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% addresses
\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}
\affiliation { \LCPQ }
\author { Michel Caffarel}
\affiliation { \LCPQ }
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\author { David P. Tew}
\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}
\affiliation { \LCPQ }
\begin { abstract}
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.
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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\section { Introduction}
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%----------------------------------------------------------------
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\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,
\\
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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.
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}
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}
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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.
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}
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However, for the sake of generality, we will discuss here the dressing of the full CI (FCI) matrix.
Atomic units are used throughout.
%----------------------------------------------------------------
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%\section{Ans{\"a}tz} : Ca prend un trema au pluriel seulement : Ans{\"a}tze
\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}
\begin { equation}
\label { eq:ansatz}
\ket { \Psi } = \kD + \kF
\end { equation}
is simply written as the sum of a \emph { ``conventional''} part
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\begin { equation}
\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}
\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
\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}
\label { eq:Ja}
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f = \sum _ { i < j} f_ { ij}
% f = \sum_{i < j} \gamma_{ij} f_{ij}
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\end { equation}
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is a (linear) 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}
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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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\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.
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}
where $ H _ { IJ } = \mel { I } { \hH } { J } $ .
Hence, we obtain the desired energy by diagonalizing the dressed Hamiltonian:
\begin { equation}
\label { eq:DrH}
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\oH _ { IJ} =
\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}
with
\begin { equation}
\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}
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 } $ .}
%\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).}
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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 $ .
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} .
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
\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}
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%%% FIG 1 %%%
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\begin { figure}
\includegraphics [width=\linewidth] { fig1}
\caption {
\label { fig:CBS}
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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%----------------------------------------------------------------
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.
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
\begin { equation}
\label { eq:RI}
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\hI = \sum _ { A \in \mA } \dyad { A} { A}
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\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}
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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}
\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 $ .
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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 $ .
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} ).:
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i) the pure doubles $ \hT _ { ij } ^ { \alpha \beta } \ket * { I } $ ,
ii) the mixed doubles $ \hT _ { ij } ^ { \alpha b } \ket * { I } $ , and
iii) the pure singles $ \hT _ { i } ^ { \alpha } \ket * { I } $ .
\alert {
The matrix element between two determinants differing by a double excitation $ \hT _ { ij } ^ { kl } $ is given by
\begin { equation}
\mel { I} { \hH f} { J} = \{ ij || kl \} - \sum _ m \{ ijm || mkl \} \Delta _ { mI} \Delta _ { mJ}
\end { equation}
where
\begin { equation}
\Delta _ { mI} = \mel { I} { a_ m^ \dagger a_ m} { I} ,
\end { equation}
\begin { equation}
\{ ijm || mkl \} = \sum _ { \alpha } \langle i j || \alpha m \rangle [ \alpha m || k l ]
+ \langle i j || m \alpha \rangle [ m \alpha || k l ],
\end { equation}
\begin { equation}
\{ ij || kl \} = \sum _ { \alpha \beta } \langle i j || \alpha \beta \rangle [ \alpha \beta || k l ] + \sum _ m \{ ijm || mkl \}
\end { equation}
The matrix element between two determinants differing by a single excitation $ \hT _ { i } ^ { k } $ is given by
\begin { equation}
\mel { I} { \hH f} { J} = \sum _ j \Delta _ { jI} \Delta _ { jJ} \qty ( \{ ij || kj \} - \sum _ m \{ ijm || mkj \} \Delta _ { mI} \Delta _ { mJ} )
\end { equation}
and the diagonal terms are
\begin { equation}
\mel { I} { \hH f} { I} = \sum _ { ij} \Delta _ { iI} \Delta _ { jI} \qty ( \{ ij || ij \} - \sum _ m \{ ijm || mij \} \Delta _ { mI} )
\end { equation}
}
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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 $ .
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}
% \\
% \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta}
% \\
% \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta}
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%\end{gather}
%
%\begin{gather}
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% \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}} =
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%\end{gather}
%
%\begin{gather}
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% \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} =
% \\
% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} =
% \\
% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} =
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%\end{gather}
%
%\begin{gather}
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% \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} =
% \\
% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} =
% \\
% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} =
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%\end{gather}
%----------------------------------------------------------------
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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.
%----------------------------------------------------------------
\section { Results}
%----------------------------------------------------------------
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%%% TABLE 1 %%%
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\begin { table}
\caption {
\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.
\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] \\
(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] \\
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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) \\
& 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) \\
& $ \infty $ & & -37.840\, 129 6 \\
& $ \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) \\
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& 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) \\
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& Q & & -75.054\, 737 (CIPSI) \\
& 5 & & -75.062\, 002 (CIPSI) \\
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& $ \infty $ & & -75.066\, 892 \fnm [7] \\
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& $ \infty $ & & -75.067\, 4 (TOTO) \\
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\hline
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\ce { F} & D & & -99.566\, 902 (CIPSI) \\
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(cc-pwCVXZ) & T & & -99.682\, 616 (CIPSI) \\
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& Q & & -99.715\, 563 (CIPSI) \\
& 5 & & -99.726\, 249 (CIPSI) \\
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& $ \infty $ & & -99.733\, 424 \fnm [7] \\
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& $ \infty $ & & -99.734\, 1 (TOTO) \\
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\hline
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\ce { Ne} & D & & -128.721\, 575 (CIPSI) \\
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(cc-pwCVXZ) & T & & -128.869\, 425 (CIPSI) \\
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& Q & & -128.913\, 064 (CIPSI) \\
& 5 & & -128.927\, 705 (CIPSI) \\
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& $ \infty $ & & -128.937\, 274 \fnm [7] \\
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& $ \infty $ & & -128.938\, 3 (TOTO) \\
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\end { tabular}
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\end { ruledtabular}
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\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} }
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\end { table}
%%%
%%% TABLE 2 %%%
\begin { table*}
\caption {
\label { tab:molecules}
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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.
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The corresponding cc-pVXZ\_ OPTRI auxiliary basis is used as CABS.}
\begin { ruledtabular}
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\begin { tabular} { lcdddd}
Molecule & X & \tabc { CIPSI} & \tabc { FCI-F12} & \tabc { i-FCIQMC} & \tabc { FCI} \\
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\hline
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\ce { H2} & D & & & \\
(cc-pVXZ) & T & & & \\
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& Q & & & \\
& 5 & & & \\
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& $ \infty $ & & & & -1.174\, 476\fnm [1] \\
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\hline
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\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] \\
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\hline
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\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] \\
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\end { tabular}
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\end { ruledtabular}
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\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} }
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\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.
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In all calculations, the associated OPTRI basis is used as CABS. \cite { Yousaf08, Yousaf09}
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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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%----------------------------------------------------------------
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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.
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}
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}