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Manuscript/rsdft-cipsi-qmc.tex
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@ -21,6 +21,7 @@
\newcommand{\EPT}{E_{\text{PT2}}}
\newcommand{\EDMC}{E_{\text{FN-DMC}}}
\newcommand{\Ndet}{N_{\text{det}}}
\newcommand{\hartree}{$E_h$}
\newcommand{\LCT}{Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France}
\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}
@ -318,18 +319,17 @@ determinants.
We can follow this path by performing FCI calculations using the
RS-DFT Hamiltonian with different values of $\mu$. In this work, we
have used the CIPSI algorithm to peform approximate FCI calculations
with the RS-DFT Hamiltonians,\cite{GinPraFerAssSavTou-JCP-18} as shown
in figure~\ref{fig:algo}. In the outer loop (red), a CIPSI selection
is performed with a RS-Hamiltonian parameterized using the current
density. An inner loop (blue) is introduced to accelerate the
with the RS-DFT Hamiltonians,\cite{GinPraFerAssSavTou-JCP-18}
$\hat{H}^\mu$ as shown in figure~\ref{fig:algo}. In the outer loop
(red), a CIPSI selection is performed with a RS-Hamiltonian
parameterized using the current density.
An inner loop (blue) is introduced to accelerate the
convergence of the self-consistent calculation, in which the set of
determinants is kept fixed, and only the diagonalization of the
RS-Hamiltonian is performed iteratively.
RS-Hamiltonian is performed iteratively with the updated density.
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 mentioned above, the convergence criterion for CIPSI was set to
$\EPT < 1$~m$E_h$.
\section{Computational details}
\label{sec:comp-details}
@ -349,7 +349,7 @@ and correlation functionals of
Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
The convergence criterion for stopping the CIPSI calculations
was $\EPT < 1$~m$E_h \vee \Ndet > 10^7$.
was $\EPT < 1$~m\hartree{} $\vee \Ndet > 10^7$.
Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
in the determinant localization approximation (DLA),\cite{Zen_2019}
@ -418,52 +418,79 @@ $\infty$ & $8302442$ & $-48.437(3)$ \\
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.} $0 < \mu < +\infty$).
We generated trial wave functions $\Psi^\mu$ with multiple
values of $\mu$, and computed the associated fixed node energy
keeping fixed all the parameters having an impact on the nodal surface.
We generated trial wave functions $\Psi^\mu$ with multiple values of
$\mu$, and computed the associated fixed node energy keeping all the
parameters having an impact on the nodal surface fixed.
We considered two weakly correlated molecular systems: the water
molecule and fluorine dimer, near their equilibrium
molecule and the fluorine dimer, near their equilibrium
geometry\cite{Caffarel_2016}.
We report the FN-DMC energies of the water molecule in
table~\ref{tab:h2o-dmc} and figure~\ref{fig:h2o-dmc}.
\eg{From table~\ref{tab:h2o-dmc} and figure~\ref{fig:h2o-dmc} one can clearly observe that }using FCI trial wave functions gives FN-DMC energies which are lower
than the energies obtained with a single Kohn-Sham determinant:
\eg{a gain of} 3~m$E_h$ at the double-zeta level and 7~m$E_h$ at the triple-zeta
level \eg{are obtained}. Interestingly, with the double-zeta basis one can obtain a
FN-DMC energy 2.5~m$E_h$ lower than the energy obtained with the FCI
From table~\ref{tab:h2o-dmc} and figures~\ref{fig:h2o-dmc}
and~\ref{fig:f2-dmc}, one can clearly observe that using FCI trial
wave functions gives FN-DMC energies which are lower than the energies
obtained with a single Kohn-Sham determinant:
a gain of $3.2 \pm 0.6$~m\hartree{} at the double-zeta level and $7.2 \pm
0.3$~m\hartree{} at the triple-zeta level are obtained for water, and
a gain of $18 \pm 3$~m\hartree{} for F$_2$. Interestingly, with the
double-zeta basis one can obtain for water a FN-DMC energy $2.6 \pm
0.7$~m\hartree{} lower than the energy obtained with the FCI
trial wave function, using the RSDFT-CIPSI with a range-separation
parameter $\mu=1.75$. This can be explained by the inability of the
basis set to properly describe short-range correlation, shifting
the nodes from their optimal position. Using DFT to take account of
short-range correlation frees the determinant expansion from describing
short-range effects, and enables a better placement of the nodes.
At the triple-zeta level, the short-range correlations can be better
described, and the improvement due to DFT is insignificant. However,
it is important to note that the same FN-DMC energy can be
obtained with a CI expansion which is eight times smaller when sr-DFT
is introduced. One can also remark that the minimum has been
shifted towards the FCI, which is consistent with the fact that
in the CBS limit we expect the minimum of the FN-DMC energy to be
obtained for the FCI wave function, at $\mu=\infty$.
\eg{To further study the behaviour of the FN-DMC energy as a function of $\mu$,
we report in table ???? and figure~\ref{fig:f2-dmc} the FN-DMC 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 FN-DMC 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.
}
parameter $\mu=1.75$~bohr$^{-1}$. This can be explained by the
inability of the basis set to properly describe the short-range
correlation effects, shifting the nodes from their optimal
position. Using DFT to take account of short-range correlation frees
the determinant expansion from describing short-range effects, and
enables a placement of the nodes closer to the optimum. In the case
of F$_2$, a similar behavior with a gain of $8 \pm 4$ m\hartree{} is
observed for $\mu\sim 5$~bohr$^{-1}$.
The optimal value of $\mu$ is larger than in the case of water. This
is probably the signature of the fact that the average
electron-electron distance in the valence is smaller in F$_2$ than in
H$_2$O due to the larger nuclear charge shrinking the electron
density. At the triple-zeta level, the short-range correlations can be
better described by the determinant expansion, and the improvement due
to DFT is insignificant. However, it is important to note that the
same FN-DMC energy can be obtained with a CI expansion which is eight
times smaller when sr-DFT is introduced. One can also remark that the
minimum has been slightly shifted towards the FCI, which is consistent
with the fact that in the CBS limit we expect the minimum of the
FN-DMC energy to be obtained for the FCI wave function, i.e. at
$\mu=\infty$.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{overlap.pdf}
\caption{Overlap of the RSDFT-CIPSI wave functions with the
wave function reoptimized in the presence of a Jastrow factor.}
\caption{Overlap of the RSDFT CI expansion with the
CI expansion optimized in the presence of a Jastrow factor.}
\label{fig:overlap}
\end{figure}
This data confirms that RSDFT/CIPSI can give improved CI coefficients
with small basis sets, similarly to the common practice of
re-optimizing the wave function in the presence of the Jastrow
factor. To confirm that the introduction of RS-DFT has the same impact
that the Jastrow factor on the CI coefficients, 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.
Within this set of determinants, we diagonalize self-consistently the
RSDFT Hamiltonian with different values of $\mu$. This gives the CI
expansions $\Psi^\mu$. Then, within the same set of determinants we
optimize the CI coefficients in the presence of a simple one- and
two-body Jastrow factor. This gives the CI expansion $\Psi^J$.
In figure~\ref{fig:overlap}, we plot the overlaps
$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer.
In the case of H$_2$O, there is a clear maximum of overlap at
$\mu=1$~bohr$^{-1}$. This confirms that introducing short-range
correlation with DFT has the same impact on the CI coefficients than
with the Jastrow factor. In the case of F$_2$, the Jastrow factor has
very little effect on the CI coefficients, as the overlap
$\braket{\Psi^J}{\Psi^{\mu=\infty}$ is very close to
$1$. Nevertheless, a slight maximum is obtained for
$\mu=5$~bohr$^{-1}$.
\section{Atomization energy benchmarks}
\label{sec:atomization}
@ -621,7 +648,7 @@ DMC@RSDFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom{0.}34) & 5.30
The number of determinants in the wave functions are shown in
figure~\ref{fig:n2-ndet}. For all the calculations, the stopping
criterion of the CIPSI algorithm was $\EPT < 1$~m$E_h$ or $\Ndet >
criterion of the CIPSI algorithm was $\EPT < 1$~m\hartree{} or $\Ndet >
10^7$.
For FCI, we have given extrapolated values at $\EPT\rightarrow 0$.
At $\mu=0$ the number of determinants is not equal to one because