662 lines
32 KiB
TeX
662 lines
32 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem,xspace,subfigure}
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\usepackage[
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breaklinks=true
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]{hyperref}
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\usepackage[normalem]{ulem}
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\newcommand{\EPT}{E_{\text{PT2}}}
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\newcommand{\Ndet}{N_{\text{det}}}
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\newcommand{\LCT}{Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, 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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\begin{document}
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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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\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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\end{abstract}
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\maketitle
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\section{Introduction}
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\label{sec:intro}
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The full configuration interaction (FCI) method within a finite atomic
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basis set leads to an approximate solution of the Schrödinger
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equation.
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This solution is the eigenpair of an approximate Hamiltonian, which is
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the projection of the exact Hamiltonian onto the finite basis of all
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possible Slater determinants.
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The FCI wave function can be interpreted as the constrained solution of the
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true Hamiltonian, where the solution is forced to span the space
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provided by the basis.
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At the complete basis set (CBS) limit, the constraint vanishes and the
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exact solution is obtained.
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Hence the FCI method enables a systematic improvement of the
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calculations by improving the quality of the basis set.
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Nevertheless, its exponential scaling with the number of electrons and
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with the size of the basis is prohibitive for large systems.
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In recent years, the introduction of new algorithms\cite{Booth_2009}
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and the
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revival\cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016}
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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, but
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the scaling remains exponential unless some bias is introduced leading
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to a loss of size extensivity.
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The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain
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the exact solution of the Schrödinger equation with an additional
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constraint, imposing the solution to have the same nodal hypersurface
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as a given trial wave function.
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Within this so-called \emph{fixed-node} approximation,
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the 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 DMC method 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 fixed-node 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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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.
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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 adequation with massively parallel architectures make it a
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serious alternative for high-accuracy simulations on large systems.
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As it is not possible to minimize directly the 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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DMC.\cite{Petruzielo_2012}
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Another approach consists in considering the 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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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}
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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 can't 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 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 DFT can be viewed as a very cheap way of introducing the effect of
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correlation in the parameters determining the nodal surface. But in
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the general case, even at the complete basis set limit a fixed-node
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error will remain because the single-determinant ans\"atz does not
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have enough freedom to describe the exact nodal surface.
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If one wants to have to exact CBS limit, a multi-determinant
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parameterization 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 is a continuous
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connection between the Hartree-Fock and FCI wave functions.
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Multiple paths exist: one can for example use
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CI methods increasing the maximum degree of excitation (CISD, CISDT,
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CISDTQ, \emph{etc}), or use increasingly large 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.
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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 $|\EPT| <
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1$~mE$_h$.
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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 electron-electron interaction is split into a short-range
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(sr) and a long range (lr) interaction as
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\begin{equation}
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\frac{1}{r_{ij}} = w_{\text{ee}}^{\text{lr}, \mu}(r_{ij}) + \qty(
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\frac{1}{r_{ij}} - w_{\text{ee}}^{\text{lr}, \mu}(r_{ij}) )
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\end{equation}
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where
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\begin{equation}
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w_{\text{ee}}^{\text{lr},\mu}(r_{ij}) = \frac{\erf \qty( \mu\, r_{ij})}{r_{ij}}
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\end{equation}
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The main idea is to treat the short-range electron-electron
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interaction with DFT, and the long range with wave function theory.
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The parameter $\mu$ controls the range of the separation, and allows
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to go continuously from the Kohn-Sham Hamiltonian ($\mu=0$) to
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the FCI Hamiltinoan ($\mu = \infty$).
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The universal density functional is decomposed as
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\begin{equation}
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\mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n],
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\label{Fdecomp}
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\end{equation}
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where $n$ is a one-particle density,
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$\mathcal{F}^{\mathrm{lr},\mu}$ is a long-range universal density
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functional and $\bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}$ is the
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complementary short-range Hartree-exchange-correlation (Hxc) density
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functional.
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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} \left\{
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\left
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\langle\Psi|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi\right
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\rangle
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+ \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_\Psi]\right\}
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\end{equation}
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with $\hat{T}$ the kinetic energy operator,
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$\hat{W}_\mathrm{ee}^{\mathrm{lr}}$ the long-range
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electron-electron interaction,
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$n_\Psi$ the one-particle density associated with $\Psi$,
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and $\hat{V}_{\mathrm{ne}}$ the electron-nucleus potential.
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The minimizing multi-determinant 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}_{\mathrm{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}+ \hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}],
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\end{equation}
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where
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$\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}$
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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 by
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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}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}}{\Psi^{\mu}}+\bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_{\Psi^\mu}].
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\end{equation}
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Note that, for $\mu=0$, the long-range interaction vanishes,
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$w_{\mathrm{ee}}^{\mathrm{lr},\mu=0}(r_{12}) = 0$, and thus
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range-separated DFT (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,
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$w_{\mathrm{ee}}^{\mathrm{lr},\mu\to\infty}(r_{12}) = 1/r_{12}$, and
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thus RS-DFT reduces to standard wave-function theory 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.}
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\label{fig:algo}
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\end{figure*}
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Hence we have a continuous path connecting the KS determinant to the
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FCI wave function, and as 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 using an increasing number of selected
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determinants.
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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
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have used the CIPSI algorithm to peform approximate FCI calculations
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with the RS-DFT Hamiltonians,\cite{GinPraFerAssSavTou-JCP-18} as shown
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in figure~\ref{fig:algo}. In the outer loop (red), a CIPSI selection
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is performed with a RS-Hamiltonian parameterized using the current
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density. An inner loop (blue) is introduced to accelerate the
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calculation, in which the set of determinants is kept fixed, and only
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the diagonalization of the RS-Hamiltonian is performed iteratively.
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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.
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As always, the convergence criterion for CIPSI was set to $\EPT <
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1$~m$E_h$.
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\subsection{Approximations}
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In this work, we use the short-range version of the
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Perdew-Burke-Ernzerhof (PBE)~\cite{PerBurErn-PRL-96} exchange and
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correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06}
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(see also Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
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\subsection{RSDFT-CIPSI}
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\begin{enumerate}
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\item Total energies and nodal quality:
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\begin{itemize}
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\item Facts: KS occupied orbitals closer to NOs than HF
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\item Even if exact functional, complete basis set, still approximated nodes for KS
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\item KS -> exponentially fast convergence (as HF) with basis because of non divergence of effective KS potential (citer le papier de Gill)
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\item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS
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\item With FCI, good limit at CBS ==> exact energy
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\item But slow convergence with basis set because of divergence of the e-e interaction not well represented in atom centered basis set
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\item Exponential increase of number of Slater determinants
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\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)
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\item Question: does such a scheme provide better nodal quality ?
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\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC
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\item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI
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\begin{itemize}
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\item less determinants $\Rightarrow$ large systems
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\item only one parameter to optimize $\Rightarrow$ deterministic
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\item $\Rightarrow$ reproducible
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\end{itemize}
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\item with the optimal $\mu$:
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\begin{itemize}
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\item Direct optimization of FNDMC with one parameter
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\item Do we improve energy differences ?
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\item system dependent
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\item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$
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\item large wave functions
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\end{itemize}
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\begin{itemize}
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\item plot $N_{det}$ en fonction de $\mu$
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\end{itemize}
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\end{itemize}
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\end{enumerate}
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% Overlap with reoptimized
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% Plot Ndets as a function of mu
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\section{Influence of the range-separation parameter on the fixed-node
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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 of the water molecule.}
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\label{tab:h2o-dmc}
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\centering
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\begin{tabular}{crlrl}
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\hline
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& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ}\\
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$\mu$ & $N_{\text{det}}$ & E(DMC) & $N_{\text{det}}$ & E(DMC)\\
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\hline
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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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\hline
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\end{tabular}
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\end{table}
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
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\caption{Fixed-node energies of the water molecule for different
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values of $\mu$.}
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\label{fig:h2o-dmc}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{f2-dmc.pdf}
|
|
\caption{Fixed-node energies of difluorine for different
|
|
values of $\mu$.}
|
|
\label{fig:f2-dmc}
|
|
\end{figure}
|
|
The water molecule was taken at the equilibrium
|
|
geometry,\cite{Caffarel_2016} and RSDFT-CIPSI wave functions were
|
|
generated with BFD pseudopotentials and the corresponding double-zeta
|
|
basis set using multiple values of the range-separation parameter
|
|
$\mu$. The convergence criterion for stopping the CIPSI calculation
|
|
was set to 1~m$E_h$ on the PT2 correction. Then, these wave functions
|
|
were used as trial wave functions for FN-DMC calculations, and the
|
|
corresponding energies are shown in table~\ref{tab:h2o-dmc} and
|
|
figure~\ref{fig:h2o-dmc}.
|
|
|
|
Using FCI trial wave functions gives FN-DMC energies which are lower
|
|
than the energies obtained with a single Kohn-Sham determinant:
|
|
3~m$E_h$ at the double-zeta level and 7~m$E_h$ at the triple-zeta
|
|
level. Interestingly, with the double-zeta basis one can obtain a
|
|
FN-DMC energy 2.5~m$E_h$ lower than the energy obtained with the FCI
|
|
trial wave function, using the RSDFT-CIPSI with a range-separation
|
|
parameter $\mu=1.75$. This can be explained by the inability of the
|
|
basis set to properly describe short-range correlation, shifting
|
|
the nodes from their optimal position. Using DFT to take account of
|
|
short-range correlation frees the determinant expansion from describing
|
|
short-range effects, and enables a better placement of the nodes.
|
|
At the triple-zeta level, the short-range correlations can be better
|
|
described, and the improvement due to DFT is insignificant. However,
|
|
it is important to note that the same FN-DMC energy can be
|
|
obtained with a CI expansion which is eight times smaller when sr-DFT
|
|
is introduced. One can also remark that the minimum has been
|
|
shifted towards the FCI, which is consistent with the fact that
|
|
in the CBS limit we expect the minimum of the FN-DMC energy to be
|
|
obtained for the FCI wave function, at $\mu=\infty$.
|
|
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{overlap.pdf}
|
|
\caption{Overlap of the RSDFT-CIPSI wave functions with the
|
|
wave function reoptimized in the presence of a Jastrow factor.}
|
|
\label{fig:overlap}
|
|
\end{figure}
|
|
\section{Computational details}
|
|
\label{sec:comp-details}
|
|
|
|
All the calculations were made using BFD
|
|
pseudopotentials\cite{Burkatzki_2008} with the associated double, triple
|
|
and quadruple zeta basis sets.
|
|
CCSD(T) and DFT calculations were made with
|
|
\emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock
|
|
determinant as a reference for open-shell systems. All the CIPSI
|
|
calculations and range-separated CIPSI calculations were made with
|
|
\emph{Quantum Package}.\cite{Garniron_2019,qp2_2020}
|
|
Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
|
|
in the determinant localization approximation.\cite{Zen_2019}
|
|
|
|
In the determinant localization approximation, 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,
|
|
the pseudopotential operator does not depend on the Jastrow factor, as
|
|
it is the case in all-electron calculations. This improves the
|
|
reproducibility of the results, as they depend only on parameters
|
|
optimized in a deterministic framework.
|
|
|
|
|
|
\section{Atomization energy benchmarks}
|
|
\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.
|
|
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 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.
|
|
|
|
Another important feature required to get accurate atomization
|
|
energies is size-extensivity, since the numbers of correlated electrons
|
|
in the isolated atoms are different from the number of correlated
|
|
electrons in the molecule.
|
|
In the context of selected CI calculations, when the variational energy is
|
|
extrapolated to the FCI energy\cite{Holmes_2017} there is obviously no
|
|
size-consistence error. But when the selected wave function is used
|
|
for as a reference for post-Hartree-Fock methods or QMC calculations,
|
|
there is a residual size-consistence error originating from the
|
|
truncation of the determinant space.
|
|
|
|
% Invariance with m_s
|
|
|
|
QMC calculations 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-consistence error can be reduced by choosing the number of
|
|
selected determinants such that the sum of the PT2 corrections on the
|
|
atoms is equal to the PT2 correction of the molecule, enforcing that
|
|
the variational dissociation potential energy surface (PES) is
|
|
parallel to the perturbatively corrected PES, which is an accurate
|
|
estimate of the FCI PES.\cite{Giner_2015}
|
|
|
|
Another source of size-consistence error in QMC calculation may originate
|
|
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}
|
|
$a$ 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
|
|
interation is not size-consistent. The dissociation of a
|
|
heteroatomic diatomic molecule $AB$ with a parameter $b_{AB}$
|
|
will lead to two different two-body Jastrow factors on each atom, each
|
|
with its own optimal value $b_A$ and $b_B$. To remove the
|
|
size-consistence 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 pseudopotential on the complete wave
|
|
function. If the wave function is not size-consistent, so will be the
|
|
locality approximation. Recently, the determinant localization
|
|
approximation was introduced.\cite{Zen_2019} This approximation
|
|
consists in removing the Jastrow factor from the wave function on
|
|
which the pseudopotential is localized.
|
|
The great advantage of this approximation is that the FN-DMC energy
|
|
within this approximation only depends on the parameters of the
|
|
determinantal component. Using a size-inconsistent Jastrow factor, or
|
|
a non-optimal Jastrow factor will not introduce an additional
|
|
size-consistence error in FN-DMC calculations, although it will
|
|
reduce the statistical errors by reducing the variance of the local energy.
|
|
|
|
|
|
The energy computed within density functional theory is extensive, 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, the convergence of the FCI energies to the CBS
|
|
limit will be slower because of the description of short-range electron
|
|
correlation with atom-centered functions, but ultimately the exact
|
|
energy will be reached.
|
|
|
|
|
|
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{table*}
|
|
\caption{Mean absolute error (MAE), mean signed errors (MSE) and
|
|
standard deviations (RMSD) obtained with the different 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 \\
|
|
\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 & --- & --- & --- & --- & --- & --- \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
%\end{squeezetable}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{g2-dmc.pdf}
|
|
\caption{Errors in the 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}
|
|
|
|
\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 wave functions are shown in
|
|
figure~\ref{fig:n2-ndet}. For all the calculations, the stopping
|
|
criterion of the CIPSI algorithm was $\EPT < 1$~m$E_h$ or $\Ndet >
|
|
10^7$.
|
|
|
|
|
|
|
|
|
|
%%---------------------------------------
|
|
\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.
|
|
\end{acknowledgments}
|
|
|
|
|
|
\bibliography{rsdft-cipsi-qmc}
|
|
|
|
\end{document}
|