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1st screening of Sec IV done

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Manuscript/rsdft-cipsi-qmc.tex
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@ -540,6 +540,7 @@ functional give very similar FN-DMC energies with respect
to those obtained with srPBE, even if the
RS-DFT energies obtained with these two functionals differ by several
tens of m\hartree{}.
Accordingly, all the RS-DFT calculations are performed with the srPBE functional in the remaining of this paper.
Another important aspect here is 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/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.
@ -558,7 +559,7 @@ 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 $J(\br_1, \ldots , \br_N)$ (where $\br_i$ is the position of the $i$th electron),
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 a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$,
where
\begin{equation}
@ -591,7 +592,7 @@ 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 VDZ-BFD basis. Within this set of determinants,
we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
with different values of $\mu$. This gives the CI expansions $\Psi^\mu$.
for different values of $\mu$ \titou{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
a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and
\begin{subequations}
@ -604,8 +605,8 @@ a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{
\end{subequations}
The one-body Jastrow factor $J_\text{eN}$ contains the electron-nucleus terms (where $\Nat$ is the number of nuclei) with a single parameter
$\alpha_A$ per nucleus.
The two-body Jastrow factor $J_\text{ee}$ contains the electron-electron terms
where the sum over $i < j$ loops over all electron pairs.
The two-body Jastrow factor $J_\text{ee}$ gathers the electron-electron terms
where the sum over $i < j$ loops over all unique electron pairs.
In Eqs.~\eqref{eq:jast-eN} and \eqref{eq:jast-ee}, $r_{iA}$ is the distance between the $i$th electron and the $A$th nucleus while $r_{ij}$ is the interlectronic distance between electrons $i$ and $j$.
The parameters $a=1/2$
and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$
@ -625,7 +626,7 @@ We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
on the same set of Slater determinants.
In Fig.~\ref{fig:overlap}, we plot the overlaps
$\braket*{\Psi^J}{\Psi^\mu}$ obtained for water,
and in Fig.~\ref{dmc_small} the FN-DMC energy of the wave functions
and in Fig.~\ref{fig:dmc_small} the FN-DMC energy of the wave functions
$\Psi^\mu$ together with that of $\Psi^J$.
%%% FIG 3 %%%
@ -641,21 +642,21 @@ $\Psi^\mu$ together with that of $\Psi^J$.
%%% FIG 4 %%%
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
\caption{\ce{H2O}, double-zeta basis set, 200 most important
determinants of the FCI expansion (see Sec.~\ref{sec:rsdft-j}).
FN-DMC energies of $\Psi^\mu$ (red curve), together with
the FN-DMC energy of $\Psi^J$ (blue line). The width of the lines
represent the statistical error bars.}
\label{dmc_small}
\caption{
FN-DMC energies of $\Psi^\mu$ (red curve) as a function of $\mu$, together with
the FN-DMC energy of $\Psi^J$ (blue line) for \ce{H2O}. The width of the lines
represent the statistical error bars.
For these two trial wave functions, the CI expansion consists of the 200 most important
determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
\label{fig:dmc_small}
\end{figure}
%%% %%% %%% %%%
There is a clear maximum of overlap at $\mu=1$~bohr$^{-1}$, which
coincides with the minimum of the FN-DMC energy of $\Psi^\mu$.
Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is compatible
with that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that
with that of $\Psi^\mu$ for $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that
introducing short-range correlation with DFT has
an impact on the CI coefficients similar to the Jastrow factor.
@ -664,7 +665,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2(\br,\br) }$
for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
\titou{Please remove table and merge data in the Fig. 5.}}
\label{table_on_top}
\label{tab:table_on_top}
\begin{ruledtabular}
\begin{tabular}{cc}
$\mu$ & $\expval{ n_2(\br,\br) }$ \\
@ -685,10 +686,11 @@ an impact on the CI coefficients similar to the Jastrow factor.
%%% FIG 5 %%%
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{density-mu.pdf}
\caption{\ce{H2O}, \titou{srLDA?} double-zeta basis set. One-electron density $n(\br)$ along
the \ce{O-H} axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
\caption{One-electron density $n(\br)$ along
the \ce{O-H} axis of \ce{H2O} as a function of $\mu$ for $\Psi^J$ (dashed curve) and $\Psi^\mu$.
For these two trial wave functions, the CI expansion consists of the 200 most important
determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
\label{fig:n1}
\end{figure}
%%% %%% %%% %%%
@ -696,31 +698,32 @@ an impact on the CI coefficients similar to the Jastrow factor.
%%% FIG 6 %%%
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{on-top-mu.pdf}
\caption{\ce{H2O}, \titou{srLDA?} double-zeta basis set. On-top pair
density $n_2(\br,\br)$ along the \ce{O-H} axis,
for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
\caption{On-top pair
density $n_2(\br,\br)$ along the \ce{O-H} axis of \ce{H2O} as a function of $\mu$
for $\Psi^J$ (dashed curve) and $\Psi^\mu$.
For these two trial wave functions, the CI expansion consists of the 200 most important
determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
\label{fig:n2}
\end{figure}
%%% %%% %%% %%%
In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we
report several quantities related to the one- and two-body density of
report several quantities related to the one- and two-body densities of
$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we
report in Table~\ref{table_on_top} the integrated on-top pair density
\begin{equation}
\expval{ n_2(\br,\br) } = \int d\br \,\,n_2(\br,\br)
\end{equation}
where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $N(N-1)$ where $N$ is the number of electrons]
where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $\Nelec(\Nelec-1)$]
obtained for both $\Psi^\mu$ and $\Psi^J$.
Then, in order to have a pictorial representation of both the on-top
pair density and the density, we report in Figs.~\ref{fig:n1} and \ref{fig:n2}
the plots of the total density $n(\br)$ and on-top pair density
$n_2(\br,\br)$ along one \ce{O-H} axis of the water molecule.
$n_2(\br,\br)$ along one of these \ce{O-H} axis of the water molecule.
From these data, one can clearly notice several trends.
First, from Table~\ref{table_on_top}, we can observe that the overall
First, from Table~\ref{tab:table_on_top}, we can observe that the overall
on-top pair density decreases when $\mu$ increases, which is expected
as the two-electron interaction increases in $H^\mu[n]$.
Second, the relative variations of the on-top pair density with $\mu$
@ -734,7 +737,7 @@ $\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is
the closest to $\mu=\infty$. The integrated on-top pair density
obtained with $\Psi^J$ lies between the values obtained with
$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies
and the overlap curve.
and the overlap curve depicted in Figs.~\ref{fig:overlap} and \ref{fig:dmc_small}
These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
@ -743,24 +746,24 @@ one can notice that the differences with respect to the usual Hamiltonian come
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.
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 $\Psi^\mu$,
while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
provided that it is exact, maintains the exact one-body density.
This is clearly what has been observed from the plots in
This is clearly what has been observed from
Figs.~\ref{fig:n1} and \ref{fig:n2}.
Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No,\cite{Ten-no2000Nov}
Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-no,\cite{Ten-no2000Nov}
the effective two-body interaction induced by the presence of a Jastrow factor
can be non-divergent when a proper Jastrow factor is chosen.
Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Jastrow-Slater optimization:
Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Slater-Jastrow optimization:
they both deal with an effective non-divergent interaction but still
produce a reasonable one-body density.
As a conclusion of the first part of this study, we can notice that:
\begin{itemize}
\item with respect to the nodes of a KS determinant or a FCI wave function,
one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller
fixed node error by properly choosing an optimal value of $\mu$
one can obtain a multi-determinant trial wave function $\Psi^\mu$ with a smaller
fixed-node error by properly choosing an optimal value of $\mu$
in RS-DFT calculations,
\item the optimal value of $\mu$ depends on the system and the
basis set, and the larger the basis set, the larger the optimal value
@ -770,7 +773,7 @@ As a conclusion of the first part of this study, we can notice that:
that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
\ie, mimicking short-range correlation effects. The latter
statement can be qualitatively understood by noticing that both RS-DFT
and transcorrelated approaches deal with an effective non-divergent
and the trans-correlated approach deal with an effective non-divergent
electron-electron interaction, while keeping the density constant.
\end{itemize}
@ -780,12 +783,12 @@ As a conclusion of the first part of this study, we can notice that:
\label{sec:atomization}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Atomization energies are challenging for post-Hartree-Fock methods
Atomization energies are challenging for post-HF methods
because their calculation requires a perfect balance in the
description of atoms and molecules. Basis sets used in molecular
calculations are atom-centered, so they are always better adapted to
atoms than molecules and atomization energies usually tend to be
underestimated with variational methods.
underestimated by variational methods.
In the context of FN-DMC calculations, the nodal surface is imposed by
the trial wavefunction which is expanded on an atom-centered basis
set, so we expect the fixed-node error to be also tightly related to