462 lines
24 KiB
TeX
462 lines
24 KiB
TeX
\documentclass[aip,jcp,reprint,showkeys]{revtex4-1}
|
|
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,hyperref,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem}
|
|
|
|
\usepackage{mathpazo,libertine}
|
|
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
|
\newcommand{\cdash}{\multicolumn{1}{c}{---}}
|
|
\newcommand{\mc}{\multicolumn}
|
|
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
|
\newcommand{\mr}{\multirow}
|
|
\newcommand{\fnt}{\footnotetext}
|
|
\newcommand{\fnm}{\footnotemark}
|
|
\urlstyle{same}
|
|
|
|
\newcommand{\eps}{\varepsilon}
|
|
\newcommand{\la}{\lambda}
|
|
|
|
\newcommand{\br}{\bm{r}}
|
|
|
|
\newcommand{\oH}{\mathring{H}}
|
|
\newcommand{\EPT}{E_\text{PT2}}
|
|
\newcommand{\EHF}{E_\text{HF}}
|
|
\newcommand{\NGG}{N_\text{GG}}
|
|
|
|
\newcommand{\hH}{\Hat{H}}
|
|
\newcommand{\hF}{\Hat{F}}
|
|
\newcommand{\hO}{\Hat{O}}
|
|
\newcommand{\hT}{\Hat{T}}
|
|
\newcommand{\hV}{\Hat{V}}
|
|
\newcommand{\hU}{\Hat{U}}
|
|
\newcommand{\hQ}{\Hat{Q}}
|
|
\newcommand{\hS}{\Hat{S}}
|
|
\newcommand{\hP}{\Hat{P}}
|
|
\newcommand{\hI}{\Hat{1}}
|
|
|
|
\newcommand{\mA}{\mathcal{A}}
|
|
\newcommand{\mC}{\mathcal{C}}
|
|
\newcommand{\mD}{\mathcal{D}}
|
|
\newcommand{\mE}{\mathcal{E}}
|
|
\newcommand{\mK}{\mathcal{K}}
|
|
\newcommand{\mF}{\mathcal{F}}
|
|
\newcommand{\mL}{\mathcal{L}}
|
|
\newcommand{\mJ}{\mathcal{J}}
|
|
|
|
\newcommand{\kA}{\ket{A}}
|
|
\newcommand{\kD}{\ket{D}}
|
|
\newcommand{\kI}{\ket{I}}
|
|
\newcommand{\kJ}{\ket{J}}
|
|
\newcommand{\kE}{\ket{E}}
|
|
\newcommand{\kK}{\ket{K}}
|
|
\newcommand{\kL}{\ket{L}}
|
|
\newcommand{\kF}{\ket{F}}
|
|
\newcommand{\kO}{\ket{0}}
|
|
|
|
\newcommand{\cD}[1]{c_{#1}}
|
|
\newcommand{\cF}[1]{a_{#1}}
|
|
|
|
% addresses
|
|
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
|
|
|
\begin{document}
|
|
|
|
\title{Dressing the configuration interaction matrix with explicit correlation}
|
|
|
|
\author{Anthony Scemama}
|
|
\affiliation{\LCPQ}
|
|
\author{Michel Caffarel}
|
|
\affiliation{\LCPQ}
|
|
\author{Pierre-Fran\c{c}ois Loos}
|
|
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
|
|
\affiliation{\LCPQ}
|
|
|
|
\begin{abstract}
|
|
We present an explicitly-correlated version of the configuration interaction (CI) method.
|
|
An explicitly-correlated term is introduced via a dressing of the CI matrix.
|
|
The dressing is guided by electron-electron cusp conditions.
|
|
This greatly enhances the convergence with respect to the one-electron basis set compared to conventional CI methods.
|
|
The performance of the newly-designed explicitly-correlated CI-F12 method is illustrated on atoms and molecules.
|
|
\end{abstract}
|
|
|
|
\keywords{configuration interaction; explicitly-correlated methods; effective Hamiltonian theory}
|
|
|
|
\maketitle
|
|
|
|
%----------------------------------------------------------------
|
|
\section{Introduction}
|
|
%----------------------------------------------------------------
|
|
\begin{quotation}
|
|
\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.''}
|
|
\flushright --- Werner Kutzelnigg,
|
|
\\
|
|
\textit{Theor.~Chim.~Acta} (1985) \textbf{68} 445--469.
|
|
\end{quotation}
|
|
One of the most fundamental and severe error in electronic structure methods is the (one-electron) basis set incompleteness.
|
|
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.
|
|
This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kutzelnigg85}
|
|
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}
|
|
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).
|
|
This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}
|
|
The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno12a, Tenno12b, Hattig12, Kong12}
|
|
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}
|
|
|
|
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.
|
|
This method, involving effective Hamiltonian theory, \cite{Malrieu85} has been shown to be successful in other scenarios. \cite{Heully92, Garniron18}
|
|
Compared to other explicitly-correlated strategies, this dressing strategy has the advantage of introducing the explicit correlation at a relatively low computational cost.
|
|
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}
|
|
However, for the sake of generality, we will discuss here the dressing of the full CI (FCI) matrix.
|
|
Atomic units are used throughout.
|
|
|
|
%----------------------------------------------------------------
|
|
\section{Ans{\"a}tz}
|
|
%----------------------------------------------------------------
|
|
Inspired by a number of previous research (see Ref.~\onlinecite{Shiozaki11} and references therein), our electronic wave function \emph{ans{\"a}tz}
|
|
\begin{equation}
|
|
\label{eq:ansatz}
|
|
\ket{\Psi} = \kD + \kF
|
|
\end{equation}
|
|
is simply written as the sum of a \emph{``conventional''} part
|
|
\begin{equation}
|
|
\label{eq:D}
|
|
\kD = \sum_{I} \cD{I} \kI
|
|
\end{equation}
|
|
composed by a linear combination of determinants $\kI$ with CI coefficients $\cD{I}$ and an \emph{``explicitly-correlated''} part
|
|
\begin{equation}
|
|
\label{eq:WF-F12-CIPSI}
|
|
\kF = \sum_{I} \cF{I} \hQ f \kI
|
|
\end{equation}
|
|
with coefficients $\cF{I}$.
|
|
The projector
|
|
\begin{equation}
|
|
\hQ = \hI - \sum_{I} \dyad{I}
|
|
\end{equation}
|
|
ensures the orthogonality between $\kD$ and $\kF$ (where $\hI$ is the identity operator), and
|
|
\begin{equation}
|
|
\label{eq:Ja}
|
|
f = \sum_{i < j} \gamma_{ij} f_{ij}
|
|
\end{equation}
|
|
is a correlation factor with
|
|
\begin{equation}
|
|
\gamma_{ij} =
|
|
\begin{cases}
|
|
1/2, & \text{for opposite-spin electrons},
|
|
\\
|
|
1/4, & \text{for same-spin electrons}.
|
|
\end{cases}
|
|
\end{equation}
|
|
\alert{The correlation factor \eqref{eq:Ja} is not size-consistent.}
|
|
|
|
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
|
|
\begin{equation}
|
|
f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}.
|
|
\end{equation}
|
|
|
|
%----------------------------------------------------------------
|
|
\section{Effective Hamiltonian}
|
|
%----------------------------------------------------------------
|
|
Our primary goal is to introduce the explicit correlation between electrons at relatively low computational cost.
|
|
Therefore, assuming that $\hH \ket{\Psi} = E\,\Psi$, one can write, by projection over $\bra{I}$,
|
|
\begin{equation}
|
|
\cD{I} \qty[ H_{II} + \cD{I}^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} \cD{J} H_{IJ} = 0.
|
|
\end{equation}
|
|
where $H_{IJ} = \mel{I}{\hH}{J}$.
|
|
Hence, we obtain the desired energy by diagonalizing the dressed Hamiltonian:
|
|
\begin{equation}
|
|
\label{eq:DrH}
|
|
\oH_{IJ} =
|
|
\begin{cases}
|
|
H_{II} + \cD{I}^{-1}\mel*{I}{\hH}{F}, & \text{if $I = J$},
|
|
\\
|
|
H_{IJ}, & \text{otherwise},
|
|
\end{cases}
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\label{eq:IHF}
|
|
\mel{I}{\hH}{F} = \sum_J \cF{J} \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ],
|
|
\end{equation}
|
|
and $f_{IJ} = \mel{I}{f}{J}$.
|
|
We refer to this strategy as diagonal dressing as only the diagonal of $\hH$ is modified in Eq.~\eqref{eq:DrH}.
|
|
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.
|
|
|
|
At this stage, two key comments are in order.
|
|
First, as one may have realized, the coefficients $\cF{I}$ are unknown.
|
|
\alert{However, they can be set to ensure the $s$- and $p$-wave electron-electron cusp conditions (SP ansatz). \cite{Tenno04a}}
|
|
|
|
\alert{This yields the following linear system of equations
|
|
\begin{equation}
|
|
\label{eq:tI}
|
|
\sum_J (\delta_{IJ} + f_{IJ}) \cF{J} = \cD{I},
|
|
\end{equation}
|
|
which can be easily solved using standard linear algebra packages (where $\delta_{IJ}$ is the Kronecker delta).}
|
|
|
|
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 $\oH$.
|
|
At each iteration, we solve Eq.~\eqref{eq:tI} to obtain the coefficients $\cF{I}$ and dress the Hamiltonian [see Eq.~\eqref{eq:DrH}].
|
|
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.
|
|
We refer the interested reader to Ref.~\onlinecite{Garniron18} for additional details about our dressing scheme.
|
|
|
|
Note that the present formalism is state-specific and only focus on the ground state.
|
|
Multi-state can potentially developed following our work in Ref.~\onlinecite{Garniron18}.
|
|
In the state-specific case, it is possible to avoid the potentially troublesome division by $\cD{I}^{-1}$ shuffling around the dressing term.
|
|
Assuming, witout loss of generality that $\cD{0}$ is the largest coefficient $\cD{I}$, we have
|
|
\begin{equation}
|
|
\label{eq:DrH}
|
|
\oH_{IJ} =
|
|
\begin{cases}
|
|
H_{I0} + \frac{\mel*{I}{\hH}{F}}{\cD{0}}, & I > 0 \wedge J = 0
|
|
\\
|
|
H_{0J} + \frac{\mel*{J}{\hH}{F}}{\cD{0}}, & I = 0 \wedge J > 0
|
|
\\
|
|
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
|
|
\\
|
|
H_{IJ}, & \text{otherwise}.
|
|
\end{cases}
|
|
\end{equation}
|
|
It is important to mention that, because the CI-F12 energy is obtained via projection, the present method is not variational.
|
|
|
|
%%% FIG 1 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{fig1}
|
|
\caption{
|
|
\label{fig:CBS}
|
|
Schematic representation of the various orbital spaces and their notation.
|
|
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).}
|
|
\end{figure}
|
|
%%% %%%
|
|
|
|
%----------------------------------------------------------------
|
|
\section{Matrix elements}
|
|
%----------------------------------------------------------------
|
|
Compared to a conventional CI calculation, new matrix elements are required.
|
|
The simplest of them $f_{IJ}$ --- required in Eqs.~\eqref{eq:IHF} and \eqref{eq:tI} --- can be easily computed by applying Condon-Slater rules. \cite{SzaboBook}
|
|
They involve two-electron integrals over the correlation factor $f_{12}$.
|
|
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}
|
|
These can be more or less expensive to compute depending on the choice of the correlation factor.
|
|
|
|
As shown in Eq.~\eqref{eq:IHF}, the present explicitly-correlated CI method also requires matrix elements of the form $\mel{I}{\hH f}{ J}$.
|
|
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}
|
|
We have recently developed recurrence relations and efficient upper bounds in order to compute these types of integrals. \cite{3ERI1, 3ERI2, 4eRR, IntF12}
|
|
|
|
However, we will here explore a different route.
|
|
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).
|
|
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}).
|
|
|
|
In the CBS, one can write
|
|
\begin{equation}
|
|
\label{eq:RI}
|
|
\hI = \sum_{A \in \mA} \dyad{A}{A}
|
|
\end{equation}
|
|
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 $\qty{a} \cup \qty{\alpha}$.
|
|
Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eqref{eq:IHF}, one gets
|
|
\begin{equation}
|
|
\label{eq:IHF-RI}
|
|
\begin{split}
|
|
\mel{I}{\hH}{F}
|
|
& = \sum_J \cF{J} \qty[ \sum_{A \in \mA} H_{IA} f_{AJ} - \sum_{K \in \mD} H_{IK} f_{KJ} ]
|
|
\\
|
|
& = \sum_J \cF{J} \sum_{A \in \mC} H_{IA} f_{AJ},
|
|
\end{split}
|
|
\end{equation}
|
|
where $\mD$ is the set of ``conventional'' determinants obtained by excitations from the occupied space $\qty{i}$ to the virtual one $\qty{a}$, and $\mC = \mA \setminus \mD$.
|
|
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}
|
|
|
|
The set $\mC$ is defined by two simple rules.
|
|
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 $\kJ$.
|
|
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.
|
|
Three types of determinants have these two properties (see Fig.~\ref{fig:CBS}).:
|
|
i) the pure doubles $\ket*{_{ij}^{\alpha \beta}}$,
|
|
ii) the mixed doubles $\ket*{_{ij}^{a \beta}}$, and
|
|
iii) the pure singles $\ket*{_{i}^{\alpha}}$.
|
|
|
|
Although $\mel{0}{\hH}{_{i}^{a}} = 0$, note that the Brillouin theorem does not hold in the CABS, i.e.~$\mel{0}{\hH}{_{i}^{\alpha}} \neq 0$.
|
|
Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero. \cite{Kutzelnigg91}
|
|
|
|
%\begin{gather}
|
|
% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j}
|
|
% \\
|
|
% \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta}
|
|
% \\
|
|
% \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta}
|
|
%\end{gather}
|
|
%
|
|
%\begin{gather}
|
|
% \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j}
|
|
% \\
|
|
% \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0
|
|
% \\
|
|
% \mel*{_k^c}{\hH}{_{ij}^{a\beta}} =
|
|
%\end{gather}
|
|
%
|
|
%\begin{gather}
|
|
% \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} =
|
|
% \\
|
|
% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} =
|
|
% \\
|
|
% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} =
|
|
%\end{gather}
|
|
%
|
|
%\begin{gather}
|
|
% \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} =
|
|
% \\
|
|
% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} =
|
|
% \\
|
|
% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} =
|
|
%\end{gather}
|
|
|
|
%----------------------------------------------------------------
|
|
\section{Computational details}
|
|
%----------------------------------------------------------------
|
|
In all the CI-F12 calculations presented below, we consider the following Slater-type correlation factor \cite{Tenno04a}
|
|
\begin{equation}
|
|
f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la},
|
|
\end{equation}
|
|
which is fitted using $N_\text{GG}$ Gaussian geminals for computational convenience, \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05} i.e.
|
|
\begin{equation}
|
|
\exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} d_\nu \exp( - \la_\nu r_{12}^2 ).
|
|
\end{equation}
|
|
The contraction coefficients $d_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study.
|
|
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.
|
|
|
|
%----------------------------------------------------------------
|
|
\section{Results}
|
|
%----------------------------------------------------------------
|
|
|
|
|
|
%%% TABLE 1 %%%
|
|
\begin{table}
|
|
\caption{
|
|
\label{tab:atoms}
|
|
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.
|
|
\alert{The corresponding cc-pVXZ\_OPTRI or cc-pCVXZ\_OPTRI auxiliary basis is used as CABS.}}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{lcdd}
|
|
Atom & X & \tabc{FCI-F12} & \tabc{FCI} \\
|
|
\hline
|
|
\ce{He} & D & & -2.887\,595 \fnm[1] \\
|
|
(cc-pVXZ) & T & & -2.900\,232 \fnm[1] \\
|
|
& Q & & -2.902\,411 \fnm[1] \\
|
|
& 5 & & -2.903\,152 \fnm[1] \\
|
|
& 6 & & -2.903\,432 \fnm[1] \\
|
|
& $\infty$ & & -2.903\,724 \fnm[2] \\
|
|
\hline
|
|
\ce{Li} & D & & -7.466\,025 (FCI) \\
|
|
(cc-pCVXZ) & T & & -7.474\,251 (FCI) \\
|
|
& Q & & -7.476\,373 (FCI) \\
|
|
& $\infty$ & & -7.478\,060 \fnm[3] \\
|
|
\hline
|
|
\ce{Be} & D & & -14.651\,833 (FCI) \\
|
|
(cc-pCVXZ) & T & & -14.662\,368 (FCI) \\
|
|
& Q & & -14.665\,566 (CIPSI) \\
|
|
& $\infty$ & & -14.667\,356 \fnm[4] \\
|
|
& $\infty$ & & -14.667\,39 (TOTO) \\
|
|
\hline
|
|
\ce{B} & D & & -24.619\,101 (FCI) \\
|
|
(cc-pwCVXZ) & T & & -24.643\,222 (CIPSI) \\
|
|
& Q & & -24.650\,331 (CIPSI) \\
|
|
& 5 & & -24.652\,309 (CIPSI) \\
|
|
& $\infty$ & & -24.653\,866 \fnm[5] \\
|
|
& $\infty$ & & -24.653\,90 (TOTO) \\
|
|
\hline
|
|
\ce{C} & D & & -37.792\,469 (FCI) \\
|
|
(cc-pwCVXZ) & T & & -37.829\,847 (CIPSI) \\
|
|
& Q & & -37.839\,816 (CIPSI) \\
|
|
& 5 & & -37.842\,731 (CIPSI) \\
|
|
& $\infty$ & & -37.840\,129 6 \\
|
|
& $\infty$ & & -37.845\,0 (TOTO) \\
|
|
\hline
|
|
\ce{N} & D & & -54.517\,650 (FCI) \\
|
|
(cc-pwCVXZ) & T & & -54.567\,764 (CIPSI) \\
|
|
& Q & & -54.581\,885 (CIPSI) \\
|
|
& 5 & & -54.585\,926 (CIPSI) \\
|
|
& $\infty$ & & -54.588\,917 \fnm[7] \\
|
|
& $\infty$ & & -54.589\,3 (TOTO) \\
|
|
\hline
|
|
\ce{O} & D & & -74.946\,393 (CIPSI) \\
|
|
(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
|
|
\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}}
|
|
\fnt[2]{Reference \onlinecite{Nakashima07}}
|
|
\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}}
|
|
\end{table}
|
|
%%%
|
|
|
|
%%% TABLE 2 %%%
|
|
\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}
|
|
|
|
%----------------------------------------------------------------
|
|
\section{Conclusion}
|
|
%----------------------------------------------------------------
|
|
We have introduced a dressed version of the well-established CI method to incorporate explicitly the correlation between electrons.
|
|
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.
|
|
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.
|
|
In particular, one eschew the computation of four-electron integrals as well as some types of three-electron integrals.
|
|
We believe that the present approach is a significant step towards the development of an accurate and efficient explicitly-correlated FCI method.
|
|
|
|
%----------------------------------------------------------------
|
|
\begin{acknowledgments}
|
|
The authors would like to thank the \emph{Centre National de la Recherche Scientifique} (CNRS) for funding.
|
|
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), and CALMIP (Toulouse) under allocation 2019-18005.
|
|
\end{acknowledgments}
|
|
%----------------------------------------------------------------
|
|
|
|
\bibliography{CI-F12,CI-F12-control}
|
|
|
|
\end{document}
|