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Manuscript/CI-F12.aux
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@ -22,34 +22,36 @@
\citation{Tenno12a,Tenno12b,Hattig12,Kong12}
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\@writefile{toc}{\contentsline {title}{Dressing the configuration interaction matrix with explicit correlation}{1}{section*.2}}
\@writefile{toc}{\contentsline {abstract}{Abstract}{1}{section*.1}}
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\bibcite{SzaboBook}{{37}{1989}{{Szabo\ and\ Ostlund}}{{}}}
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\bibcite{Klopper02}{{39}{2002}{{Klopper\ and\ Samson}}{{}}}
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\@writefile{lot}{\contentsline {table}{\numberline {I}{\ignorespaces FCI-F12, CIPSI and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pVXZ basis set. The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}}{3}{table.1}}
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@ -6,7 +6,7 @@
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Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
Date-Added = {2019-03-10 21:06:09 +0100},
Date-Modified = {2019-03-10 21:06:15 +0100},
Journal = {J. Chem. Theory Comput.},
Title = {Quantum Package 2.0: a open-source determinant-driven suite of programs},
Volume = {in press},
Year = {2019}}
@article{Scemama18a,
Author = {A. Scemama and Y. Garniron and M. Caffarel and P. F. Loos},
Date-Added = {2019-03-10 21:04:59 +0100},
Date-Modified = {2019-03-10 21:05:07 +0100},
Doi = {10.1021/acs.jctc.7b01250},
Journal = {J. Chem. Theory Comput.},
Pages = {1395},
Title = {Deterministic construction of nodal surfaces within quantum Monte Carlo: the case of FeS},
Volume = {14},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b01250}}
@article{Scemama18b,
Author = {Anthony Scemama and Anouar Benali and Denis Jacquemin and Michel Caffarel and Pierre-Fran{\c{c}}ois Loos},
Date-Added = {2019-03-10 21:04:59 +0100},
Date-Modified = {2019-03-10 21:05:12 +0100},
Doi = {10.1063/1.5041327},
Journal = {J. Chem. Phys.},
Month = {jul},
Number = {3},
Pages = {034108},
Publisher = {{AIP} Publishing},
Title = {Excitation energies from diffusion Monte Carlo using selected configuration interaction nodes},
Url = {https://doi.org/10.1063%2F1.5041327},
Volume = {149},
Year = 2018,
Bdsk-Url-1 = {https://doi.org/10.1063%2F1.5041327},
Bdsk-Url-2 = {https://doi.org/10.1063/1.5041327}}
@article{Veril_2018,
Author = {M. Veril and P. Romaniello and J. A. Berger and P. F. Loos},
Date-Added = {2019-03-10 21:04:59 +0100},
Date-Modified = {2019-03-10 21:04:59 +0100},
Doi = {10.1021/acs.jctc.8b00745},
Journal = {J. Chem. Theory Comput.},
Pages = {5220},
Title = {Unphysical Discontinuities in {{GW}} Methods},
Volume = {14},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00745}}
@article{Garniron18,
Author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos},
Date-Added = {2019-03-10 21:04:56 +0100},
Date-Modified = {2019-03-10 21:05:47 +0100},
Doi = {10.1063/1.5044503},
Journal = {J. Chem. Phys.},
Pages = {064103},
Title = {Selected Configuration Interaction Dressed by Perturbation},
Volume = {149},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}}
@article{Loos18a,
Author = {P. F. Loos and P. Romaniello and J. A. Berger},
Date-Added = {2019-03-10 21:04:56 +0100},
Date-Modified = {2019-03-10 21:05:22 +0100},
Doi = {10.1021/acs.jctc.8b00260},
Journal = {J. Chem. Theory Comput.},
Pages = {3071},
Title = {Green Functions and Self-Consistency: Insights From the Spherium Model},
Volume = {14},
Year = {2018},
Bdsk-Url-1 = {https://dx.doi.org/10.1021/acs.jctc.8b00260}}
@article{Loos18b,
Author = {Pierre-Fran{\c{c}}ois Loos and Anthony Scemama and Aymeric Blondel and Yann Garniron and Michel Caffarel and Denis Jacquemin},
Date-Added = {2019-03-10 21:04:56 +0100},
Date-Modified = {2019-03-10 21:05:31 +0100},
Doi = {10.1021/acs.jctc.8b00406},
Journal = {J. Chem. Theory Comput.},
Month = {jul},
Number = {8},
Pages = {4360--4379},
Publisher = {American Chemical Society ({ACS})},
Title = {A Mountaineering Strategy to Excited States: Highly Accurate Reference Energies and Benchmarks},
Url = {https://doi.org/10.1021%2Facs.jctc.8b00406},
Volume = {14},
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Bdsk-Url-1 = {https://doi.org/10.1021%2Facs.jctc.8b00406},
Bdsk-Url-2 = {https://doi.org/10.1021/acs.jctc.8b00406}}
@article{Loos19,
Author = {P. F. Loos and M. Boggio-Pasqua and A. Scemama and M. Caffarel and D. Jacquemin},
Date-Added = {2019-03-10 21:04:56 +0100},
Date-Modified = {2019-03-10 21:05:26 +0100},
Doi = {10.1021/acs.jctc.8b01205},
Journal = {J. Chem. Theory Comput.},
Pages = {in press},
Title = {Reference energies for double excitations},
Volume = {15},
Year = {2019},
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@article{Garniron17b,
Author = {Yann Garniron and Anthony Scemama and Pierre-Fran{\c{c}}ois Loos and Michel Caffarel},
Date-Added = {2019-03-10 21:04:41 +0100},
Date-Modified = {2019-03-10 21:05:44 +0100},
Doi = {10.1063/1.4992127},
Journal = {J. Chem. Phys.},
Month = {jul},
Number = {3},
Pages = {034101},
Publisher = {{AIP} Publishing},
Title = {Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory},
Url = {https://doi.org/10.1063%2F1.4992127},
Volume = {147},
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Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4992127},
Bdsk-Url-2 = {https://doi.org/10.1063/1.4992127}}
@article{PT2,
Author = {Y. Garniron and A. Scemama and P. F. Loos and M. Caffarel.},
Date-Added = {2017-07-21 19:57:06 +0000},

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\BOOKMARK [0][-]{section*.2}{Dressing the configuration interaction matrix with explicit correlation}{}% 2
\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1
\BOOKMARK [1][-]{section*.3}{Introduction}{section*.2}% 3
\BOOKMARK [1][-]{section*.4}{Ansatz}{section*.2}% 4
\BOOKMARK [1][-]{section*.5}{Dressing}{section*.2}% 5
\BOOKMARK [1][-]{section*.6}{Matrix elements}{section*.2}% 6
\BOOKMARK [1][-]{section*.7}{Conclusion}{section*.2}% 7
\BOOKMARK [1][-]{section*.8}{Acknowledgments}{section*.2}% 8

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@ -74,28 +74,28 @@ The performance of the newly-designed explicitly-correlated dressing CI method i
\maketitle
%----------------------------------------------------------------
\textit{Introduction.---}
\section{Introduction}
%----------------------------------------------------------------
One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
This problem was already spotted thirty years ago by Kutzelnigg \cite{Kutzelnigg85} who 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{Kutzelnigg91, Termath91, Klopper91a, Klopper91b, Noga94}.
This problem was already spotted thirty years ago by Kutzelnigg \cite{Kutzelnigg85} who 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{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}.
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 \cite{Huron73, Evangelisti83}.
This method, involving effective Hamiltonian theory, has been shown to be successful in other scenarios \cite{Heully92}.
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. \cite{Huron73, Evangelisti83}
This method, involving effective Hamiltonian theory, has been shown to be successful in other scenarios. \cite{Heully92, Garniron18}
Compared to other explicitly-correlated methods, this dressing strategy has the advantage of introducing the explicit correlation at a low computational cost.
The present explicitly-correlated dressing CI method is completely general and can be applied to any type of truncated, full, or even selected CI methods \cite{Giner13, Scemama13a, Scemama13b, Scemama14, Giner15, Caffarel16}.
The present explicitly-correlated dressed CI method is completely general and can be applied to any type of truncated, full, or even selected CI methods. \cite{Giner13, Scemama13a, Scemama13b, Scemama14, Giner15, Caffarel16, Loos18b, Loos19}
However, for the sake of generality, we will discuss here the dressing of the full CI (FCI) matrix.
%Here, we focus on systems well described by a single (reference) determinant $\kO$ assumed to be a Hartree-Fock (HF) determinant.
%The multireference version of the present method will be reported in a separate study.
Atomic units are used throughout.
%----------------------------------------------------------------
\textit{Ansatz.---}
\section{Ansatz}
%----------------------------------------------------------------
Inspired by a number of previous research \cite{Shiozaki11}, our electronic wave function ansatz $\ket{\Psi} = \kD + \kF$ is simply written as the sum of a ``conventional'' part
Inspired by a number of previous research, \cite{Shiozaki11} our electronic wave function ansatz $\ket{\Psi} = \kD + \kF$ is simply written as the sum of a ``conventional'' part
\begin{equation}
\label{eq:D}
\kD = \sum_{I} c_I \kI
@ -131,7 +131,7 @@ As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various
\end{equation}
%----------------------------------------------------------------
\textit{Dressing.---}
\section{Dressing}
%----------------------------------------------------------------
Our primary goal is to introduce the explicit correlation between electrons at low computational cost.
Therefore, assuming that $\hH \ket{\Psi} = E \Psi$, one can write, by projection over $\bra{I}$,
@ -159,7 +159,7 @@ It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} =
At this stage, two key comments are in order.
First, as one may have realized, the coefficients $t_I$ are unknown.
\alert{However, they can be set to ensure the $s$- and $p$-wave electron-electron cusp conditions (SP ansatz) \cite{Tenno04a}.}
\alert{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}
@ -171,7 +171,7 @@ which can be easily solved using standard linear algebra packages.}
Second, because Eq.~\eqref{eq:DrH} depends on the CI coefficient $c_I$, 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 $t_I$ 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.
For pathological cases, a DIIS-like procedure may be employed \cite{Pulay82}.
For pathological cases, a DIIS-like procedure may be employed. \cite{Pulay82}
%%% FIG 1 %%%
%\begin{figure}
@ -184,21 +184,21 @@ For pathological cases, a DIIS-like procedure may be employed \cite{Pulay82}.
%%% %%%
%----------------------------------------------------------------
\textit{Matrix elements.---}
\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}.
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 geminal 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}.
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 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}.
These are more problematic, as they involve the computation of numerous three-electron integrals over 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 explore here a different route.
We propose to compute them using the resolution of the identity (RI) approximation \cite{Kutzelnigg91, Klopper02, Valeev04, Werner07, Hattig12}, which requires 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}).
We propose to compute them using the resolution of the identity (RI) approximation, \cite{Kutzelnigg91, Klopper02, Valeev04, Werner07, Hattig12} which requires 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}
@ -217,7 +217,7 @@ Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eq
\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{PT2}.
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 $\kJ$.
@ -228,7 +228,7 @@ Because $f$ is a two-electron operator, the way to compute efficiently Eq.~\eqre
%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}.
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}
@ -269,12 +269,12 @@ In all the calculations presented below, we consider the following Slater-type c
\begin{equation}
f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la},
\end{equation}
which is fitted using $N_\text{GG}$ Gaussian geminals fo computational convenience \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}, i.e.
which is fitted using $N_\text{GG}$ Gaussian geminals fo computational convenience, \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05} i.e.
\begin{equation}
\exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} a_\nu \exp( - \la_\nu r_{12}^2 ).
\end{equation}
The coefficients $a_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study.
All the calculations have been performed with Quantum Package \cite{QP}.
All the calculations have been performed with Quantum Package. \cite{Garniron19}
%%% TABLE 1 %%%
\begin{table}
@ -283,68 +283,68 @@ All the calculations have been performed with Quantum Package \cite{QP}.
FCI-F12, CIPSI and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pVXZ basis set.
The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
\begin{ruledtabular}
\begin{tabular}{lcccd}
Atom & $N$ & FCI-F12 & CIPSI & \text{FCI} \\
\begin{tabular}{lcdd}
Atom & $N$ & \mcc{FCI-F12} & \mcc{FCI} \\
\hline
\ce{He} & D & & & -2.887\,595 \footnotemark[1] \\
(cc-pV$N$Z) & T & & & -2.900\,232 \footnotemark[1] \\
& Q & & & -2.902\,411 \footnotemark[1] \\
& 5 & & & -2.903\,152 \footnotemark[1] \\
& 6 & & & -2.903\,432 \footnotemark[1] \\
& $\infty$ & & & -2.903\,724 \footnotemark[2] \\
\ce{He} & D & & -2.887\,595 \footnotemark[1] \\
(cc-pV$N$Z) & T & & -2.900\,232 \footnotemark[1] \\
& Q & & -2.902\,411 \footnotemark[1] \\
& 5 & & -2.903\,152 \footnotemark[1] \\
& 6 & & -2.903\,432 \footnotemark[1] \\
& $\infty$ & & -2.903\,724 \footnotemark[2] \\
\hline
\ce{Li} & D & & & -7.466\,025 (FCI) \\
(cc-pCV$N$Z) & T & & & -7.474\,251 (FCI) \\
& Q & & & -7.476\,373 (FCI) \\
& $\infty$ & & & -7.478\,060 \footnotemark[3] \\
\ce{Li} & D & & -7.466\,025 (FCI) \\
(cc-pCV$N$Z) & T & & -7.474\,251 (FCI) \\
& Q & & -7.476\,373 (FCI) \\
& $\infty$ & & -7.478\,060 \footnotemark[3] \\
\hline
\ce{Be} & D & & & -14.651\,833 (FCI) \\
(cc-pCV$N$Z) & T & & & -14.662\,368 (FCI) \\
& Q & & & -14.665\,566 (CIPSI) \\
& $\infty$ & & & -14.667\,356 \footnotemark[4] \\
& $\infty$ & & & -14.667\,39 (TOTO) \\
\ce{Be} & D & & -14.651\,833 (FCI) \\
(cc-pCV$N$Z) & T & & -14.662\,368 (FCI) \\
& Q & & -14.665\,566 (CIPSI) \\
& $\infty$ & & -14.667\,356 \footnotemark[4] \\
& $\infty$ & & -14.667\,39 (TOTO) \\
\hline
\ce{B} & D & & & -24.619\,101 (FCI) \\
(cc-pwCV$N$Z) & T & & & -24.643\,222 (CIPSI) \\
& Q & & & -24.650\,331 (CIPSI) \\
& 5 & & & -24.652\,309 (CIPSI) \\
& $\infty$ & & & -24.653\,866 \footnotemark[5] \\
& $\infty$ & & & -24.653\,90 (TOTO) \\
\ce{B} & D & & -24.619\,101 (FCI) \\
(cc-pwCV$N$Z) & T & & -24.643\,222 (CIPSI) \\
& Q & & -24.650\,331 (CIPSI) \\
& 5 & & -24.652\,309 (CIPSI) \\
& $\infty$ & & -24.653\,866 \footnotemark[5] \\
& $\infty$ & & -24.653\,90 (TOTO) \\
\hline
\ce{C} & D & & & -37.792\,469 (FCI) \\
(cc-pwCV$N$Z) & 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) \\
\ce{C} & D & & -37.792\,469 (FCI) \\
(cc-pwCV$N$Z) & 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-pwCV$N$Z) & T & & & \\
& Q & & & \\
& 5 & & & -54.585\,926 (CIPSI) \\
& $\infty$ & & & -54.588\,917 \footnotemark[7] \\
& $\infty$ & & & -54.589\,3 (TOTO) \\
\ce{N} & D & & -54.517\,650 (FCI) \\
(cc-pwCV$N$Z) & T & & \\
& Q & & \\
& 5 & & -54.585\,926 (CIPSI) \\
& $\infty$ & & -54.588\,917 \footnotemark[7] \\
& $\infty$ & & -54.589\,3 (TOTO) \\
\hline
\ce{O} & D & & & \\
(cc-pwCV$N$Z) & T & & & \\
& Q & & & -75.054\,737 (CIPSI) \\
& 5 & & & -75.062\,002 (CIPSI) \\
& $\infty$ & & & -75.066\,892 \footnotemark[7] \\
& $\infty$ & & & -75.067\,4 (TOTO) \\
\ce{O} & D & & \\
(cc-pwCV$N$Z) & T & & \\
& Q & & -75.054\,737 (CIPSI) \\
& 5 & & -75.062\,002 (CIPSI) \\
& $\infty$ & & -75.066\,892 \footnotemark[7] \\
& $\infty$ & & -75.067\,4 (TOTO) \\
\hline
\ce{F} & D & & & -99.566\,902 (CIPSI) \\
(cc-pwCV$N$Z) & T & & & -99.682\,616 (CIPSI) \\
& Q & & & -99.715\,563 (CIPSI) \\
& 5 & & & -99.726\,249 (CIPSI) \\
& $\infty$ & & & -99.733\,424 \footnotemark[7] \\
& $\infty$ & & & -99.734\,1 (TOTO) \\
\ce{F} & D & & -99.566\,902 (CIPSI) \\
(cc-pwCV$N$Z) & T & & -99.682\,616 (CIPSI) \\
& Q & & -99.715\,563 (CIPSI) \\
& 5 & & -99.726\,249 (CIPSI) \\
& $\infty$ & & -99.733\,424 \footnotemark[7] \\
& $\infty$ & & -99.734\,1 (TOTO) \\
\hline
\ce{Ne} & D & & & \\
(cc-pwCV$N$Z) & T & & & \\
& Q & & & \\
& 5 & & & \\
& $\infty$ & & & -128.937\,274 \footnotemark[7] \\
& $\infty$ & & & -128.938\,3 (TOTO) \\
\ce{Ne} & D & & \\
(cc-pwCV$N$Z) & T & & \\
& Q & & \\
& 5 & & \\
& $\infty$ & & -128.937\,274 \footnotemark[7] \\
& $\infty$ & & -128.938\,3 (TOTO) \\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Reference \onlinecite{Kong12}}
@ -391,12 +391,12 @@ Molecule & cc-pVXZ & \mcc{CIPSI} & \mcc{FCI-F12} &
%%%
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 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}.
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}
%----------------------------------------------------------------
\textit{Conclusion.---}
\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.
@ -404,8 +404,10 @@ In particular, one eschew the computation of four-electron integrals as well as
We believe that the present approach is a significant step towards the development of an accurate and efficient explicitly-correlated full CI methods.
%----------------------------------------------------------------
\textit{Acknowledgments.---}
This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2016-0510 and from GENCI-TGCC (Grant 2016-08s015).
\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 allocations 2018-0510, 2018-18005 and 2019-18005.
\end{acknowledgments}
%----------------------------------------------------------------
\bibliography{CI-F12}