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Manuscript/rsdft-cipsi-qmc.tex
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@ -23,15 +23,16 @@
\newcommand{\Ndet}{N_{\text{det}}}
\newcommand{\hartree}{$E_h$}
\newcommand{\LCT}{Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\begin{document}
\title{Enabling high accuracy diffusion Monte Carlo calculations with
range-separated density functional theory and selected configuration interaction}
\title{Taming the fixed-node error in diffusion Monte Carlo via range separation}
%\title{Enabling high accuracy diffusion Monte Carlo calculations with
% range-separated density functional theory and selected configuration interaction}
\author{Anthony Scemama}
\affiliation{\LCPQ}
@ -55,32 +56,27 @@
\section{Introduction}
\label{sec:intro}
The full configuration interaction (FCI) method within a finite atomic
basis set leads to an approximate solution of the Schrödinger
equation.
This solution is the eigenpair of an approximate Hamiltonian, which is
the projection of the exact Hamiltonian onto the finite basis of all
possible Slater determinants.
The FCI wave function can be interpreted as the constrained solution of the
true Hamiltonian, where the solution is forced to span the space
provided by the basis.
At the complete basis set (CBS) limit, the constraint vanishes and the
exact solution is obtained.
Hence the FCI method enables a systematic improvement of the
calculations by improving the quality of the basis set.
Nevertheless, its exponential scaling with the number of electrons and
with the size of the basis is prohibitive for large systems.
In recent years, the introduction of new algorithms\cite{Booth_2009}
and the
revival\cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016}
Within a finite one-electron basis, full configuration interaction (FCI)
delivers only an approximate solution of the Schr\"odinger equation.
This solution is the eigenpair of an approximate Hamiltonian defined as
the projection of the exact Hamiltonian onto the finite many-electron basis of
all possible Slater determinants generated within this finite one-electron basis.
The FCI wave function can be interpreted as a constrained solution of the
true Hamiltonian forced to span the restricted space provided by the one-electron basis.
In the complete basis set (CBS) limit, the constraint is lifted and the
exact solution is recovered.
Hence, the accuracy of a FCI calculation can be systematically improved by increasing the size of the one-electron basis set.
Nevertheless, its exponential scaling with the number of electrons and with the size of the basis is prohibitive for most chemical systems.
In recent years, the introduction of new algorithms \cite{Booth_2009} and the
revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016}
of selected configuration interaction (sCI)
methods\cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
the sizes of the systems that could be computed at the FCI level, but
the scaling remains exponential unless some bias is introduced leading
to a loss of size consistency.
methods \cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
the sizes of the systems that could be computed at the FCI level.
However, the scaling remains exponential unless some bias is introduced leading
to a loss of size consistency. % CITE CIPSI 3-CLASS AND INITIATOR APPROXIMATION.
The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain
the exact solution of the Schrödinger equation with an additional
Diffusion Monte Carlo (DMC) is a numerical scheme to obtain
the exact solution of the Schr\"odinger equation with a different
constraint, imposing the solution to have the same nodal hypersurface
as a given trial wave function.
Within this so-called \emph{fixed-node} (FN) approximation,