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@ -13,27 +13,40 @@
]{hyperref}
\urlstyle{same}
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\newcommand{\eg}[1]{\textcolor{blue}{#1}}
\definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem}
\newcommand{\toto}[1]{\textcolor{blue}{#1}}
\newcommand{\trashAS}[1]{\textcolor{blue}{\sout{#1}}}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\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{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\begin{document}
\title{Enabling high accuracy diffusion Monte Carlo calculations with
range-separated density functional theory and selected configuration interaction}
\title{Taming the fixed-node error in diffusion Monte Carlo via range separation}
%\title{Enabling high accuracy diffusion Monte Carlo calculations with
% range-separated density functional theory and selected configuration interaction}
\author{Anthony Scemama}
\email{scemama@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr}
@ -55,58 +68,64 @@
\section{Introduction}
\label{sec:intro}
The full configuration interaction (FCI) method within a finite atomic
basis set leads to an approximate solution of the Schrödinger
equation.
This solution is the eigenpair of an approximate Hamiltonian, which is
the projection of the exact Hamiltonian onto the finite basis of all
possible Slater determinants.
The FCI wave function can be interpreted as the constrained solution of the
true Hamiltonian, where the solution is forced to span the space
provided by the basis.
At the complete basis set (CBS) limit, the constraint vanishes and the
exact solution is obtained.
Hence the FCI method enables a systematic improvement of the
calculations by improving the quality of the basis set.
Nevertheless, its exponential scaling with the number of electrons and
with the size of the basis is prohibitive for large systems.
In recent years, the introduction of new algorithms\cite{Booth_2009}
and the
revival\cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016}
of selected configuration interaction (sCI)
methods\cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
the sizes of the systems that could be computed at the FCI level, but
the scaling remains exponential unless some bias is introduced leading
to a loss of size consistency.
Solving the Schr\"odinger equation for atoms and molecules is a complex task that has kept theoretical and computational chemists busy for almost hundred years now. \cite{Schrodinger_1926}
In order to achieve this formidable endeavour, various strategies have been carefully designed and implemented in quantum chemistry software packages.
The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain
the exact solution of the Schrödinger equation with an additional
constraint, imposing the solution to have the same nodal hypersurface
as a given trial wave function.
One of this strategies consists in relying on wave function theory and, in particular, on the full configuration interaction (FCI) method.
However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite one-electron basis.
This solution is the eigenpair of an approximate Hamiltonian defined as
the projection of the exact Hamiltonian onto the finite many-electron basis of
all possible Slater determinants generated within this finite one-electron basis.
The FCI wave function can be interpreted as a constrained solution of the
true Hamiltonian forced to span the restricted space provided by the one-electron basis.
In the complete basis set (CBS) limit, the constraint is lifted and the
exact solution is recovered.
Hence, the accuracy of a FCI calculation can be systematically improved by increasing the size of the one-electron basis set.
Nevertheless, its exponential scaling with the number of electrons and with the size of the basis is prohibitive for most chemical systems.
In recent years, the introduction of new algorithms \cite{Booth_2009} and the
revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016,Garniron_2018}
of selected configuration interaction (SCI)
methods \cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
the sizes of the systems that could be computed at the FCI level. \cite{Booth_2010,Cleland_2010,Daday_2012,Chien_2018,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
However, the scaling remains exponential unless some bias is introduced leading
to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017}
Diffusion Monte Carlo (DMC) is another numerical scheme to obtain
the exact solution of the Schr\"odinger equation with a different
constraint. In DMC, the solution is imposed to have the same nodes (or zeroes)
as a given trial (approximate) wave function.
Within this so-called \emph{fixed-node} (FN) approximation,
the FN-DMC energy associated with a given trial wave function is an upper
bound to the exact energy, and the latter is recovered only when the
nodes of the trial wave function coincide with the nodes of the exact
wave function.
The polynomial scaling with the number of electrons and with the size
of the trial wave function makes the FN-DMC method attractive.
of the trial wave function makes the FN-DMC method particularly attractive.
In addition, the total energies obtained are usually far below
those obtained with the FCI method in computationally tractable basis
sets because the constraints imposed by the fixed-node approximation
sets because the constraints imposed by the FN approximation
are less severe than the constraints imposed by the finite-basis
approximation.
%However, it is usually harder to control the FN error in DMC, and this
%might affect energy differences such as atomization energies.
%Moreover, improving systematically the nodal surface of the trial wave
%function can be a tricky job as \trashAS{there is no variational
%principle for the nodes}\toto{the derivatives of the FN-DMC energy
%with respect to the variational parameters of the wave function can't
%be computed}.
The qualitative picture of the electronic structure of weakly
correlated systems, such as organic molecules near their equilibrium
geometry, is usually well represented with a single Slater
determinant. This feature is in part responsible for the success of
density functional theory (DFT) and coupled cluster.
density-functional theory (DFT) and coupled cluster theory.
DMC with a single-determinant trial wave function can be used as a
single-reference post-Hatree-Fock method, with an accuracy comparable
to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
The favorable scaling of QMC, its very low memory requirements and
its adequation with massively parallel architectures make it a
serious alternative for high-accuracy simulations on large systems.
its adequacy with massively parallel architectures make it a
serious alternative for high-accuracy simulations of large systems.
As it is not possible to minimize directly the FN-DMC energy with respect
to the variational parameters of the trial wave function, the
@ -115,7 +134,7 @@ finite-basis approximation.
The conventional approach consists in multiplying the trial wave
function by a positive function, the \emph{Jastrow factor}, taking
account of the electron-electron cusp and the short-range correlation
effects. The wave function is then re-optimized within Variational
effects. The wave function is then re-optimized within variational
Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal
surface is expected to be improved. Using this technique, it has been
shown that the chemical accuracy could be reached within
@ -125,13 +144,13 @@ Another approach consists in considering the FN-DMC method as a
\emph{post-FCI method}. The trial wave function is obtained by
approaching the FCI with a selected configuration interaction (sCI)
method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}
When the basis set is increased, the trial wave function gets closer
\titou{When the basis set is increased, the trial wave function gets closer
to the exact wave function, so the nodal surface can be systematically
improved.\cite{Caffarel_2016}
improved.\cite{Caffarel_2016} WRONG}
This technique has the advantage that using FCI nodes in a given basis
set is well defined, so the calculations are reproducible in a
black-box way without needing any expertise in QMC.
But this technique can't be applied to large systems because of the
But this technique cannot be applied to large systems because of the
exponential scaling of the size of the trial wave function.
Extrapolation techniques have been used to estimate the FN-DMC energies
obtained with FCI wave functions,\cite{Scemama_2018} and other authors
@ -318,7 +337,7 @@ 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
have used the CIPSI algorithm to perform approximate FCI calculations
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
@ -339,9 +358,9 @@ post-HF method of interest.
All the calculations were made using BFD
pseudopotentials\cite{Burkatzki_2008} with the associated double,
triple and quadruple zeta basis sets (BFD-V$n$Z).
CCSD(T) and DFT calculations were made with
pseudopotentials\cite{Burkatzki_2008} with the associated double-,
triple-, and quadruple-$\zeta$ basis sets (BFD-VXZ).
CCSD(T) and KS-DFT calculations were made with
\emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock
determinant as a reference for open-shell systems.
@ -361,7 +380,7 @@ in the determinant localization approximation (DLA),\cite{Zen_2019}
where only the determinantal component of the trial wave
function is present in the expression of the wave function on which
the pseudopotential is localized. Hence, in the DLA the fixed-node
energy is independent of the Jatrow factor, as in all-electron
energy is independent of the Jastrow factor, as in all-electron
calculations. Simple Jastrow factors were used to reduce the
fluctuations of the local energy.
@ -372,36 +391,37 @@ fluctuations of the local energy.
error}
\label{sec:mu-dmc}
\begin{table}
\caption{Fixed-node energies and number of determinants in the water
molecule and the fluorine dimer with different trial wave functions.}
\caption{Fixed-node energies (in hartree) and number of determinants in \ce{H2O} and \ce{F2} with various trial wave functions.}
\label{tab:h2o-dmc}
\centering
\begin{ruledtabular}
\begin{tabular}{crlrl}
& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
\begin{tabular}{ccrlrl}
& & \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
\cline{3-4} \cline{5-6}
System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
\hline
& \multicolumn{4}{c}{H$_2$O} \\
$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
& \multicolumn{3}{c}{F$_2$} \\
$0.00$ & $23$ & $-48.419\,5(4)$ \\
$0.25$ & $8$ & $-48.421\,9(4)$ \\
$0.50$ & $1743$ & $-48.424\,8(8)$ \\
$1.00$ & $11952$ & $-48.432\,4(3)$ \\
$2.00$ & $829438$ & $-48.441\,0(7)$ \\
$5.00$ & $5326459$ & $-48.445(2)$ \\
$\infty$ & $8302442$ & $-48.437(3)$ \\
\ce{H2O}
& $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
& $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
& $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
& $0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
& $0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
& $1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
& $1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
& $2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
& $3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
& $5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
& $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
& $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
\\
\ce{F2}
& $0.00$ & $23$ & $-48.419\,5(4)$ \\
& $0.25$ & $8$ & $-48.421\,9(4)$ \\
& $0.50$ & $1743$ & $-48.424\,8(8)$ \\
& $1.00$ & $11952$ & $-48.432\,4(3)$ \\
& $2.00$ & $829438$ & $-48.441\,0(7)$ \\
& $5.00$ & $5326459$ & $-48.445(2)$ \\
& $\infty$ & $8302442$ & $-48.437(3)$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
@ -410,7 +430,8 @@ $\infty$ & $8302442$ & $-48.437(3)$ \\
\centering
\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
\caption{Fixed-node energies of the water molecule for different
values of $\mu$.}
values of $\mu$, using the sr-LDA or sr-PBE short-range density
functionals to build the trial wave function.}
\label{fig:h2o-dmc}
\end{figure}
@ -423,7 +444,7 @@ $\infty$ & $8302442$ & $-48.437(3)$ \\
\end{figure}
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.} $0 < \mu < +\infty$).
range separation parameter (\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 all the
parameters having an impact on the nodal surface fixed.
@ -433,80 +454,76 @@ geometry\cite{Caffarel_2016}.
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:
wave functions ($\mu = \infty$) gives FN-DMC energies which are lower
than the energies obtained with a single Kohn-Sham determinant ($\mu=0$):
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 wave function $\Psi^{\mu}$ with a range-separation
parameter of $\mu=1.75$~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
$\mu=\infty$.
a gain of $18 \pm 3$~m\hartree{} for F$_2$.
Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with intermediate values of $\mu$,
the figures~\ref{fig:h2o-dmc} show that a smooth behaviour is obtained:
starting from $\mu=0$ (\textit{i.e.} the KS determinant),
the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$,
and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function).
For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set,
with the optimal value of $\mu$, one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
and $8 \pm 3$~m\hartree{} for the water and difluorine molecules, respectively.
The optimal value of $\mu$ is $\mu=1.75$~bohr$^{-1}$ and $\mu=5$~bohr$^{-1}$ for the water and fluorine dimer, respectively.
When the basis set is increased, the gain in FN-DMC energy with respect to the FCI trial wave function is reduced, and the optimal value of $\mu$ is shifted towards large $\mu$.
Eventually, the nodes of the wave functions $\Psi^\mu$ obtained with short-range
LDA exchange-correlation functional give very similar FN-DMC energy with respect
to those obtained with the short-range PBE functional, even if the RS-DFT energies obtained
with these two functionals differ by several tens of m\hartree{}.
These observations call some important comments for the present study.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{overlap.pdf}
\caption{Overlap of the RSDFT CI expansion with the
\caption{Overlap of the RS-DFT 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
This data confirms that RS-DFT-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
re-optimizing the trial wave function in the presence of the Jastrow
factor.
To confirm that the introduction of sr-DFT has an impact on
the CI coefficients similar to the Jastrow factor, 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 RS-DFT 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
correlation with DFT has the an impact on the CI coefficients similar to
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$^{-1}$.
These data suggest that the RS-DFT effective Hamiltonian is somehow similar
to the effective Hamiltonian obtained with a usual Jastrow-Slater
optimization and a one- and two-body Jastrow factor.
These data suggest that the eigenfunctions of $H^\mu$ and that of the effective
Hamiltonian obtained with a simple one- and two-body jastrow factor are similar, and therefore that the operators
themselves contain similar physics.
Considering the form of $\hat{H}^\mu[n_{\Psi^{\mu}}]$ (see Eq.~\eqref{H_mu}),
one can notice that the differences with respect to the usual Hamiltonian come
from the non-divergent two-body interaction $\hat{W}_{\mathrm{ee}}^{\mathrm{lr},\mu}$
and the effective one-body potential $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}]$ which is the functional derivative of the Hartree-exchange-correlation functional.
The role of these two terms are therefore very different: the non divergent two body interaction tend to make electrons closer at short-range distance while the effective one-body potential, provided that it is exact, maintains the exact one-body density. Considering that
The role of these two terms are therefore very different: with respect
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 $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}]$,
provided that it is exact, maintains the exact one-body density.
As pointed out by Ten-No in the context of transcorrelated approaches\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, it is
Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the explicitely correlated parts.
As a conclusion of the first part of this study, we can notice that:
@ -518,6 +535,8 @@ on the system and the basis set, and the larger the basis set, the larger the op
iii) numerical experiments (such as computation of overlap) indicates
that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
\textit{i.e.} mimicking short-range correlation effects.
Nevertheless, a slight maximum is obtained for
$\mu=5$~bohr$^{-1}$.
\section{Atomization energies}
@ -531,14 +550,14 @@ atoms than molecules and atomization energies usually tend to be
underestimated with 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
set, so we expect the fixed-node error to be also tightly related to
the basis set incompleteness error.
Increasing the size of the basis set improves the description of
the density and of electron correlation, but also reduces the
imbalance in the quality of the description of the atoms and the
molecule, leading to more accurate atomization energies.
\subsection{Size-consistency}
\subsection{Size consistency}
An extremely important feature required to get accurate
atomization energies is size-consistency (or strict separability),
@ -579,7 +598,7 @@ one-electron, two-electron and one-nucleus-two-electron terms.
The problematic part is the two-electron term, whose simplest form can
be expressed as
\begin{equation}
J_\text{ee} = \sum_i \sum_{j<i} \frac{a r_{ij}}{1 + b r_{ij}}.
J_\text{ee} = \sum_i \sum_{j<i} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.
\end{equation}
The parameter
$a$ is determined by cusp conditions, and $b$ is obtained by energy
@ -610,104 +629,216 @@ are computed analytically and the computational cost of the
pseudo-potential is dramatically reduced (for more detail, see
Ref.~\onlinecite{Scemama_2015}).
In this section, we make a numerical verification that the produced
wave functions are size-consistent. We have computed the energy of the
dissocited fluorine dimer, where the two atoms are at a distance of 50~\AA.
We expect that the energy of this system is equal to twice the energy
of the fluorine atom.
%\begin{squeezetable}
\begin{table}
\caption{FN-DMC Energies of the fluorine atom and the dissociated fluorine
dimer, and size-consistency error.}
\caption{FN-DMC energies (in hartree) using the VDZ-BFD basis set
and pseudo-potential of the fluorine atom and the dissociated fluorine
dimer, and size-consistency error. }
\label{tab:size-cons}
\begin{ruledtabular}
\begin{tabular}{clll}
$\mu$ & F & Dissociated F$_2$ & Size-consistency error \\
\begin{tabular}{cccc}
$\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\
\hline
0.00 & & & \\
0.25 & & & \\
0.50 & & & \\
1.00 & & & \\
2.00 & & & \\
5.00 & & & \\
$\infty$ & & & \\
0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\
0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\
0.50 & $-24.188\,8(1)$ & $-48.376\,9(4)$ & $+0.000\,7(4)$ \\
1.00 & $-24.189\,7(1)$ & $-48.380\,2(4)$ & $-0.000\,8(4)$ \\
2.00 & $-24.194\,1(3)$ & $-48.388\,4(4)$ & $-0.000\,2(5)$ \\
5.00 & $-24.194\,7(4)$ & $-48.388\,5(7)$ & $+0.000\,9(8)$ \\
$\infty$ & $-24.193\,5(2)$ & $-48.386\,9(4)$ & $+0.000\,1(5)$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
In this section, we make a numerical verification that the produced
wave functions are size-consistent for a given range-separation
parameter.
We have computed the energy of the dissociated fluorine dimer, where
the two atoms are at a distance of 50~\AA. We expect that the energy
of this system is equal to twice the energy of the fluorine atom.
The data in table~\ref{tab:size-cons} shows that it is indeed the
case, so we can conclude that the proposed scheme provides
size-consistent FN-DMC energies for all values of $\mu$ (within
$2\times$ statistical error bars).
\subsection{Spin-invariance}
Closed-shell molecules usually dissociate into open-shell
\subsection{Spin invariance}
Closed-shell molecules often dissociate into open-shell
fragments. To get reliable atomization energies, it is important to
have a theory which is of comparable quality for open-shell and
closed-shell systems.
FCI wave functions are invariant with respect to the spin quantum
number $m_s$, but the introduction of a
Jastrow factor breaks this spin-invariance if the parameters
closed-shell systems. A good test is to check that all the components
of a spin multiplet are degenerate.
FCI wave functions have this property and give degenrate energies with
respect to the spin quantum number $m_s$, but the multiplication by a
Jastrow factor introduces spin contamination if the parameters
for the same-spin electron pairs are different from those
for the opposite-spin pairs.\cite{Tenno_2004}
Again, using pseudo-potentials this error is transferred in the DMC
calculation unless the determinant localization approximation is used.
Again, when pseudo-potentials are used this tiny error is transferred
in the FN-DMC energy unless the determinant localization approximation
is used.
Within DFT, the common density functionals make a difference for
same-spin and opposite-spin interactions. As DFT is a
single-determinant theory, the density functionals are designed to be
used with the highest value of $m_s$, and therefore different values
of $m_s$ lead to different energies.
So in the context of RS-DFT, the determinantal expansions will be
impacted by this spurious effect, as opposed to FCI.
\begin{table}
\caption{FN-DMC energies (in hartree) of the triplet carbon atom (BFD-VDZ) with
different values of $m_s$.}
\label{tab:spin}
\begin{ruledtabular}
\begin{tabular}{cccc}
$\mu$ & $m_s=1$ & $m_s=0$ & Spin-invariance error \\
\hline
0.00 & $-5.416\,8(1)$ & $-5.414\,9(1)$ & $+0.001\,9(2)$ \\
0.25 & $-5.417\,2(1)$ & $-5.416\,5(1)$ & $+0.000\,7(1)$ \\
0.50 & $-5.422\,3(1)$ & $-5.421\,4(1)$ & $+0.000\,9(2)$ \\
1.00 & $-5.429\,7(1)$ & $-5.429\,2(1)$ & $+0.000\,5(2)$ \\
2.00 & $-5.432\,1(1)$ & $-5.431\,4(1)$ & $+0.000\,7(2)$ \\
5.00 & $-5.431\,7(1)$ & $-5.431\,4(1)$ & $+0.000\,3(2)$ \\
$\infty$ & $-5.431\,6(1)$ & $-5.431\,3(1)$ & $+0.000\,3(2)$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
In this section, we investigate the impact of the spin contamination
due to the short-range density functional on the FN-DMC energy. We have
computed the energies of the carbon atom in its triplet state
with BFD pseudo-potentials and the corresponding double-zeta basis
set. The calculation was done with $m_s=1$ (3 $\uparrow$ electrons
and 1 $\downarrow$ electrons) and with $m_s=0$ (2 $\uparrow$ and 2
$\downarrow$ electrons).
The results are presented in table~\ref{tab:spin}.
Although using $m_s=0$ the energy is higher than with $m_s=1$, the
bias is relatively small, more than one order of magnitude smaller
than the energy gained by reducing the fixed-node error going from the single
determinant to the FCI trial wave function. The highest bias, close to
2~m\hartree, is obtained for $\mu=0$, but the bias decreases quickly
below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$
there is no bias (within the error bars), and the bias is not
noticeable with $\mu=5$~bohr$^{-1}$.
To check that the RSDFT-CIPSI are spin-invariant, we compute the
FN-DMC energies of the ?? dimer with different values of the spin
quantum number $m_s$.
\subsection{Benchmark}
The 55 molecules of the benchmark for the Gaussian-1
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the quality
of the RSDFT-CIPSI trial wave functions for energy differences.
%\begin{squeezetable}
\begin{squeezetable}
\begin{table*}
\caption{Mean absolute error (MAE), mean signed errors (MSE) and
standard deviations (RMSD) obtained with the different methods and
\caption{Mean absolute errors (MAE), mean signed errors (MSE) and
standard deviations (RMSD) obtained with various methods and
basis sets.}
\label{tab:mad}
\begin{ruledtabular}
\begin{tabular}{ll rrr rrr rrr}
Method & \(\mu\) & \phantom{} & VDZ-BFD & \phantom{} & \phantom{} & VTZ-BFD & \phantom{} & \phantom{} & VQZ-BFD & \phantom{} \\
\phantom{} & \phantom{} & MAE & MSE & RMSD & MAE & MSE & RMSD & MAE & MSE & RMSD \\
\begin{tabular}{ll ddd ddd ddd}
& & \mc{3}{c}{VDZ-BFD} & \mc{3}{c}{VTZ-BFD} & \mc{3}{c}{VQZ-BFD} \\
\cline{3-5} \cline{6-8} \cline{9-11}
Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} \\
\hline
PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\
BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\
PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\
B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\
\hline
CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
\hline
RSDFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
\phantom{} & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
\phantom{} & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
\phantom{} & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & --- & --- & --- \\
\phantom{} & 2 & 19.20 & -19.05 & 10.91 & --- & --- & --- & --- & --- & --- \\
\phantom{} & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\
\phantom{} & \(\infty\) & 23.62 & -23.48 & 12.81 & --- & --- & --- & --- & --- & --- \\
\hline
DMC@RSDFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom{0.}34) & 5.30 & 3.52(19) & -1.03(19) & 4.39 & 3.16(26) & -0.12(26) & 4.12 \\
\phantom{} & 1/4 & 4.04(37) & -3.13(\phantom{0.}37) & 4.88 & 3.39(77) & -0.59(77) & 4.44 & 2.90(25) & 0.25(25) & 3.74 \\
\phantom{} & 1/2 & 3.74(35) & -3.53(\phantom{0.}35) & 4.03 & 2.46(18) & -1.72(18) & 3.02 & 2.06(35) & -0.44(35) & 2.74 \\
\phantom{} & 1 & 5.42(29) & -5.14(\phantom{0.}29) & 4.55 & 4.38(94) & -4.24(94) & 5.11 & --- & --- & --- \\
\phantom{} & 2 & 5.98(83) & -5.91(\phantom{0.}83) & 4.79 & --- & --- & --- & --- & --- & --- \\
\phantom{} & 5 & 6.18(84) & -6.13(\phantom{0.}84) & 4.87 & --- & --- & --- & --- & --- & --- \\
\phantom{} & \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67 & --- & --- & --- & --- & --- & --- \\
\phantom{} & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79 & --- & --- & --- & --- & --- & --- \\
PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\
BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\
PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\
B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\
\\
CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
\\
RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
& 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
& 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
& 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & & & \\
& 2 & 19.20 & -19.05 & 10.91 & & & & & & \\
& 5 & 22.93 & -22.79 & 13.24 & & & & & & \\
& \(\infty\) & 23.63(4) & -23.49(4) & 12.81(4) & 8.43(39) & -8.43(39) & 4.87(7) & 4.51(78) & -4.18(78) & 4.19(20) \\
\\
DMC@ & 0 & 4.61(34) & -3.62(34) & 5.30(09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\
RS-DFT-CIPSI & 1/4 & 4.04(37) & -3.13(37) & 4.88(10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\
& 1/2 & 3.74(35) & -3.53(35) & 4.03(23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\
& 1 & 5.42(29) & -5.14(29) & 4.55(03) & 4.38(94) & -4.24(94) & 5.11(31) & & & \\
& 2 & 5.98(83) & -5.91(83) & 4.79(71) & & & & & & \\
& 5 & 6.18(84) & -6.13(84) & 4.87(55) & & & & & & \\
& \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(68) & & & & & & \\
& Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & & & & & & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
%\end{squeezetable}
\end{squeezetable}
The 55 molecules of the benchmark for the Gaussian-1
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the
performance of the RS-DFT-CIPSI trial wave functions in the context of
energy differences. Calculations were made in the double-, triple-
and quadruple-zeta basis sets with different values of $\mu$, and using
natural orbitals of a preliminary CIPSI calculation.
For comparison, we have computed the energies of all the atoms and
molecules at the DFT level with different density functionals, and at
the CCSD(T) level. Table~\ref{tab:mad} gives the corresponding mean
absolute errors (MAE), mean signed errors (MSE) and standard
deviations (RMSD). For FCI (RS-DFT-CIPSI, $\mu=\infty$) we have
given extrapolated values at $\EPT\rightarrow 0$, and the error bars
correspond to the difference between the energies computed with a
two-point and with a three-point linear extrapolation.
In this benchmark, the great majority of the systems are well
described by a single determinant. Therefore, the atomization energies
calculated at the DFT level are relatively accurate, even when
the basis set is small. The introduction of exact exchange (B3LYP and
PBE0) make the results more sensitive to the basis set, and reduce the
accuracy. Thanks to the single-reference character of these systems,
the CCSD(T) energy is an excellent estimate of the FCI energy, as
shown by the very good agreement of the MAE, MSE and RMSD of CCSDT(T)
and FCI energies.
The imbalance of the quality of description of molecules compared
to atoms is exhibited by a very negative value of the MSE for
CCSD(T) and FCI/VDZ-BFD, which is reduced by a factor of two
when going to the triple-zeta basis, and again by a factor of two when
going to the quadruple-zeta basis.
This large imbalance at the double-zeta level affects the nodal
surfaces, because although the FN-DMC energies obtained with near-FCI
trial wave functions are much lower than the single-determinant FN-DMC
energies, the MAE obtained with FCI (7.38~$\pm$ 1.08~kcal/mol) is
larger than the single-determinant MAE (4.61~$\pm$ 0.34 kcal/mol).
Using the FCI trial wave function the MSE is equal to the
negative MAE which confirms that all the atomization energies are
underestimated. This confirms that some of the basis-set
incompleteness error is transferred in the fixed-node error.
Within the double-zeta basis set, the calculations could be done for the
whole range of values of $\mu$, and the optimal value of $\mu$ for the
trial wave function was estimated for each system by searching for the
minimum of the spline interpolation curve of the FN-DMC energy as a
function of $\mu$.
This corresponds the the line of the table labelled by the \emph{Opt}
value of $\mu$. Using the optimal value of $\mu$ clearly improves the
MAE, the MSE an the RMSD compared the the FCI wave function. This
result is in line with the common knowledge that re-optimizing
the determinantal component of the trial wave function in the presence
of electron correlation reduces the errors due to the basis set incompleteness.
These calculations were done only for the smallest basis set
because of the expensive computational cost of the QMC calculations
when the trial wave function is expanded on more than a few million
determinants.
At the RS-DFT-CIPSI level, we can remark that with the triple-zeta
basis set the MAE are larger for $\mu=1$~bohr$^{-1}$ than for the
FCI. For the largest systems, as shown in figure~\ref{fig:g2-ndet}
there are many systems which did not reach the threshold
$\EPT<1$~m\hartree{}, and the number of determinants exceeded
10~million so the calculation stopped. In this regime, there is a
small size-consistency error originating from the imbalanced
truncation of the wave functions, which is not present in the
extrapolated FCI energies. The same comment applies to
$\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{g2-dmc.pdf}
\caption{Errors in the DMC atomization energies with the different
\caption{Errors in the FN-DMC atomization energies with the different
trial wave functions. Each dot corresponds to an atomization
energy.
The boxes contain the data between first and third quartiles, and
@ -716,6 +847,18 @@ DMC@RSDFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom{0.}34) & 5.30
\label{fig:g2-dmc}
\end{figure}
Searching for the optimal value of $\mu$ may be too costly, so we have
computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
are illustrated in figure~\ref{fig:g2-dmc}. As seen on the figure and
in table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is
0.5~bohr$^{-1}$ for all three basis sets. It is the value for which
the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23),
3.02(06) and 2.74(13)~kcal/mol) are minimal. Note that these values
are even lower than those obtained with the optimal value of
$\mu$. Although the FN-DMC energies are higher, the numbers show that
they are more consistent from one system to another, giving improved
cancellations of errors.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{g2-ndet.pdf}
@ -727,61 +870,75 @@ DMC@RSDFT-CIPSI & 0 & 4.61(34) & -3.62(\phantom{0.}34) & 5.30
\label{fig:g2-ndet}
\end{figure}
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\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
we have used the natural orbitals of a first CIPSI calculation.
The number of determinants in the trial wave functions are shown in
figure~\ref{fig:g2-ndet}. As expected, the number of determinants
is smaller when $\mu$ is small and larger when $\mu$ is large.
It is important to remark that the median of the number of
determinants when $\mu=0.5$~bohr$^{-1}$ is below 100~000 determinants
with the quadruple-zeta basis set, making these calculations feasilble
with such a large basis set. At the double-zeta level, compared to the
FCI trial wave functions the median of the number of determinants is
reduced by more than two orders of magnitude.
Moreover, going to $\mu=0.25$~bohr$^{-1}$ gives a median close to 100
determinants at the double-zeta level, and close to 1~000 determinants
at the quadruple-zeta level for only a slight increase of the
MAE. Hence, RS-DFT-CIPSI trial wave functions with small values of
$\mu$ could be very useful for large systems to go beyond the
single-determinant approximation at a very low computational cost
while keeping the size-consistency.
Note that when $\mu=0$ the number of determinants is not equal to one because
we have used the natural orbitals of a first CIPSI calculation, and
not the sr-PBE orbitals.
So the Kohn-Sham determinant is expressed as a linear combination of
determinants built with natural orbitals. Note that it is possible to add
determinants built with natural orbitals. It is possible to add
an extra step to the algorithm to compute the natural orbitals from the
RS-DFT/CIPSI wave function, and re-do the RS-DFT/CIPSI calculation with
these orbitals to get an even more compact expansion. In that case, we would
have obtained the Kohn-Sham orbitals with $\mu=0$, and the solution would have
been the KS single determinant.
have converged to the Kohn-Sham orbitals with $\mu=0$, and the
solution would have been the PBE single determinant.
We could have obtained
single-determinant wave functions by using the natural orbitals of a first
\section{Conclusion}
We have seen that introducing short-range correation via
a range-separated Hamiltonian in a full CI expansion yields improved
nodes, especially with small basis sets. The effect is similar to the
effect of re-optimizing the CI coefficients in the presence of a
Jastrow factor, but without the burden of performing a stochastic
optimization.
The proposed procedure provides a method to optimize the
FN-DMC energy via a single parameter, namely the range-separation
parameter $\mu$. The size-consistency error is controlled, as well as the
invariance with respect to the spin projection $m_s$.
Finding the optimal value of $\mu$ gives the lowest FN-DMC energies
within basis set. However, if one wants to compute an energy
difference, one should not minimize the
FN-DMC energies of the reactants independently. It is preferable to
choose a value of $\mu$ for which the fixed-node errors are well
balanced, leading to a good cncellation of errors. We found that a
value of $\mu=0.5$~bohr${^-1}$ is the value where the errors are the
smallest. Moreover, such a small value of $\mu$ gives extermely
nodal surfaces, especially with small basis sets. The effect of sr-DFT
on the determinant expansion is similar to the effect of re-optimizing
the CI coefficients in the presence of a Jastrow factor, but without
the burden of performing a stochastic optimization.
Varying the range-separation parameter $\mu$ and approaching the
RS-DFT-FCI with CIPSI provides a way to adapt the number of
determinants in the trial wave function, leading always to
size-consistent FN-DMC energies.
We propose two methods. The first one is for the computation of
accurate total energies by a one-parameter optimization of the FN-DMC
energy via the variation of the parameter $\mu$.
The second method is for the computation of energy differences, where
the target is not the lowest possible FN-DMC energies but the best
possible cancellation of errors. Using a fixed value of $\mu$
increases the consistency of the trial wave functions, and we have found
that $\mu=0.5$~bohr$^{-1}$ is the value where the cancellation of
errors is the most effective.
Moreover, such a small value of $\mu$ gives extermely
compact wave functions, making this recipe a good candidate for
accurate calcultions of large systems with a multi-reference character.
the accurate description of the whole potential energy surfaces of
large systems. If the number of determinants is still too large, the
value of $\mu$ can be further reduced to $0.25$~bohr$^{-1}$ to get
extremely compact wave functions at the price of less efficient
cancellations of errors.
%%---------------------------------------
%%
\begin{acknowledgments}
An award of computer time was provided by the Innovative and Novel
Computational Impact on Theory and Experiment (INCITE) program. This
research has used resources of the Argonne Leadership Computing
Facility, which is a DOE Office of Science User Facility supported
under Contract DE-AC02-06CH11357. AB, was supported by the
U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division, as part of the
Computational Materials Sciences Program and Center for Predictive
Simulation of Functional Materials.
This work was performed using HPC resources from GENCI-TGCC (Grand
Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
2019-0510.
\end{acknowledgments}
@ -790,15 +947,16 @@ Simulation of Functional Materials.
\begin{enumerate}
\item Total energies and nodal quality:
\begin{itemize}
\item Facts: KS occupied orbitals closer to NOs than HF
\item Even if exact functional, complete basis set, still approximated nodes for KS
% \item Facts: KS occupied orbitals closer to NOs than HF
% \item Even if exact functional, complete basis set, still approximated nodes for KS
\item KS -> exponentially fast convergence (as HF) with basis because of non divergence of effective KS potential (citer le papier de Gill)
\item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS
\item With FCI, good limit at CBS ==> exact energy
\item But slow convergence with basis set because of divergence of the e-e interaction not well represented in atom centered basis set
\item Exponential increase of number of Slater determinants
% \item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS
% \item With FCI, good limit at CBS ==> exact energy
% \item But slow convergence with basis set because of divergence of the e-e interaction not well represented in atom centered basis set
% \item Exponential increase of number of Slater determinants
\item Cite papiers RS-DFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté)
\item Question: does such a scheme provide better nodal quality ?
%<<<<<<< HEAD
\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC
\item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI
\begin{itemize}
@ -815,6 +973,23 @@ Simulation of Functional Materials.
\item large wave functions
\end{itemize}
% \item Invariance with m_s
%=======
\item In RS-DFT we cannot optimize energy with $\mu$ , but in FNDMC
% \item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI
% \begin{itemize}
% \item less determinants $\Rightarrow$ large systems
% \item only one parameter to optimize $\Rightarrow$ deterministic
% \item $\Rightarrow$ reproducible
% \end{itemize}
% \item with the optimal $\mu$:
% \begin{itemize}
% \item Direct optimization of FNDMC with one parameter
% \item Do we improve energy differences ?
% \item system dependent
% \item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$
% \item large wave functions
% \end{itemize}
%>>>>>>> 44470b89936d1727b638aefd982ce83be9075cc8
\end{itemize}
\end{enumerate}
@ -822,3 +997,18 @@ Simulation of Functional Materials.
\end{document}
% * Recouvrement avec Be : Optimization tous electrons
% impossible. Abandon. On va prendre H2O.
% * Manu doit faire des programmes pour des plots de ensite a 1 et 2
% corps le long des axes de liaison, et l'integrale de la densite a
% 2 corps a coalescence.
% 1 Manu calcule Be en cc-pvdz tous electrons: FCI -> NOs -> FCI ->
% qp edit -n 200
% 2 Manu calcule qp_cipsi_rsh avec mu = [ 1.e-6 , 0.25, 0.5, 1.0, 2.0, 5.0, 1e6 ]
% 3 Manu fait tourner ses petits programmes
% 4 Manu envoie a toto un tar avec tous les ezfio
% 5 Toto optimise les coefs en presence e jastrow
% 6 Toto renvoie a manu psicoef
% 7 Manu fait tourner ses petits programmes avec psi_J