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@ -13,27 +13,40 @@
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]{hyperref}
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\urlstyle{same}
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\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\eg}{\textit{e.g.}}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\newcommand{\eg}[1]{\textcolor{blue}{#1}}
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\toto}[1]{\textcolor{blue}{#1}}
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\newcommand{\trashAS}[1]{\textcolor{blue}{\sout{#1}}}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\EPT}{E_{\text{PT2}}}
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\newcommand{\EDMC}{E_{\text{FN-DMC}}}
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\newcommand{\Ndet}{N_{\text{det}}}
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\newcommand{\hartree}{$E_h$}
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\newcommand{\LCT}{Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
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\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\begin{document}
|
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|
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\title{Enabling high accuracy diffusion Monte Carlo calculations with
|
||||
range-separated density functional theory and selected configuration interaction}
|
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\title{Taming the fixed-node error in diffusion Monte Carlo via range separation}
|
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%\title{Enabling high accuracy diffusion Monte Carlo calculations with
|
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% range-separated density functional theory and selected configuration interaction}
|
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|
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\author{Anthony Scemama}
|
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\email{scemama@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
|
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\author{Emmanuel Giner}
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\email{emmanuel.giner@lct.jussieu.fr}
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@ -55,58 +68,64 @@
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\section{Introduction}
|
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\label{sec:intro}
|
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|
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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.
|
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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
|
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surface is expected to be improved. Using this technique, it has been
|
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shown that the chemical accuracy could be reached within
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@ -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
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||||
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
|
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obtained with FCI wave functions,\cite{Scemama_2018} and other authors
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@ -318,7 +337,7 @@ determinants.
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||||
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We can follow this path by performing FCI calculations using the
|
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RS-DFT Hamiltonian with different values of $\mu$. In this work, we
|
||||
have used the CIPSI algorithm to peform approximate FCI calculations
|
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have used the CIPSI algorithm to perform approximate FCI calculations
|
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with the RS-DFT Hamiltonians,\cite{GinPraFerAssSavTou-JCP-18}
|
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$\hat{H}^\mu$ as shown in figure~\ref{fig:algo}. In the outer loop
|
||||
(red), a CIPSI selection is performed with a RS-Hamiltonian
|
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@ -339,9 +358,9 @@ post-HF method of interest.
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||||
|
||||
|
||||
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
|
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determinant as a reference for open-shell systems.
|
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|
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@ -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
|
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energy is independent of the Jastrow factor, as in all-electron
|
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calculations. Simple Jastrow factors were used to reduce the
|
||||
fluctuations of the local energy.
|
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@ -372,36 +391,37 @@ fluctuations of the local energy.
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error}
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\label{sec:mu-dmc}
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\begin{table}
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\caption{Fixed-node energies and number of determinants in the water
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molecule and the fluorine dimer with different trial wave functions.}
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\caption{Fixed-node energies (in hartree) and number of determinants in \ce{H2O} and \ce{F2} with various trial wave functions.}
|
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\label{tab:h2o-dmc}
|
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\centering
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\begin{ruledtabular}
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\begin{tabular}{crlrl}
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& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
|
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$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
|
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\begin{tabular}{ccrlrl}
|
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& & \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
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\cline{3-4} \cline{5-6}
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System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
|
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\hline
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& \multicolumn{4}{c}{H$_2$O} \\
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$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
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$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
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$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
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$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
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$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
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$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
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$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
|
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$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
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$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
|
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$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
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$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
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$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
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& \multicolumn{3}{c}{F$_2$} \\
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$0.00$ & $23$ & $-48.419\,5(4)$ \\
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$0.25$ & $8$ & $-48.421\,9(4)$ \\
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$0.50$ & $1743$ & $-48.424\,8(8)$ \\
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$1.00$ & $11952$ & $-48.432\,4(3)$ \\
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$2.00$ & $829438$ & $-48.441\,0(7)$ \\
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$5.00$ & $5326459$ & $-48.445(2)$ \\
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$\infty$ & $8302442$ & $-48.437(3)$ \\
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\ce{H2O}
|
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& $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
|
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& $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
|
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& $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
|
||||
& $0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
|
||||
& $0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
|
||||
& $1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
|
||||
& $1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
|
||||
& $2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
|
||||
& $3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
|
||||
& $5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
|
||||
& $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
|
||||
& $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
|
||||
\\
|
||||
\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 & --- & --- & --- & --- & --- & --- \\
|
||||
\\
|
||||
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
|
||||
|
Loading…
Reference in New Issue
Block a user