491 lines
26 KiB
TeX
491 lines
26 KiB
TeX
\documentclass[aip,jcp,reprint]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,hyperref,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem}
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\usepackage{mathpazo,libertine}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\newcommand{\cdash}{\multicolumn{1}{c}{---}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\mr}{\multirow}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\fnm}{\footnotemark}
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\urlstyle{same}
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\newcommand{\eps}{\varepsilon}
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\newcommand{\la}{\lambda}
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\newcommand{\br}{\bm{r}}
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\newcommand{\oH}{\mathring{H}}
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\newcommand{\EPT}{E_\text{PT2}}
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\NGG}{N_\text{GG}}
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\newcommand{\hH}{\Hat{H}}
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\newcommand{\hF}{\Hat{F}}
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\newcommand{\hO}{\Hat{O}}
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hV}{\Hat{V}}
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\newcommand{\hU}{\Hat{U}}
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\newcommand{\hQ}{\Hat{Q}}
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\newcommand{\hS}{\Hat{S}}
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\newcommand{\hP}{\Hat{P}}
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\newcommand{\hI}{\Hat{1}}
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\newcommand{\mA}{\mathcal{A}}
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\newcommand{\mC}{\mathcal{C}}
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\newcommand{\mD}{\mathcal{D}}
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\newcommand{\mE}{\mathcal{E}}
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\newcommand{\mK}{\mathcal{K}}
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\newcommand{\mF}{\mathcal{F}}
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\newcommand{\mL}{\mathcal{L}}
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\newcommand{\mJ}{\mathcal{J}}
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\newcommand{\kA}{\ket{A}}
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\newcommand{\kD}{\ket{D}}
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\newcommand{\kI}{\ket{I}}
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\newcommand{\kJ}{\ket{J}}
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\newcommand{\kE}{\ket{E}}
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\newcommand{\kK}{\ket{K}}
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\newcommand{\kL}{\ket{L}}
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\newcommand{\kF}{\ket{F}}
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\newcommand{\kO}{\ket{0}}
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\newcommand{\cD}[1]{c_{#1}}
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\newcommand{\cF}[1]{a_{#1}}
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% addresses
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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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\newcommand{\MPI}{Max-Planck-Institut f\"ur Festk\"orperforschung, Heisenbergstra{\ss}e 1, 70569 Stuttgart, Germany}
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\begin{document}
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\title{Dressing the configuration interaction matrix with explicit correlation}
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\author{Anthony Scemama}
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\affiliation{\LCPQ}
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\author{Michel Caffarel}
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\affiliation{\LCPQ}
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\author{David P. Tew}
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\affiliation{\MPI}
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\author{Pierre-Fran\c{c}ois Loos}
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\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\begin{abstract}
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We present an explicitly-correlated version of the configuration interaction (CI) method.
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An explicitly-correlated term is introduced via a dressing of the CI matrix.
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The dressing is guided by electron-electron cusp conditions.
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This greatly enhances the convergence with respect to the one-electron basis set compared to conventional CI methods.
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The performance of the newly-designed explicitly-correlated CI-F12 method is illustrated on atoms and molecules.
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\end{abstract}
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%\keywords{configuration interaction; explicitly-correlated methods; effective Hamiltonian theory}
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\maketitle
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%----------------------------------------------------------------
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\section{Introduction}
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%----------------------------------------------------------------
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\begin{quotation}
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\textit{``The key idea is that traditional CI is not really bad, it only has difficulties to represent the wave function at those regions of configuration space where one interelectronic distance $r_{ij}$ approaches zero.''}
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\flushright --- Werner Kutzelnigg,
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\\
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\textit{Theor.~Chim.~Acta} \textbf{68} (1985) 445--469.
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\end{quotation}
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One of the most fundamental and severe error in electronic structure methods is the (one-electron) basis set incompleteness.
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In particular, conventional quantum chemistry wave function methods typically display a slow energy convergence with respect to the size of the one-electron basis set.
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This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kutzelnigg85}
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To palliate this, he proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kutzelnigg85, Kutzelnigg91, Termath91, Klopper91a, Klopper91b, Noga94}
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This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis).
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This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}
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The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno12a, Tenno12b, Hattig12, Kong12}
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For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis. \cite{Tew07b}
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In the present study, following Kutzelnigg's idea, we propose to introduce the explicit correlation between electrons within the configuration interaction (CI) method via a dressing of the CI matrix.
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This method, involving effective Hamiltonian theory, \cite{Malrieu85} has been shown to be successful in other scenarios. \cite{Heully92, Garniron18}
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Compared to other explicitly-correlated strategies, this dressing strategy has the advantage of introducing the explicit correlation at a relatively low computational cost.
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The present explicitly-correlated dressed CI method is completely general and can be applied to any type of truncated, full, \cite{Knowles84, Knowles89} or even selected CI methods. \cite{Huron73, Giner13, Giner15, Caffarel16, Scemama18a, Scemama18b, Loos18b, Loos19}
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However, for the sake of generality, we will discuss here the dressing of the full CI (FCI) matrix.
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Atomic units are used throughout.
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%----------------------------------------------------------------
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%\section{Ans{\"a}tz} : Ca prend un trema au pluriel seulement : Ans{\"a}tze
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\section{Ansatz}
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%----------------------------------------------------------------
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Inspired by a number of previous research (see Ref.~\onlinecite{Shiozaki11} and references therein), our electronic wave function \emph{ans{\"a}tz}
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\begin{equation}
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\label{eq:ansatz}
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\ket{\Psi} = \kD + \kF
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\end{equation}
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is simply written as the sum of a \emph{``conventional''} part
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\begin{equation}
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\label{eq:D}
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\kD = \sum_{I} \cD{I} \kI
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\end{equation}
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composed by a linear combination of determinants $\kI$ with CI coefficients $\cD{I}$ and an \emph{``explicitly-correlated''} part
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\begin{equation}
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\label{eq:WF-F12-CIPSI}
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\kF = \sum_{I} \cF{I} \hQ f \kI
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\end{equation}
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with coefficients $\cF{I}$.
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The projector
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\begin{equation}
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\hQ = \hI - \sum_{I} \dyad{I}
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\end{equation}
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ensures the orthogonality between $\kD$ and $\kF$ (where $\hI$ is the identity operator), and
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\begin{equation}
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\label{eq:Ja}
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f = \sum_{i < j} f_{ij}
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% f = \sum_{i < j} \gamma_{ij} f_{ij}
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\end{equation}
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is a (linear) correlation factor.% with
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%\begin{equation}
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% \gamma_{ij} =
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% \begin{cases}
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% 1/2, & \text{for opposite-spin electrons},
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% \\
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% 1/4, & \text{for same-spin electrons}.
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% \end{cases}
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%\end{equation}
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As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various authors \cite{Pack66, Morgan93, Tew08, ExSpherium10, eee15}), for small $r_{12}$, the two-electron correlation factor $f_{12}$ in Eq.~\eqref{eq:Ja} must behave as
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\begin{equation}
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f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}.
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\end{equation}
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%----------------------------------------------------------------
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\section{Effective Hamiltonian}
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%----------------------------------------------------------------
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Our primary goal is to introduce the explicit correlation between electrons at relatively low computational cost.
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Therefore, assuming that $\hH \ket{\Psi} = E\,\Psi$, one can write, by projection over $\bra{I}$,
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\begin{equation}
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\cD{I} \qty[ H_{II} + \cD{I}^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} \cD{J} H_{IJ} = 0,
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\end{equation}
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where $H_{IJ} = \mel{I}{\hH}{J}$.
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Hence, we obtain the desired energy by diagonalizing the dressed Hamiltonian:
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\begin{equation}
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\label{eq:DrH}
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\oH_{IJ} =
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\begin{cases}
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H_{II} + \cD{I}^{-1}\mel*{I}{\hH}{F}, & \text{if $I = J$},
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\\
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H_{IJ}, & \text{otherwise},
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\end{cases}
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\end{equation}
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with
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\begin{equation}
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\label{eq:IHF}
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\mel{I}{\hH}{F} = \sum_J \cF{J} \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ],
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\end{equation}
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and $f_{IJ} = \mel{I}{f}{J}$.
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Because only the diagonal of $\hH$ is modified in Eq.~\eqref{eq:DrH}, we refer to this strategy as diagonal dressing.
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It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} = 0$, which demonstrates that the dressed term vanishes in the limit of a complete one-electron basis, as one would expect.
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Moreover, because the CI-F12 energy is obtained via projection, the present method is not variational.
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At this stage, two key comments are in order.
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First, as one may have realized, the coefficients $\cF{I}$ are unknown.
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However, they can be set to ensure the $s$- and $p$-wave electron-electron cusp conditions (SP ansatz). \cite{Tenno04a}
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\alert{T2: Here include the rules to determine the coefficients $\cF{I}$.}
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%\alert{This yields the following linear system of equations
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%\begin{equation}
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%\label{eq:tI}
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% \sum_J (\delta_{IJ} + f_{IJ}) \cF{J} = \cD{I},
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%\end{equation}
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%which can be easily solved using standard linear algebra packages (where $\delta_{IJ}$ is the Kronecker delta).}
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Second, because Eq.~\eqref{eq:DrH} depends on the CI coefficient $\cD{I}$, one must iterate the diagonalization process self-consistently until convergence of the desired eigenvalues of the dressed Hamiltonian $\oH$.
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At each iteration, we solve Eq.~\eqref{eq:tI} to obtain the coefficients $\cF{I}$ and dress the Hamiltonian [see Eq.~\eqref{eq:DrH}].
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In practice, we initially start with a CI vector obtained by the diagonalization of the undressed Hamiltonian, and convergence is usually reached within few cycles.
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We refer the interested reader to Ref.~\onlinecite{Garniron18} for additional details about our dressing scheme.
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Note that the present formalism is state-specific and only focus on the ground state.
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A multi-state strategy can be applied following our work in Ref.~\onlinecite{Garniron18}.
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In the state-specific case, it is possible to avoid the potentially troublesome division by $\cD{I}^{-1}$ by shuffling around the dressing term.
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Assuming, witout loss of generality that $\cD{0}$ is the largest coefficient $\cD{I}$, we have
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\begin{equation}
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\label{eq:DrH}
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\oH_{IJ} =
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\begin{cases}
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H_{I0} + \frac{\mel*{I}{\hH}{F}}{\cD{0}}, & I > 0 \wedge J = 0
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\\
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H_{0J} + \frac{\mel*{J}{\hH}{F}}{\cD{0}}, & I = 0 \wedge J > 0
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\\
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H_{00} + \frac{2 \mel*{0}{\hH}{F}}{\cD{0}} - \sum_I \frac{\cD{I}}{\cD{0}} \frac{\mel*{I}{\hH}{F}}{\cD{0}}, & I = 0 \wedge J = 0
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\\
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H_{IJ}, & \text{otherwise}.
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\end{cases}
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\end{equation}
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%%% FIG 1 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{fig1}
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\caption{
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\label{fig:CBS}
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Schematic representation of the various orbital spaces and their notation.
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The arrows represent the three types of excited determinants contributing to the dressing term: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).}
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\end{figure}
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%%% %%%
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%----------------------------------------------------------------
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\section{Matrix elements}
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%----------------------------------------------------------------
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Compared to a conventional CI calculation, new matrix elements are required.
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The simplest of them $f_{IJ}$ --- required in Eq.~\eqref{eq:IHF} --- can be easily computed by applying Slater-Condon rules. \cite{SzaboBook}
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They involve two-electron integrals over the correlation factor $f_{12}$.
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Their computation has been thoroughly studied in the literature in the last thirty years. \cite{Kutzelnigg91, Klopper92, Persson97, Klopper02, Manby03, Werner03, Klopper04, Tenno04a, Tenno04b, May05, Manby06, Tenno07, Komornicki11, Reine12, GG16}
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These can be more or less expensive to compute depending on the choice of the correlation factor.
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As shown in Eq.~\eqref{eq:IHF}, the present explicitly-correlated CI method also requires matrix elements of the form $\mel{I}{\hH f}{ J}$.
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These are more problematic, as they involve the computation of numerous three-electron integrals over, for instance, the operator $r_{12}^{-1}f_{13}$, as well as new two-electron integrals. \cite{Kutzelnigg91, Klopper92}
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We have recently developed recurrence relations and efficient upper bounds in order to compute these types of integrals. \cite{3ERI1, 3ERI2, 4eRR, IntF12}
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However, we will here explore a different route.
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We propose to compute them using the resolution of the identity (RI) approximation, \cite{Kutzelnigg91, Klopper02, Valeev04, Werner07, Hattig12} which requires (at least formally) a complete basis set (CBS).
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This CBS is built as the union of the orbital basis set (OBS) $\qty{p}$ (divided as occupied $\qty{i}$ and virtual $\qty{a}$ subspaces) augmented by a complementary auxiliary basis set (CABS) $\qty{\alpha}$, such as $ \qty{p} \cap \qty{\alpha} = \varnothing$ and $\braket{p}{\alpha} = 0$. \cite{Klopper02, Valeev04} (see Fig.~\ref{fig:CBS}).
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In the CBS, one can write
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\begin{equation}
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\label{eq:RI}
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\hI = \sum_{A \in \mA} \dyad{A}{A}
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\end{equation}
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where $\mA$ is the set of all the determinants $\kA$ corresponding to electronic excitations from occupied orbitals $\qty{i}$ to the extended virtual orbital space $\qty{a} \cup \qty{\alpha}$.
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Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eqref{eq:IHF}, one gets
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\begin{equation}
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\label{eq:IHF-RI}
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\begin{split}
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\mel{I}{\hH}{F}
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& = \sum_J \cF{J} \qty[ \sum_{A \in \mA} H_{IA} f_{AJ} - \sum_{K \in \mD} H_{IK} f_{KJ} ]
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\\
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& = \sum_J \cF{J} \sum_{A \in \mC} H_{IA} f_{AJ},
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\end{split}
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\end{equation}
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where $\mD$ is the set of ``conventional'' determinants obtained by excitations from the occupied space $\qty{i}$ to the virtual one $\qty{a}$, and $\mC = \mA \setminus \mD$.
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Because $f$ is a two-electron operator, the way to compute efficiently Eq.~\eqref{eq:IHF-RI} is actually very similar to what is done within second-order multireference perturbation theory. \cite{Garniron17b}
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The set $\mC$ is defined by two simple rules.
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First, because $f$ is a two-electron operator [and thanks to the matrix element $f_{AJ}$ in \eqref{eq:IHF-RI}], we know that the sum over $A$ is restricted to the singly- or doubly-excited determinants with respect to the determinant $\kJ$.
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Second, to ensure that $H_{IA} \neq 0$, $A$ must be connected to $\kI$, i.e.~differs from $\kI$ by no more than two spin orbitals.
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Three types of determinants have these two properties (see Fig.~\ref{fig:CBS}).:
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i) the pure doubles $\hT_{ij}^{\alpha \beta}\ket*{I}$,
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ii) the mixed doubles $\hT_{ij}^{\alpha b}\ket*{I}$, and
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iii) the pure singles $\hT_{i}^{\alpha}\ket*{I}$.
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\alert{
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The matrix element between two determinants differing by a double excitation $\hT_{ij}^{kl}$ is given by
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\begin{equation}
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\mel{I}{\hH f}{J} = \{ ij || kl \} - \sum_m \{ ijm || mkl \} \Delta_{mI} \Delta_{mJ}
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\end{equation}
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where
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\begin{equation}
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\Delta_{mI} = \mel{I}{a_m^\dagger a_m}{I},
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\end{equation}
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\begin{equation}
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\{ ijm || mkl \} = \sum_{\alpha} \langle i j || \alpha m \rangle [ \alpha m || k l ]
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+ \langle i j || m \alpha \rangle [ m \alpha || k l ],
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\end{equation}
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\begin{equation}
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\{ ij || kl \} = \sum_{\alpha \beta} \langle i j || \alpha \beta \rangle [ \alpha \beta || k l ] + \sum_m \{ ijm || mkl \}
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\end{equation}
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The matrix element between two determinants differing by a single excitation $\hT_{i}^{k}$ is given by
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\begin{equation}
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\mel{I}{\hH f}{J} = \sum_j \Delta_{jI} \Delta_{jJ} \qty( \{ ij || kj \} - \sum_m \{ ijm || mkj \} \Delta_{mI} \Delta_{mJ} )
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\end{equation}
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and the diagonal terms are
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\begin{equation}
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\mel{I}{\hH f}{I} = \sum_{ij} \Delta_{iI} \Delta_{jI} \qty( \{ ij || ij \} - \sum_m \{ ijm || mij \} \Delta_{mI} )
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\end{equation}
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}
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Although $\mel{0}{\hH}{_{i}^{a}} = 0$, note that the Brillouin theorem does not hold in the CABS, i.e.~$\mel{0}{\hH}{_{i}^{\alpha}} \neq 0$.
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Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero. \cite{Kutzelnigg91}
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%\begin{gather}
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% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j}
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% \\
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% \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta}
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% \\
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% \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta}
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%\end{gather}
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%
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%\begin{gather}
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% \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j}
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% \\
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% \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0
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% \\
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% \mel*{_k^c}{\hH}{_{ij}^{a\beta}} =
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%\end{gather}
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%
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%\begin{gather}
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% \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} =
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% \\
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% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} =
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% \\
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% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} =
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%\end{gather}
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%
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%\begin{gather}
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% \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} =
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% \\
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% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} =
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% \\
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% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} =
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%\end{gather}
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%----------------------------------------------------------------
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\section{Computational details}
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%----------------------------------------------------------------
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In all the CI-F12 calculations presented below, we consider the following Slater-type correlation factor \cite{Tenno04a}
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\begin{equation}
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f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la},
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\end{equation}
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which is fitted using $N_\text{GG}$ Gaussian geminals for computational convenience, \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05} i.e.
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\begin{equation}
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\exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} d_\nu \exp( - \la_\nu r_{12}^2 ).
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\end{equation}
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The contraction coefficients $d_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study.
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Unless otherwise stated, all the calculations have been performed with \textsc{QCaml}, an electronic structure software written in \textsc{OCaml} specifically designed for the present study.
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%----------------------------------------------------------------
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\section{Results}
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%----------------------------------------------------------------
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%%% TABLE 1 %%%
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\begin{table}
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\caption{
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\label{tab:atoms}
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|
FCI-F12 and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pCVXZ and cc-pVXZ basis sets.
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\alert{The corresponding cc-pVXZ\_OPTRI or cc-pCVXZ\_OPTRI auxiliary basis is used as CABS.}}
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\begin{ruledtabular}
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\begin{tabular}{lcdd}
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|
Atom & X & \tabc{FCI-F12} & \tabc{FCI} \\
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|
\hline
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|
\ce{He} & D & & -2.887\,595 \fnm[1] \\
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|
(cc-pVXZ) & T & & -2.900\,232 \fnm[1] \\
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|
& Q & & -2.902\,411 \fnm[1] \\
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|
& 5 & & -2.903\,152 \fnm[1] \\
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|
& 6 & & -2.903\,432 \fnm[1] \\
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|
& $\infty$ & & -2.903\,724 \fnm[2] \\
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|
\hline
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|
\ce{Li} & D & & -7.466\,025 (FCI) \\
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|
(cc-pCVXZ) & T & & -7.474\,251 (FCI) \\
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|
& Q & & -7.476\,373 (FCI) \\
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|
& $\infty$ & & -7.478\,060 \fnm[3] \\
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|
\hline
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|
\ce{Be} & D & & -14.651\,833 (FCI) \\
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|
(cc-pCVXZ) & T & & -14.662\,368 (FCI) \\
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|
& Q & & -14.665\,566 (CIPSI) \\
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|
& $\infty$ & & -14.667\,356 \fnm[4] \\
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|
& $\infty$ & & -14.667\,39 (TOTO) \\
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\hline
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|
\ce{B} & D & & -24.619\,101 (FCI) \\
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|
(cc-pwCVXZ) & T & & -24.643\,222 (CIPSI) \\
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|
& Q & & -24.650\,331 (CIPSI) \\
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& 5 & & -24.652\,309 (CIPSI) \\
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|
& $\infty$ & & -24.653\,866 \fnm[5] \\
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& $\infty$ & & -24.653\,90 (TOTO) \\
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|
\hline
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|
\ce{C} & D & & -37.792\,469 (FCI) \\
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|
(cc-pwCVXZ) & T & & -37.829\,847 (CIPSI) \\
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|
& Q & & -37.839\,816 (CIPSI) \\
|
|
& 5 & & -37.842\,731 (CIPSI) \\
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|
& $\infty$ & & -37.840\,129 6 \\
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|
& $\infty$ & & -37.845\,0 (TOTO) \\
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|
\hline
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|
\ce{N} & D & & -54.517\,650 (FCI) \\
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|
(cc-pwCVXZ) & T & & -54.567\,764 (CIPSI) \\
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|
& Q & & -54.581\,885 (CIPSI) \\
|
|
& 5 & & -54.585\,926 (CIPSI) \\
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|
& $\infty$ & & -54.588\,917 \fnm[7] \\
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& $\infty$ & & -54.589\,3 (TOTO) \\
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|
\hline
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|
\ce{O} & D & & -74.946\,393 (CIPSI) \\
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|
(cc-pwCVXZ) & T & & -75.031\,607 (CIPSI) \\
|
|
& Q & & -75.054\,737 (CIPSI) \\
|
|
& 5 & & -75.062\,002 (CIPSI) \\
|
|
& $\infty$ & & -75.066\,892 \fnm[7] \\
|
|
& $\infty$ & & -75.067\,4 (TOTO) \\
|
|
\hline
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|
\ce{F} & D & & -99.566\,902 (CIPSI) \\
|
|
(cc-pwCVXZ) & T & & -99.682\,616 (CIPSI) \\
|
|
& Q & & -99.715\,563 (CIPSI) \\
|
|
& 5 & & -99.726\,249 (CIPSI) \\
|
|
& $\infty$ & & -99.733\,424 \fnm[7] \\
|
|
& $\infty$ & & -99.734\,1 (TOTO) \\
|
|
\hline
|
|
\ce{Ne} & D & & -128.721\,575 (CIPSI) \\
|
|
(cc-pwCVXZ) & T & & -128.869\,425 (CIPSI) \\
|
|
& Q & & -128.913\,064 (CIPSI) \\
|
|
& 5 & & -128.927\,705 (CIPSI) \\
|
|
& $\infty$ & & -128.937\,274 \fnm[7] \\
|
|
& $\infty$ & & -128.938\,3 (TOTO) \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\fnt[1]{Reference \onlinecite{Kong12}}
|
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\fnt[2]{Reference \onlinecite{Nakashima07}}
|
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\fnt[3]{Reference \onlinecite{Puchalski09}}
|
|
\fnt[4]{Reference \onlinecite{Sharkey11}}
|
|
\fnt[5]{Reference \onlinecite{Bubin11}}
|
|
\fnt[6]{Reference \onlinecite{Sharkey10}}
|
|
\fnt[7]{Reference \onlinecite{Klopper10}}
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|
\end{table}
|
|
%%%
|
|
|
|
%%% TABLE 2 %%%
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|
\begin{table*}
|
|
\caption{
|
|
\label{tab:molecules}
|
|
CIPSI, FCI-F12, i-FCIQMC and FCI total ground-state energy of the \ce{H2}, \ce{F2} and \ce{H2)} molecules at experimental geometry with Dunning's cc-pVXZ basis set.
|
|
The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lcdddd}
|
|
Molecule & X & \tabc{CIPSI} & \tabc{FCI-F12} & \tabc{i-FCIQMC} & \tabc{FCI} \\
|
|
\hline
|
|
\ce{H2} & D & & & \\
|
|
(cc-pVXZ) & T & & & \\
|
|
& Q & & & \\
|
|
& 5 & & & \\
|
|
& $\infty$ & & & & -1.174\,476\fnm[1] \\
|
|
\hline
|
|
\ce{F2} & D & -199.099\,28\fnm[2] & & -199.099\,41(9)\fnm[3] \\
|
|
(cc-pVXZ)& T & -199.296\,5\fnm[2] & & -199.297\,7(1)\fnm[3] \\
|
|
\hline
|
|
\ce{H2O} & D & -76.282\,136\fnm[4] & & & -76.282\,865\fnm[5] \\
|
|
(cc-pVXZ)& T & -76.388\,287\fnm[4] & & & -76.390\,158\fnm[5] \\
|
|
& Q & -76.419\,324\fnm[4] & & & -76.421\,148\fnm[5] \\
|
|
& 5 & -76.428\,550\fnm[4] & & & -76.431\,105\fnm[5] \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\fnt[1]{Reference \onlinecite{Pachucki10}}
|
|
\fnt[2]{Reference \onlinecite{Giner15}}
|
|
\fnt[3]{Reference \onlinecite{Cleland12}}
|
|
\fnt[4]{Reference \onlinecite{Caffarel16}}
|
|
\fnt[5]{Reference \onlinecite{AlmoraDiaz14}}
|
|
\end{table*}
|
|
%%%
|
|
|
|
In Table \ref{tab:atoms}, we report the total atomic energy of the neutral atoms from $Z = 2$ to $10$ for various Dunning's basis sets.
|
|
In all calculations, the associated OPTRI basis is used as CABS. \cite{Yousaf08, Yousaf09}
|
|
|
|
In Table \ref{tab:molecules}, we report the total energy of the \ce{H2}, \ce{F2} and \ce{H2O} molecules at experimental geometry. \cite{Giner13, Giner15, Caffarel16}
|
|
|
|
%----------------------------------------------------------------
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\section{Conclusion}
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%----------------------------------------------------------------
|
|
We have introduced a dressed version of the well-established CI method to incorporate explicitly the correlation between electrons.
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We have shown that the new CI-F12 method allows to fix one of the main issue of conventional CI methods, i.e.~the slow convergence of the electronic energy with respect to the size of the one-electron basis set.
|
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Albeit not variational, our method is able to catch a large fraction of the basis set incompleteness error at a low computational cost compared to other variants.
|
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In particular, one eschew the computation of four-electron integrals as well as some types of three-electron integrals.
|
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We believe that the present approach is a significant step towards the development of an accurate and efficient explicitly-correlated FCI method.
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%----------------------------------------------------------------
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\begin{acknowledgments}
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The authors would like to thank the \emph{Centre National de la Recherche Scientifique} (CNRS) for funding.
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This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), and CALMIP (Toulouse) under allocation 2019-18005.
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\end{acknowledgments}
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%----------------------------------------------------------------
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\bibliography{CI-F12,CI-F12-control}
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\end{document}
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