919 lines
46 KiB
TeX
919 lines
46 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[T1]{fontenc}
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\usepackage[utf8]{inputenc}
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\usepackage{txfonts}
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\usepackage[
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colorlinks=true,
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citecolor=blue,
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breaklinks=true
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]{hyperref}
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\urlstyle{same}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\newcommand{\eg}[1]{\textcolor{blue}{#1}}
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\EPT}{E_{\text{PT2}}}
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\newcommand{\EDMC}{E_{\text{FN-DMC}}}
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\newcommand{\Ndet}{N_{\text{det}}}
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\newcommand{\hartree}{$E_h$}
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\newcommand{\LCT}{Laboratoire de Chimie 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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\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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\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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Within a finite one-electron basis, full configuration interaction (FCI)
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delivers only an approximate solution of the Schr\"odinger equation.
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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}
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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.
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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 CIPSI 3-CLASS AND INITIATOR APPROXIMATION.
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Diffusion Monte Carlo (DMC) is a numerical scheme to obtain
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the exact solution of the Schr\"odinger equation with a different
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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} (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 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 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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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 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 $|\EPT| <
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1$~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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To rigorously connect wave function theory 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}^{\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\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} \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}
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$\hat{H}^\mu$ as shown in figure~\ref{fig:algo}. In the outer loop
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(red), a CIPSI selection is performed with a RS-Hamiltonian
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parameterized using the current density.
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An inner 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 is kept fixed, and only the diagonalization of the
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RS-Hamiltonian is performed iteratively with the updated density.
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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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Note that any range-separated post-Hartree-Fock 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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\section{Computational details}
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\label{sec:comp-details}
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All the calculations were made using BFD
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pseudopotentials\cite{Burkatzki_2008} with the associated double,
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triple and quadruple zeta basis sets (BFD-V$n$Z).
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CCSD(T) and DFT calculations were made with
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\emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock
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determinant as a reference for open-shell systems.
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All the CIPSI calculations were made with \emph{Quantum
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|
Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version
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of the Perdew-Burke-Ernzerhof (PBE)~\cite{PerBurErn-PRL-96} exchange
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and correlation functionals of
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Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
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Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
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The convergence criterion for stopping the CIPSI calculations
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was $\EPT < 1$~m\hartree{} $\vee \Ndet > 10^7$.
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All the wave functions are eigenfunctions of the $S^2$ operator, as
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described in ref~\onlinecite{Applencourt_2018}.
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Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
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in the determinant localization approximation (DLA),\cite{Zen_2019}
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where only the determinantal component of the trial wave
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function is present in the expression of the wave function on which
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the pseudopotential is localized. Hence, in the DLA the fixed-node
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energy is independent of the Jatrow factor, as in all-electron
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calculations. Simple Jastrow factors were used to reduce the
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fluctuations of the local energy.
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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 and number of determinants in the water
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molecule and the fluorine dimer with different trial wave functions.}
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\label{tab:h2o-dmc}
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\centering
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\begin{ruledtabular}
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\begin{tabular}{crlrl}
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& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
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$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
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\hline
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& \multicolumn{4}{c}{H$_2$O} \\
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$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
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$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
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$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
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$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
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$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
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$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
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$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
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$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
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$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
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$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
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$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
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$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
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& \multicolumn{3}{c}{F$_2$} \\
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$0.00$ & $23$ & $-48.419\,5(4)$ \\
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$0.25$ & $8$ & $-48.421\,9(4)$ \\
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$0.50$ & $1743$ & $-48.424\,8(8)$ \\
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$1.00$ & $11952$ & $-48.432\,4(3)$ \\
|
|
$2.00$ & $829438$ & $-48.441\,0(7)$ \\
|
|
$5.00$ & $5326459$ & $-48.445(2)$ \\
|
|
$\infty$ & $8302442$ & $-48.437(3)$ \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
|
|
\caption{Fixed-node energies of the water molecule for different
|
|
values of $\mu$.}
|
|
\label{fig:h2o-dmc}
|
|
\end{figure}
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{f2-dmc.pdf}
|
|
\caption{Fixed-node energies of difluorine for different
|
|
values of $\mu$.}
|
|
\label{fig:f2-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 two weakly correlated molecular systems: the water
|
|
molecule and the fluorine dimer, near their equilibrium
|
|
geometry\cite{Caffarel_2016}.
|
|
|
|
From table~\ref{tab:h2o-dmc} and figures~\ref{fig:h2o-dmc}
|
|
and~\ref{fig:f2-dmc}, one can clearly observe that using FCI trial
|
|
wave functions ($\mu = \infty$) gives FN-DMC energies which are lower
|
|
than the energies obtained with a single Kohn-Sham determinant ($\mu=0$):
|
|
a gain of $3.2 \pm 0.6$~m\hartree{} at the double-zeta level and $7.2 \pm
|
|
0.3$~m\hartree{} at the triple-zeta level are obtained for water, and
|
|
a gain of $18 \pm 3$~m\hartree{} for F$_2$. Interestingly, using the
|
|
RS-DFT-CIPSI trial wave function with a range-separation parameter
|
|
$\mu=1.75$~bohr$^{-1}$ with the double-zeta basis one can obtain for
|
|
water a FN-DMC energy $2.6 \pm 0.7$~m\hartree{} lower than the energy
|
|
obtained with the FCI trial wave function. This can be explained by
|
|
the inability of the basis set to properly describe the short-range
|
|
correlation effects, shifting the nodes from their optimal
|
|
position. Using DFT to take account of short-range correlation frees
|
|
the determinant expansion from describing short-range effects, and
|
|
enables a placement of the nodes closer to the optimum. In the case
|
|
of F$_2$, a similar behavior with a gain of $8 \pm 4$ m\hartree{} is
|
|
observed for $\mu\sim 5$~bohr$^{-1}$.
|
|
The optimal value of $\mu$ is larger than in the case of water, and this
|
|
is probably the signature of the fact that the average
|
|
electron-electron distance in the valence is smaller in F$_2$ than in
|
|
H$_2$O due to the larger nuclear charge shrinking the electron
|
|
density. At the triple-zeta level, the short-range correlations can be
|
|
better described by the determinant expansion, and the effect of
|
|
sr-DFT on the trial wave function is insignificant on the
|
|
fixed-node energy. However, it is important to note that the
|
|
same FN-DMC energy can be obtained with a CI expansion which is eight
|
|
times smaller when sr-DFT is introduced. One can also remark that the
|
|
minimum has been slightly shifted towards the FCI, which is consistent
|
|
with the fact that in the CBS limit we expect the minimum of the
|
|
FN-DMC energy to be obtained for the FCI wave function, i.e. at
|
|
$\mu=\infty$.
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\columnwidth]{overlap.pdf}
|
|
\caption{Overlap of the RS-DFT CI expansion with the
|
|
CI expansion optimized in the presence of a Jastrow factor.}
|
|
\label{fig:overlap}
|
|
\end{figure}
|
|
|
|
This data confirms that RS-DFT-CIPSI can give improved CI coefficients
|
|
with small basis sets, similarly to the common practice of
|
|
re-optimizing the trial wave function in the presence of the Jastrow
|
|
factor. To confirm that the introduction of sr-DFT has an impact on
|
|
the CI coefficients similar to the Jastrow factor, we have made the
|
|
following numerical experiment. First, we extract the 200 determinants
|
|
with the largest weights in the FCI wave function out of a large CIPSI
|
|
calculation. Within this set of determinants, we diagonalize
|
|
self-consistently the RS-DFT Hamiltonian with different values of
|
|
$\mu$. This gives the CI expansions $\Psi^\mu$. Then, within the same
|
|
set of determinants we optimize the CI coefficients in the presence of
|
|
a simple one- and two-body Jastrow factor. This gives the CI expansion
|
|
$\Psi^J$. In figure~\ref{fig:overlap}, we plot the overlaps
|
|
$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer.
|
|
|
|
In the case of H$_2$O, there is a clear maximum of overlap at
|
|
$\mu=1$~bohr$^{-1}$. This confirms that introducing short-range
|
|
correlation with DFT has the an impact on the CI coefficients similar to
|
|
the Jastrow factor. In the case of F$_2$, the Jastrow factor has
|
|
very little effect on the CI coefficients, as the overlap
|
|
$\braket{\Psi^J}{\Psi^{\mu=\infty}}$ is very close to
|
|
$1$. Nevertheless, a slight maximum is obtained for
|
|
$\mu=5$~bohr$^{-1}$.
|
|
|
|
|
|
\section{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
|
|
interation 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 of the fluorine atom and the dissociated fluorine
|
|
dimer, and size-consistency error.}
|
|
\label{tab:size-cons}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{cccc}
|
|
$\mu$ & F & Dissociated F$_2$ & 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 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 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
|
|
RS-DFT-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@RS-DFT-CIPSI & 0 & 4.61(\phantom{0.}34) & -3.62(\phantom{0.}34) & 5.30(\phantom{0.}09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\
|
|
\phantom{} & 1/4 & 4.04(\phantom{0.}37) & -3.13(\phantom{0.}37) & 4.88(\phantom{0.}10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\
|
|
\phantom{} & 1/2 & 3.74(\phantom{0.}35) & -3.53(\phantom{0.}35) & 4.03(\phantom{0.}23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\
|
|
\phantom{} & 1 & 5.42(\phantom{0.}29) & -5.14(\phantom{0.}29) & 4.55(\phantom{0.}03) & 4.38(94) & -4.24(94) & 5.11(31) & --- & --- & --- \\
|
|
\phantom{} & 2 & 5.98(\phantom{0.}83) & -5.91(\phantom{0.}83) & 4.79(\phantom{0.}71) & --- & --- & --- & --- & --- & --- \\
|
|
\phantom{} & 5 & 6.18(\phantom{0.}84) & -6.13(\phantom{0.}84) & 4.87(\phantom{0.}55) & --- & --- & --- & --- & --- & --- \\
|
|
\phantom{} & \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(\phantom{0.}68) & --- & --- & --- & --- & --- & --- \\
|
|
\phantom{} & 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 $\EPT\rightarrow 0$.
|
|
|
|
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 $\mu$ 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 labelled by the \emph{Opt} value of $\mu$
|
|
in the table. 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.
|
|
|
|
\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. Is 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.
|
|
|
|
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 aproaching the
|
|
RS-DFT-FCI with CIPSI provides a way to adapt the number of
|
|
determinants in the trial wave function, leading to size-consistent
|
|
FN-DMC energies.
|
|
We propose two methods. The first one is for the computation of
|
|
accurate total energies by a one-parameter optimization of the FN-DMC
|
|
energy via the variation of the parameter $\mu$.
|
|
The second method is for the computation of energy differences, where
|
|
the target is not the lowest possible FN-DMC energies but the best
|
|
possible cancellation of errors. Using a fixed value of $\mu$
|
|
increases the consistency of the trial wave functions, and we have found
|
|
that $\mu=0.5$~bohr${^-1}$ is the value where the cancellation of
|
|
errors is the most effective.
|
|
Moreover, such a small value of $\mu$ gives extermely
|
|
compact wave functions, making this recipe a good candidate for
|
|
the accurate description of the whole potential energy surfaces of
|
|
large systems.
|
|
|
|
|
|
|
|
|
|
%%---------------------------------------
|
|
\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}
|
|
|
|
|
|
\bibliography{rsdft-cipsi-qmc}
|
|
|
|
\begin{enumerate}
|
|
\item Total energies and nodal quality:
|
|
\begin{itemize}
|
|
\item Facts: KS occupied orbitals closer to NOs than HF
|
|
\item Even if exact functional, complete basis set, still approximated nodes for KS
|
|
\item KS -> exponentially fast convergence (as HF) with basis because of non divergence of effective KS potential (citer le papier de Gill)
|
|
\item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS
|
|
\item With FCI, good limit at CBS ==> exact energy
|
|
\item But slow convergence with basis set because of divergence of the e-e interaction not well represented in atom centered basis set
|
|
\item Exponential increase of number of Slater determinants
|
|
\item Cite papiers RS-DFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté)
|
|
\item Question: does such a scheme provide better nodal quality ?
|
|
\item In RS-DFT we cannot optimize energy with $\mu$ , but in FNDMC
|
|
\item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI
|
|
\begin{itemize}
|
|
\item less determinants $\Rightarrow$ large systems
|
|
\item only one parameter to optimize $\Rightarrow$ deterministic
|
|
\item $\Rightarrow$ reproducible
|
|
\end{itemize}
|
|
\item with the optimal $\mu$:
|
|
\begin{itemize}
|
|
\item Direct optimization of FNDMC with one parameter
|
|
\item Do we improve energy differences ?
|
|
\item system dependent
|
|
\item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$
|
|
\item large wave functions
|
|
\end{itemize}
|
|
\item Invariance with $m_s$
|
|
|
|
\end{itemize}
|
|
\end{enumerate}
|
|
|
|
|
|
|
|
\end{document}
|
|
|
|
|
|
% * Recouvrement avec Be
|
|
% * Tester sr-LDA avec H2O-DZ
|
|
% * Manu doit faire des programmes pour des plots de ensite a 1 et 2
|
|
% corps le long des axes de liaison, et l'integrale de la densite a
|
|
% 2 corps a coalescence.
|
|
% 1 Manu calcule Be en cc-pvdz tous electrons: FCI -> NOs -> FCI ->
|
|
% qp edit -n 200
|
|
% 2 Manu calcule qp_cipsi_rsh avec mu = [ 1.e-6 , 0.25, 0.5, 1.0, 2.0, 5.0, 1e6 ]
|
|
% 3 Manu fait tourner es petits programmes
|
|
% 4 Manu envoie a toto un tar avec tous les ezfio
|
|
% 5 Toto optimise les coefs en presence e jastrow
|
|
% 6 Toto renvoie a manu psicoef
|
|
% 7 Manu fait tourner ses petits programmes avec psi_J
|