1090 lines
57 KiB
TeX
1090 lines
57 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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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), Universit\'e de Toulouse, CNRS, UPS, France}
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\DeclareMathOperator{\erfc}{erfc}
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\begin{document}
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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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\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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\affiliation{\LCT}
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\author{Anouar Benali}
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\email{benali@anl.gov}
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\affiliation{\ANL}
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\author{Pierre-Fran\c{c}ois Loos}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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By combining density-functional theory (DFT) and wave function theory (WFT) via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixed-node diffusion Monte Carlo (FN-DMC) energies with compact multideterminant trial wave functions.
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These compact trial wave functions are generated via the diagonalization of the RS-DFT Hamiltonian.
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In particular, we combine here short-range correlation functionals with selected configuration interaction (SCI).
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As the WFT method is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional SCI calculation.
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\titou{T2: work in progress.}
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\end{abstract}
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\maketitle
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\section{Introduction}
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\label{sec:intro}
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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}
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In order to achieve this formidable endeavour, various strategies have been carefully designed and implemented in quantum chemistry software packages.
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One of this strategies consists in relying on wave function theory and, in particular, on the full configuration interaction (FCI) method.
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However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite one-electron basis.
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This solution is the eigenpair of an approximate Hamiltonian defined as
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the projection of the exact Hamiltonian onto the finite many-electron basis of
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all possible Slater determinants generated within this finite one-electron basis.
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The FCI wave function can be interpreted as a constrained solution of the
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true Hamiltonian forced to span the restricted space provided by the one-electron basis.
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In the complete basis set (CBS) limit, the constraint is lifted and the
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exact solution is recovered.
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Hence, the accuracy of a FCI calculation can be systematically improved by increasing the size of the one-electron basis set.
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Nevertheless, its exponential scaling with the number of electrons and with the size of the basis is prohibitive for most chemical systems.
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In recent years, the introduction of new algorithms \cite{Booth_2009} and the
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revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016,Garniron_2018}
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of selected configuration interaction (SCI)
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methods \cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
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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}
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However, the scaling remains exponential unless some bias is introduced leading
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to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017}
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Diffusion Monte Carlo (DMC) is another numerical scheme to obtain
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the exact solution of the Schr\"odinger equation with a different
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constraint. In DMC, the solution is imposed to have the same nodes (or zeroes)
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as a given trial (approximate) wave function.
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Within this so-called \emph{fixed-node} (FN) approximation,
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the FN-DMC energy associated with a given trial wave function is an upper
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bound to the exact energy, and the latter is recovered only when the
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nodes of the trial wave function coincide with the nodes of the exact
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wave function.
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The polynomial scaling with the number of electrons and with the size
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of the trial wave function makes the FN-DMC method particularly attractive.
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In addition, the total energies obtained are usually far below
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those obtained with the FCI method in computationally tractable basis
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sets because the constraints imposed by the FN approximation
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are less severe than the constraints imposed by the finite-basis
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approximation.
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%However, it is usually harder to control the FN error in DMC, and this
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%might affect energy differences such as atomization energies.
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%Moreover, improving systematically the nodal surface of the trial wave
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%function can be a tricky job as \trashAS{there is no variational
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%principle for the nodes}\toto{the derivatives of the FN-DMC energy
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%with respect to the variational parameters of the wave function can't
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%be computed}.
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The qualitative picture of the electronic structure of weakly
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correlated systems, such as organic molecules near their equilibrium
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geometry, is usually well represented with a single Slater
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determinant. This feature is in part responsible for the success of
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density-functional theory (DFT) and coupled cluster theory.
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DMC with a single-determinant trial wave function can be used as a
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single-reference post-Hatree-Fock method, with an accuracy comparable
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to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
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The favorable scaling of QMC, its very low memory requirements and
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its adequacy with massively parallel architectures make it a
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serious alternative for high-accuracy simulations of large systems.
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As it is not possible to minimize directly the FN-DMC energy with respect
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to the variational parameters of the trial wave function, the
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fixed-node approximation is much more difficult to control than the
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finite-basis approximation.
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The conventional approach consists in multiplying the trial wave
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function by a positive function, the \emph{Jastrow factor}, taking
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account of the electron-electron cusp and the short-range correlation
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effects. The wave function is then re-optimized within variational
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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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FN-DMC.\cite{Petruzielo_2012}
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Another approach consists in considering the FN-DMC method as a
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\emph{post-FCI method}. The trial wave function is obtained by
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approaching the FCI with a selected configuration interaction (SCI)
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method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}
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\titou{When the basis set is increased, the trial wave function gets closer
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to the exact wave function, so the nodal surface can be systematically
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improved.\cite{Caffarel_2016} WRONG}
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This technique has the advantage that using FCI nodes in a given basis
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set is well defined, so the calculations are reproducible in a
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black-box way without needing any expertise in QMC.
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But this technique cannot be applied to large systems because of the
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exponential scaling of the size of the trial wave function.
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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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have used a combination of the two approaches where highly truncated
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CIPSI trial wave functions are re-optimized in VMC under the presence
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of a Jastrow factor to keep the number of determinants
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small,\cite{Giner_2016} and where the consistency between the
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different wave functions is kept by imposing a constant energy
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difference between the estimated FCI energy and the variational energy
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of the CI wave function.\cite{Dash_2018,Dash_2019}
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Nevertheless, finding a robust protocol to obtain high accuracy
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calculations which can be reproduced systematically, and which is
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applicable for large systems with a multi-configurational character is
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still an active field of research. The present paper falls
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within this context.
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\section{Combining CIPSI with range-separated DFT}
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\label{sec:rsdft-cipsi}
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In single-determinant DMC calculations, the degrees of freedom used to
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reduce the fixed-node error are the molecular orbitals on which the
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Slater determinant is built.
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Different molecular orbitals can be chosen:
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Hartree-Fock (HF), Kohn-Sham (KS), natural (NO) orbitals of a
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correlated wave function, or orbitals optimized under the
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presence of a Jastrow factor.
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The nodal surfaces obtained with the KS determinant are in general
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better than those obtained with the HF determinant,\cite{Per_2012} and
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of comparable quality to those obtained with a Slater determinant
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built with NOs.\cite{Wang_2019} Orbitals obtained in the presence
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of a Jastrow factor are generally superior to KS
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orbitals.\cite{Filippi_2000,Scemama_2006,HaghighiMood_2017,Ludovicy_2019}
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The description of electron correlation within DFT is very different
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from correlated methods.
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In DFT, one solves a mean field problem with a modified potential
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incorporating the effects of electron correlation, whereas in
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correlated methods the real Hamiltonian is used and the
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electron-electron interactions are considered.
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Nevertheless, as the orbitals are one-electron functions,
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the procedure of orbital optimization in the presence of the
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Jastrow factor can be interpreted as a self-consistent field procedure
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with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.
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So KS-DFT can be viewed as a very cheap way of introducing the effect of
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correlation in the orbital parameters determining the nodal surface
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of a single Slater determinant.
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Nevertheless, even when using the exact exchange correlation potential at the
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CBS limit, a fixed-node error necessarily remains because the
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single-determinant ansätz does not have enough flexibility to describe the
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nodal surface of the exact correlated wave function of a generic $N$-electron
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system.
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If one wants to have to exact CBS limit, a multi-determinant parameterization
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of the wave functions is required.
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\subsection{CIPSI}
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Beyond the single-determinant representation, the best
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multi-determinant wave function one can obtain is the FCI. FCI is
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a \emph{post-Hartree-Fock} method, and there exists several systematic
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improvements between the Hartree-Fock and FCI wave functions:
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increasing the maximum degree of excitation of CI methods (CISD, CISDT,
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CISDTQ, \emph{etc}), or increasing the complete active space
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(CAS) wave functions until all the orbitals are in the active space.
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Selected CI methods take a shorter path between the Hartree-Fock
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determinant and the FCI wave function by increasing iteratively the
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number of determinants on which the wave function is expanded,
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selecting the determinants which are expected to contribute the most
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to the FCI eigenvector. At every iteration, the lowest eigenpair is
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extracted from the CI matrix expressed in the determinant subspace,
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and the FCI energy can be estimated by computing a second-order
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perturbative correction (PT2) to the variational energy, $\EPT$.
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The magnitude of $\EPT$ is a
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measure of the distance to the exact eigenvalue, and is an adjustable
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parameter controlling the quality of the wave function.
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Within the \emph{Configuration interaction using a perturbative
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selection made iteratively} (CIPSI)\cite{Huron_1973} method, the PT2
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correction is computed along with the determinant selection. So the
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magnitude of $\EPT$ can be made the only parameter of the algorithm,
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and we choose this parameter as the convergence criterion of the CIPSI
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algorithm.
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Considering that the perturbatively corrected energy is a reliable
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estimate of the FCI energy, using a fixed value of the PT2 correction
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as a stopping criterion enforces a constant distance of all the
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calculations to the FCI energy. In this work, we target the chemical
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accuracy so all the CIPSI selections were made such that $|\EPT| <
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1$~m\hartree{}.
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\subsection{Range-separated DFT}
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\label{sec:rsdft}
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Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004}
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the Coulomb operator entering the interelectronic repulsion is split into two pieces:
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\begin{equation}
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\frac{1}{r}
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= w_{\text{ee}}^{\text{sr}, \mu}(r)
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+ w_{\text{ee}}^{\text{lr}, \mu}(r)
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\end{equation}
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where
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\begin{align}
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w_{\text{ee}}^{\text{sr},\mu}(r) & = \frac{\erfc \qty( \mu\, r)}{r},
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&
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w_{\text{ee}}^{\text{lr},\mu}(r) & = \frac{\erf \qty( \mu\, r)}{r}
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\end{align}
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are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version.
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The main idea behind RS-DFT is to treat the short-range part of the
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interaction within KS-DFT, and the long range part within a WFT method like FCI in the present case.
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The parameter $\mu$ controls the range of the separation, and allows
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to go continuously from the KS Hamiltonian ($\mu=0$) to
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the FCI Hamiltonian ($\mu = \infty$).
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To rigorously connect WFT and DFT, the universal
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Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is
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decomposed as
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\begin{equation}
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\mathcal{F}[n] = \mathcal{F}^{\text{lr},\mu}[n] + \bar{E}_{\text{Hxc}}^{\text{sr,}\mu}[n],
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\label{Fdecomp}
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\end{equation}
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where $n$ is a one-electron density,
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$\mathcal{F}^{\text{lr},\mu}$ is a long-range universal density
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functional and $\bar{E}_{\text{Hxc}}^{\text{sr,}\mu}$ is the
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complementary short-range Hartree-exchange-correlation (Hxc) density
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functional. \cite{Savin_1996,Toulouse_2004}
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One obtains the following expression for the ground-state
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electronic energy
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\begin{equation}
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\label{min_rsdft} E_0= \min_{\Psi} \qty{
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\mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi}
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+ \bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_\Psi]
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},
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\end{equation}
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with $\hat{T}$ the kinetic energy operator,
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$\hat{W}_\text{ee}^{\text{lr}}$ the long-range
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electron-electron interaction,
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$n_\Psi$ the one-electron density associated with $\Psi$,
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and $\hat{V}_{\text{ne}}$ the electron-nucleus potential.
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The minimizing multideterminant wave function $\Psi^\mu$
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can be determined by the self-consistent eigenvalue equation
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\begin{equation}
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\label{rs-dft-eigen-equation}
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\hat{H}^\mu[n_{\Psi^{\mu}}] \ket{\Psi^{\mu}}= \mathcal{E}^{\mu} \ket{\Psi^{\mu}},
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\end{equation}
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with the long-range interacting Hamiltonian
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\begin{equation}
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\label{H_mu}
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\hat{H}^\mu[n_{\Psi^{\mu}}] = \hat{T}+\hat{W}_{\text{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}+ \hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}],
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\end{equation}
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where
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$\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}$
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is the complementary short-range Hartree-exchange-correlation
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potential operator.
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Once $\Psi^{\mu}$ has been calculated, the electronic ground-state
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energy is obtained as
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\begin{equation}
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\label{E-rsdft}
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E_0= \mel{\Psi^{\mu}}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi^{\mu}}+\bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_{\Psi^\mu}].
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\end{equation}
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Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie,
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$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus
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RS-DFT reduces to standard KS-DFT and $\Psi^\mu$
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is the KS determinant. For $\mu\to\infty$, the long-range
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interaction becomes the standard Coulomb interaction, \ie,
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$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and
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thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is
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the FCI wave function.
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\begin{figure*}
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\centering
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\includegraphics[width=0.7\linewidth]{algorithm.pdf}
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\caption{Algorithm showing the generation of the RS-DFT wave
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function $\Psi^{\mu}$ starting from $\Psi^{(0)}$.
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The outer (macro-iteration) and inner (micro-iteration) loops are represented in red and blue, respectively.
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The steps common to both loops are represented in purple.
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DIIS extrapolations of the one-electron density are introduced in both the outer and inner loops in order to speed up convergence of the self-consistent process.}
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\label{fig:algo}
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\end{figure*}
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Hence, range separation creates a continuous path connecting smoothly the KS determinant to the
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FCI wave function. Because the KS nodes are of higher quality than the
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HF nodes, we expect that using wave functions built along this path
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will always provide reduced fixed-node errors compared to the path
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connecting HF to FCI which consists in increasing the number of determinants.
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We follow the KS-to-FCI path by performing FCI calculations using the
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RS-DFT Hamiltonian with different values of $\mu$.
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Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a
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single- or multi-determinant wave function $\Psi^{(0)}$ which can
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be obtained in many different ways depending on the system that one considers.
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One of the particularity of the present work is that we
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use the CIPSI algorithm to perform approximate FCI calculations
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with the RS-DFT Hamiltonian $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
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This provides a multi-determinant trial wave function $\Psi^{\mu}$ that one can ``feed'' to DMC.
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In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selection is performed
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to obtain $\Psi^{\mu\,(k)}$ with the RS Hamiltonian $\hat{H}^{\mu\,(k)}$
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parameterized using the current one-electron density $n^{(k)}$.
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At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases.
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One exits the macro-iteration loop when the absolute energy difference between two successive iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to \titou{???} in the present study.
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An inner (micro-iteration) loop (blue) is introduced to accelerate the
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convergence of the self-consistent calculation, in which the set of
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determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of
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$\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$.
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The micro-iteration loop is exited when the absolute energy difference between two successive iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been set to \titou{???} in the present study.
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The convergence of the algorithm was further improved
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by introducing a direct inversion in the iterative subspace (DIIS)
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step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982}
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Note that any range-separated post-HF method can be
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implemented using this scheme by just replacing the CIPSI step by the
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post-HF method of interest.
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Note that, thanks to the self-consistent nature of the algorithm,
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the final trial wave function $\Psi^{\mu}$ is independent of the starting wave function $\Psi^{(0)}$.
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\section{Computational details}
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\label{sec:comp-details}
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\titou{The geometries for the G2 data set are provided as {\SI}.}
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All the calculations have been performed using Burkatzki-Filippi-Dolg (BFD)
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pseudopotentials \cite{Burkatzki_2007,Burkatzki_2008} with the associated double-,
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triple-, and quadruple-$\zeta$ basis sets (BFD-VXZ).
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The small-core BFD pseudopotentials include scalar relativistic effects.
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CCSD(T) and KS-DFT energies have been computed with
|
|
\emph{Gaussian09},\cite{g16} using the unrestricted formalism for open-shell systems.
|
|
|
|
All the CIPSI calculations have been performed with \emph{Quantum
|
|
Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version
|
|
of the Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
|
|
and correlation functionals defined in
|
|
Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
|
|
Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
|
|
The convergence criterion for stopping the CIPSI calculations
|
|
has been set to $\EPT < 1$~m\hartree{} or $ \Ndet > 10^7$.
|
|
All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as
|
|
described in Ref.~\onlinecite{Applencourt_2018}.
|
|
|
|
Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
|
|
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 Jastrow factor, as in all-electron
|
|
calculations. Simple Jastrow factors were used to reduce the
|
|
fluctuations of the local energy.
|
|
The FN-DMC simulations are performed with the stochastic reconfiguration
|
|
algorithm developed by Assaraf \textit{et al.}, \cite{Assaraf_2000}
|
|
with a time step of $5 \times 10^{-4}$ a.u.
|
|
\titou{All-electron move DMC?}
|
|
|
|
|
|
\section{Influence of the range-separation parameter on the fixed-node
|
|
error}
|
|
\label{sec:mu-dmc}
|
|
|
|
%%% TABLE I %%%
|
|
\begin{table}
|
|
\caption{Fixed-node energies $\EDMC$ (in hartree) and number of determinants $\Ndet$ in \ce{H2O} 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} \\
|
|
\cline{2-3} \cline{4-5}
|
|
$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
|
|
\hline
|
|
$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
|
|
$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
|
|
$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
|
|
$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
|
|
$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
|
|
$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
|
|
$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
|
|
$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
|
|
$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
|
|
$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
|
|
$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
|
|
$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
%%% %%% %%% %%%
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
|
|
\caption{Fixed-node energies of \ce{H2O} for different
|
|
values of $\mu$, using the srLDA or srPBE density
|
|
functionals to build the trial wave function.}
|
|
\label{fig:h2o-dmc}
|
|
\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 (\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.
|
|
We considered a weakly correlated molecular systems: the water
|
|
molecule near its equilibrium geometry.\cite{Caffarel_2016}
|
|
|
|
\subsection{Fixed-node energy of $\Psi^\mu$}
|
|
\label{sec:fndmc_mu}
|
|
From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc},
|
|
one can clearly observe that using a FCI trial
|
|
wave functions ($\mu = \infty$) give an FN-DMC energies 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.
|
|
Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with
|
|
intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows 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{}, with an optimal value at
|
|
$\mu=1.75$~bohr$^{-1}$.
|
|
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 slightly shifted towards large $\mu$. Eventually, the nodes
|
|
of the wave functions $\Psi^\mu$ obtained with the short-range
|
|
LDA exchange-correlation functional give very similar FN-DMC energies 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{}.
|
|
|
|
\subsection{Link between RS-DFT and Jastrow factors }
|
|
\label{sec:rsdft-j}
|
|
The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients
|
|
giving trial wave functions with better nodes than FCI wave functions.
|
|
Such behaviour can be compared to the common practice of
|
|
re-optimizing the Slater part of a trial wave function in the presence of the Jastrow factor.
|
|
In the present paragraph, we would like therefore to elaborate some more on the link between RS-DFT
|
|
and wave function optimization within the presence of a Jastrow factor.
|
|
|
|
Let us assume a fixed Jastrow factor $J({\bf{r}_1}, \hdots , {\bf{r}_N})$,
|
|
and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$,
|
|
where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determinants $D_I$.
|
|
The only remaining variational parameters in $\Phi$ are therefore the Slater part $\Psi$.
|
|
Let us call $\Psi^J$ the linear combination of Slater determinant minimizing the variational energy
|
|
\begin{equation}
|
|
\Psi^J = \text{argmin}_{\Psi}\frac{ \mel{ \Psi }{ e^{J} H e^{J} }{ \Psi } }{\mel{ \Psi }{ e^{2J} }{ \Psi } }.
|
|
\end{equation}
|
|
Such a wave function $\Psi^J$ satisfies the generalized hermitian eigenvalue equation
|
|
\begin{equation}
|
|
e^{J} H e^{J} \Psi^J = E e^{2J} \Psi^J,
|
|
\end{equation}
|
|
but also the non-hermitian transcorrelated eigenvalue problem\cite{many_things}
|
|
\begin{equation}
|
|
\label{eq:transcor}
|
|
e^{-J} H e^{J} \Psi^J = E \Psi^J,
|
|
\end{equation}
|
|
which is much easier to handle despite its non-hermicity.
|
|
Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$.
|
|
In a finite basis set and with a quite accurate Jastrow factor, it is known that the nodes
|
|
of $\Psi^J$ may be better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$.
|
|
|
|
To do so, we have made the following numerical experiment.
|
|
First, we extract the 200 determinants with the largest weights in the FCI wave
|
|
function out of a large CIPSI calculation. Within this set of determinants,
|
|
we solve the self-consistent equations of RS-DFT (see Eq.~\eqref{rs-dft-eigen-equation})
|
|
with different values of $\mu$. This gives the CI expansions $\Psi^\mu$.
|
|
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 $\Psi^J$.
|
|
Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
|
|
on the same Slater determinant basis.
|
|
In figure~\ref{fig:overlap}, we plot the overlaps
|
|
$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer,
|
|
and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions
|
|
$\Psi^\mu$ together with that of $\Psi^J$.
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{overlap.pdf}
|
|
\caption{\ce{H2O}, double-zeta basis set, 200 most important
|
|
determinants of the FCI expansion (see \ref{sec:rsdft-j}).
|
|
Overlap of the RS-DFT CI expansions $\Psi^\mu$ with the CI
|
|
expansion optimized in the presence of a Jastrow factor $\Psi^J$.}
|
|
\label{fig:overlap}
|
|
\end{figure}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{h2o-200-dmc.pdf}
|
|
\caption{\ce{H2O}, double-zeta basis set, 200 most important
|
|
determinants of the FCI expansion (see \ref{sec:rsdft-j}).
|
|
FN-DMC energies of $\Psi^\mu$ (red curve), together with
|
|
the FN-DMC energy of $\Psi^J$ (blue line). The width of the lines
|
|
represent the statistical error bars.}
|
|
\label{dmc_small}
|
|
\end{figure}
|
|
|
|
|
|
|
|
There is a clear maximum of overlap at $\mu=1$~bohr$^{-1}$, which
|
|
coincides with the minimum of the FN-DMC energy of $\Psi^\mu$.
|
|
Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is compatible
|
|
with that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{-1}$. This confirms that
|
|
introducing short-range correlation with DFT has
|
|
an impact on the CI coefficients similar to the Jastrow factor.
|
|
|
|
\begin{table}
|
|
\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$
|
|
for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
|
|
\label{table_on_top}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{cc}
|
|
$\mu$ & $\expval{ n_2({\bf r},{\bf r}) }$ \\
|
|
\hline
|
|
0.00 & 1.443 \\
|
|
0.25 & 1.438 \\
|
|
0.50 & 1.423 \\
|
|
1.00 & 1.378 \\
|
|
2.00 & 1.325 \\
|
|
5.00 & 1.288 \\
|
|
$\infty$ & 1.277 \\
|
|
\hline
|
|
$\Psi^J$ & 1.404 \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{on-top-mu.pdf}
|
|
\caption{\ce{H2O}, double-zeta basis set. On-top pair
|
|
density $n_2({\bf r},{\bf r})$ along the O---H axis,
|
|
for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
|
|
\label{fig:n2}
|
|
\end{figure}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{density-mu.pdf}
|
|
\caption{\ce{H2O}, double-zeta basis set. Density $n({\bf r})$ along
|
|
the O---H axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
|
|
\label{fig:n1}
|
|
\end{figure}
|
|
|
|
|
|
In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we
|
|
report several quantities related to the one- and two-body density of
|
|
$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we
|
|
report in table~\ref{table_on_top} the integrated on-top pair density
|
|
$\expval{ n_2({\bf r},{\bf r}) }$
|
|
\begin{equation}
|
|
\expval{ n_2({\bf r},{\bf r}) } = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})
|
|
\end{equation}
|
|
where $n_2({\bf r}_1,{\bf r}_2)$ is the two-body density (normalized to $N(N-1)$ where $N$ is the number of electrons)
|
|
obtained for both $\Psi^\mu$ and $\Psi^J$.
|
|
Then, in order to have a pictorial representation of both the on-top
|
|
pair density and the density, we report in Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}
|
|
the plots of the total density $n({\bf r})$ and on-top pair density
|
|
$n_2({\bf r},{\bf r})$ along one O---H axis of the water molecule.
|
|
|
|
From these data, one can clearly notice several trends.
|
|
First, from Tab.~\ref{table_on_top}, we can observe that the overall
|
|
on-top pair density decreases when $\mu$ increases. This is expected
|
|
as the two-electron interaction increases in $H^\mu[n]$.
|
|
Second, the relative variations of the on-top pair density with $\mu$
|
|
are much more important than that of the one-body density, the latter
|
|
being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the
|
|
former can vary by about 10$\%$ in some regions.
|
|
%TODO TOTO
|
|
In the high-density region of the O---H bond, the value of the on-top
|
|
pair density obtained from $\Psi^J$ is superimposed with
|
|
$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is
|
|
the closest to $\mu=\infty$. The integrated on-top pair density
|
|
obtained with $\Psi^J$ lies between the values obtained with
|
|
$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies
|
|
and the overlap curve.
|
|
|
|
These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
|
|
and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
|
|
Considering the form of $\hat{H}^\mu[n]$ (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 $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
|
|
and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional.
|
|
The roles of these two terms are therefore very different: with respect
|
|
to the exact ground state wave function $\Psi$, the non divergent two body interaction
|
|
increases the probability to find electrons at short distances in $\Psi^\mu$,
|
|
while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
|
|
provided that it is exact, maintains the exact one-body density.
|
|
This is clearly what has been observed from the plots in
|
|
Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}.
|
|
Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No,\cite{Ten-no2000Nov}
|
|
the effective two-body interaction induced by the presence of a Jastrow factor
|
|
can be non-divergent when a proper Jastrow factor is chosen.
|
|
Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Jastrow-Slater optimization:
|
|
they both deal with an effective non-divergent interaction but still
|
|
produce a reasonable one-body density.
|
|
|
|
As a conclusion of the first part of this study, we can notice that:
|
|
\begin{itemize}
|
|
\item with respect to the nodes of a KS determinant or a FCI wave function,
|
|
one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller
|
|
fixed node error by properly choosing an optimal value of $\mu$
|
|
in RS-DFT calculations,
|
|
\item the optimal value of $\mu$ depends on the system and the
|
|
basis set, and the larger the basis set, the larger the optimal value
|
|
of $\mu$,
|
|
\item numerical experiments (overlap $\braket{\Psi^\mu}{\Psi^J}$,
|
|
one-body density, on-top pair density, and FN-DMC energy) indicate
|
|
that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
|
|
\textit{i.e.} mimicking short-range correlation effects. The latter
|
|
statement can be qualitatively understood by noticing that both RS-DFT
|
|
and transcorrelated approaches deal with an effective non-divergent
|
|
electron-electron interaction, while keeping the density constant.
|
|
\end{itemize}
|
|
|
|
|
|
\section{Energy differences in FN-DMC: atomization energies}
|
|
\label{sec:atomization}
|
|
|
|
Atomization energies are challenging for post-Hartree-Fock methods
|
|
because their calculation requires a perfect balance in the
|
|
description of atoms and molecules. Basis sets used in molecular
|
|
calculations are atom-centered, so they are always better adapted to
|
|
atoms than molecules and atomization energies usually tend to be
|
|
underestimated with variational methods.
|
|
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
|
|
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}
|
|
|
|
An extremely important feature required to get accurate
|
|
atomization energies is size-consistency (or strict separability),
|
|
since the numbers of correlated electron pairs in the isolated atoms
|
|
are different from those of the molecules.
|
|
The energy computed within density functional theory is size-consistent, and
|
|
as it is a mean-field method the convergence to the complete basis set
|
|
(CBS) limit is relatively fast. Hence, DFT methods are very well adapted to
|
|
the calculation of atomization energies, especially with small basis
|
|
sets. But going to the CBS limit will converge to biased atomization
|
|
energies because of the use of approximate density functionals.
|
|
|
|
On the other hand, FCI is also size-consistent, but the convergence of
|
|
the FCI energies to the CBS limit is much slower because of the
|
|
description of short-range electron correlation using atom-centered
|
|
functions. But ultimately the exact energy will be reached.
|
|
|
|
In the context of selected CI calculations, when the variational energy is
|
|
extrapolated to the FCI energy\cite{Holmes_2017} there is no
|
|
size-consistency error. But when the truncated SCI wave function is used
|
|
as a reference for post-Hartree-Fock methods such as SCI+PT2
|
|
or for QMC calculations, there is a residual size-consistency error
|
|
originating from the truncation of the wave function.
|
|
|
|
QMC energies can be made size-consistent by extrapolating the
|
|
FN-DMC energy to estimate the energy obtained with the FCI as a trial
|
|
wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the
|
|
size-consistency error can be reduced by choosing the number of
|
|
selected determinants such that the sum of the PT2 corrections on the
|
|
fragments is equal to the PT2 correction of the molecule, enforcing that
|
|
the variational potential energy surface (PES) is
|
|
parallel to the perturbatively corrected PES, which is a relatively
|
|
accurate estimate of the FCI PES.\cite{Giner_2015}
|
|
|
|
Another source of size-consistency error in QMC calculations originates
|
|
from the Jastrow factor. Usually, the Jastrow factor contains
|
|
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}}.
|
|
\end{equation}
|
|
The parameter
|
|
$a$ is determined by cusp conditions, and $b$ is obtained by energy
|
|
or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
|
|
One can easily see that this parameterization of the two-body
|
|
interaction is not size-consistent: the dissociation of a
|
|
diatomic molecule $AB$ with a parameter $b_{AB}$
|
|
will lead to two different two-body Jastrow factors, each
|
|
with its own optimal value $b_A$ and $b_B$. To remove the
|
|
size-consistency error on a PES using this ansätz for $J_\text{ee}$,
|
|
one needs to impose that the parameters of $J_\text{ee}$ are fixed:
|
|
$b_A = b_B = b_{AB}$.
|
|
|
|
When pseudopotentials are used in a QMC calculation, it is common
|
|
practice to localize the non-local part of the pseudopotential on the
|
|
complete wave function (determinantal component and Jastrow).
|
|
If the wave function is not size-consistent,
|
|
so will be the locality approximation. Within, the determinant
|
|
localization approximation,\cite{Zen_2019} the Jastrow factor is
|
|
removed from the wave function on which the pseudopotential is localized.
|
|
The great advantage of this approximation is that the FN-DMC energy
|
|
only depends on the parameters of the determinantal component. Using a
|
|
non-size-consistent Jastrow factor, or a non-optimal Jastrow factor will
|
|
not introduce an additional error in FN-DMC calculations, although it
|
|
will reduce the statistical errors by reducing the variance of the
|
|
local energy. Moreover, the integrals involved in the pseudo-potential
|
|
are computed analytically and the computational cost of the
|
|
pseudo-potential is dramatically reduced (for more detail, see
|
|
Ref.~\onlinecite{Scemama_2015}).
|
|
|
|
%\begin{squeezetable}
|
|
\begin{table}
|
|
\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}{cccc}
|
|
$\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\
|
|
\hline
|
|
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 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. 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, 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$^{-1}$.
|
|
|
|
|
|
|
|
|
|
\subsection{Benchmark}
|
|
|
|
\begin{squeezetable}
|
|
\begin{table*}
|
|
\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 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 \\
|
|
\\
|
|
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}
|
|
|
|
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 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
|
|
the line in the box represents the median. The outliers are shown
|
|
with a cross.}
|
|
\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}
|
|
\caption{Number of determinants in the different trial wave
|
|
functions. Each dot corresponds to an atomization energy.
|
|
The boxes contain the data between first and third quartiles, and
|
|
the line in the box represents the median. The outliers are shown
|
|
with a cross.}
|
|
\label{fig:g2-ndet}
|
|
\end{figure}
|
|
|
|
|
|
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. 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 converged to the Kohn-Sham orbitals with $\mu=0$, and the
|
|
solution would have been the PBE single determinant.
|
|
|
|
|
|
|
|
\section{Conclusion}
|
|
|
|
We have seen that introducing short-range correation via
|
|
a range-separated Hamiltonian in a full CI expansion yields improved
|
|
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 extremely
|
|
compact wave functions, making this recipe a good candidate for
|
|
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}
|
|
This work was performed using HPC resources from GENCI-TGCC (Grand
|
|
Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
|
|
2019-0510.
|
|
\end{acknowledgments}
|
|
|
|
\section*{Data availability}
|
|
|
|
The data that support the findings of this study are available within the article and its {\SI}, and are openly available in [repository name] at \url{http://doi.org/[doi]}, reference number [reference number].
|
|
|
|
\bibliography{rsdft-cipsi-qmc}
|
|
|
|
\end{document}
|
|
|
|
|