of the right panel of Fig~\ref{fig:densities} the integrated ontop pair density


\begin{equation}


\expval{ P } = \int d\br \,\,n_2(\br,\br),


\expval{ P } = \int d\br \,n_2(\br,\br),


\end{equation}


where $n_2(\br_1,\br_2)$ is the twobody density [normalized to $\Nelec(\Nelec1)$]


obtained for both $\Psi^\mu$ and $\Psi^J$.


Then, in order to have a pictorial representation of both the ontop


pair density and the density, we report in Fig.~\ref{fig:densities}


the plots of the total density $n(\br)$ and ontop pair density


$n_2(\br,\br)$ along one of the \ce{OH} axis of the water molecule.


obtained for both $\Psi^\mu$ and $\Psi^J$,


where $n_2(\br_1,\br_2)$ is the twobody density [normalized to $\Nelec(\Nelec1)$].


Then, in order to have a pictorial representation of both the onebody density $n(\br)$ and the ontop


pair density $n_2(\br,\br)$, we report in Fig.~\ref{fig:densities}


the plots of $n(\br)$ and $n_2(\br,\br)$ along one of the \ce{OH} axis of the water molecule.




From these data, one can clearly notice several trends.


First, the overall ontop pair density decreases when $\mu$ increases,


First, the integrated ontop pair density $\expval{ P }$ decreases when $\mu$ increases,


which is expected as the twoelectron interaction increases in


$H^\mu[n]$.


Second, Fig.~\ref{fig:densities} shows that the relative variations of the ontop pair density with respect to $\mu$


@ 709,10 +708,10 @@ and therefore that the operators that produced these wave functions (\ie, $H^\mu


Considering the form of $\hat{H}^\mu[n]$ [see Eq.~\eqref{H_mu}],


one can notice that the differences with respect to the usual bare Hamiltonian come


from the nondivergent twobody interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$


and the effective onebody potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartreeexchangecorrelation functional.


and the effective onebody potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hxc functional.


The roles of these two terms are therefore very different: with respect


to the exact groundstate wave function $\Psi$, the nondivergent twobody interaction


increases the probability to find electrons at short distances in $\Psi^\mu$,


increases the probability of finding electrons at short distances in $\Psi^\mu$,


while the effective onebody potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,


providing that it is exact, maintains the exact onebody density.


This is clearly what has been observed in


@ 723,7 +722,7 @@ can be nondivergent when a proper twobody Jastrow factor $J_\text{ee}$ is chos


There is therefore a clear parallel between $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ in RSDFT and $J_\text{ee}$ in FNDMC.


Moreover, the onebody Jastrow term $J_\text{eN}$ ensures that the onebody density remain unchanged when the CI coefficients are reoptmized in the presence of $J_\text{ee}$.


There is then a second clear parallel between $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ in RSDFT and $J_\text{eN}$ in FNDMC.


Thus, one can understand the similarity between the eigenfunctions of $H^\mu$ and the SlaterJastrow optimization:


Thus, one can understand the similarity between the eigenfunctions of $H^\mu$ and the optimization of the SlaterJastrow wave function:


they both deal with an effective nondivergent interaction but still


produce a reasonable onebody density.





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