1st screening of Sec IV done
This commit is contained in:
parent
1488fc76fe
commit
b22dcbede2
@ -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
|
||||
|
Loading…
Reference in New Issue
Block a user