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done with Sec IV

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Manuscript/rsdft-cipsi-qmc.tex
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@ -556,7 +556,7 @@ This is a key result of the present study.
\label{sec:rsdft-j} \label{sec:rsdft-j}
%====================================================== %======================================================
The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT 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 As mentioned in Sec.~\ref{sec:SD}, such behavior can be directly compared to the common practice of
re-optimizing the multi-determinant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006,Umrigar_2007,Toulouse_2007,Toulouse_2008} re-optimizing the multi-determinant part of a trial wave function $\Psi$ (the so-called 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 RS-DFT Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT
@ -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 For the sake of simplicity, the molecular orbitals and the Jastrow
factor are kept fixed; only the CI coefficients are varied. 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 Slater-Jastrow wave function $\Phi = e^J \Psi$, and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$,
where where
\begin{equation} \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 First, we extract the 200 determinants with the largest weights in the FCI wave
function out of a large CIPSI calculation obtained with the VDZ-BFD basis. Within this set of determinants, function out of a large CIPSI calculation obtained with the VDZ-BFD basis. Within this set of determinants,
we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}] we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
for different values of $\mu$ using the srPBE functional. This gives the CI expansions $\Psi^\mu$. 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 $c_I$ [see Eq.~\eqref{eq:Slater}] in the presence of Then, within the same set of determinants we optimize the CI coefficients in the presence of
a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and
\begin{subequations} \begin{subequations}
\begin{gather} \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 $\gamma_{\text{H}}=0.35$ 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. were obtained by energy minimization of a single determinant.
The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements 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 Jastrow-correlated determinants $e^J D_i$: basis of Jastrow-correlated determinants $e^J D_i$:
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
@ -678,17 +678,16 @@ report several quantities related to the one- and two-body densities of
$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we $\Psi^J$ 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 on-top pair density report in the legend of the right panel of Fig~\ref{fig:densities} the integrated on-top pair density
\begin{equation} \begin{equation}
\expval{ P } = \int d\br \,\,n_2(\br,\br), \expval{ P } = \int d\br \,n_2(\br,\br),
\end{equation} \end{equation}
where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $\Nelec(\Nelec-1)$] obtained for both $\Psi^\mu$ and $\Psi^J$,
obtained for both $\Psi^\mu$ and $\Psi^J$. where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $\Nelec(\Nelec-1)$].
Then, in order to have a pictorial representation of both the on-top Then, in order to have a pictorial representation of both the one-body density $n(\br)$ and the on-top
pair density and the density, we report in Fig.~\ref{fig:densities} pair density $n_2(\br,\br)$, we report in Fig.~\ref{fig:densities}
the plots of the total density $n(\br)$ and on-top pair density the plots of $n(\br)$ and $n_2(\br,\br)$ along one of the \ce{O-H} axis of the water molecule.
$n_2(\br,\br)$ along one of the \ce{O-H} axis of the water molecule.
From these data, one can clearly notice several trends. From these data, one can clearly notice several trends.
First, the overall on-top pair density decreases when $\mu$ increases, First, the integrated on-top pair density $\expval{ P }$ decreases when $\mu$ increases,
which is expected as the two-electron interaction increases in which is expected as the two-electron interaction increases in
$H^\mu[n]$. $H^\mu[n]$.
Second, Fig.~\ref{fig:densities} shows that the relative variations of the on-top pair density with respect to $\mu$ Second, Fig.~\ref{fig:densities} shows that the relative variations of the on-top 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}], 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 one can notice that the differences with respect to the usual bare Hamiltonian come
from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional. and the effective one-body 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 The roles of these two terms are therefore very different: with respect
to the exact ground-state wave function $\Psi$, the non-divergent two-body interaction to the exact ground-state wave function $\Psi$, the non-divergent two-body 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 one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$, while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
providing that it is exact, maintains the exact one-body density. providing that it is exact, maintains the exact one-body density.
This is clearly what has been observed in This is clearly what has been observed in
@ -723,7 +722,7 @@ can be non-divergent when a proper two-body Jastrow factor $J_\text{ee}$ is chos
There is therefore a clear parallel between $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ in RS-DFT and $J_\text{ee}$ in FN-DMC. There is therefore a clear parallel between $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ in RS-DFT and $J_\text{ee}$ in FN-DMC.
Moreover, the one-body Jastrow term $J_\text{eN}$ ensures that the one-body density remain unchanged when the CI coefficients are re-optmized in the presence of $J_\text{ee}$. Moreover, the one-body Jastrow term $J_\text{eN}$ ensures that the one-body density remain unchanged when the CI coefficients are re-optmized 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 RS-DFT and $J_\text{eN}$ in FN-DMC. There is then a second clear parallel between $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ in RS-DFT and $J_\text{eN}$ in FN-DMC.
Thus, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Slater-Jastrow optimization: Thus, one can understand the similarity between the eigenfunctions of $H^\mu$ and the optimization of the Slater-Jastrow wave function:
they both deal with an effective non-divergent interaction but still they both deal with an effective non-divergent interaction but still
produce a reasonable one-body density. produce a reasonable one-body density.