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\@writefile{toc}{\contentsline {abstract}{Abstract}{1}{section*.1}} \@writefile{toc}{\contentsline {abstract}{Abstract}{1}{section*.1}}
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\bibcite{Shiozaki11}{{27}{2011}{{Shiozaki, Knizia,\ and\ Werner}}{{}}} \bibcite{Loos19}{{29}{2019}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}}
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\bibcite{IntF12}{{48}{2017}{{Barca\ and\ Loos}}{{}}} \bibcite{4eRR}{{50}{ress}{{Barca\ and\ Loos}}{{}}}
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\bibcite{Sharkey10}{{57}{2010}{{Sharkey, Bubin,\ and\ Adamowicz}}{{}}} \bibcite{Puchalski09}{{57}{2009}{{Puchalski, Kedziera,\ and\ Pachucki}}{{}}}
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@ -6,7 +6,7 @@
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@article{Garniron19,
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},
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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},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}}
@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},
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Year = 2017,
Bdsk-Url-1 = {https://doi.org/10.1063%2F1.4992127},
Bdsk-Url-2 = {https://doi.org/10.1063/1.4992127}}
@article{PT2, @article{PT2,
Author = {Y. Garniron and A. Scemama and P. F. Loos and M. Caffarel.}, Author = {Y. Garniron and A. Scemama and P. F. Loos and M. Caffarel.},
Date-Added = {2017-07-21 19:57:06 +0000}, 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 [0][-]{section*.2}{Dressing the configuration interaction matrix with explicit correlation}{}% 2
\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1 \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 \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. 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 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}. 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}. 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}. 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}. 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}. 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. 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. 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. %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. %The multireference version of the present method will be reported in a separate study.
Atomic units are used throughout. 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} \begin{equation}
\label{eq:D} \label{eq:D}
\kD = \sum_{I} c_I \kI \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} \end{equation}
%---------------------------------------------------------------- %----------------------------------------------------------------
\textit{Dressing.---} \section{Dressing}
%---------------------------------------------------------------- %----------------------------------------------------------------
Our primary goal is to introduce the explicit correlation between electrons at low computational cost. 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 $\bra{I}$, 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. At this stage, two key comments are in order.
First, as one may have realized, the coefficients $t_I$ are unknown. 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 \alert{This yields the following linear system of equations
\begin{equation} \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 $\oH$. 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}]. 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. 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 %%% %%% FIG 1 %%%
%\begin{figure} %\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. 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 $f_{12}$. 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. 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 $\mel{I}{\hH f}{ J}$. 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}. 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}. 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. 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). 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}). 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 In the CBS, one can write
\begin{equation} \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{split}
\end{equation} \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 $\mC = \mA \setminus \mD$. 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. %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 $\kJ$. %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 $\ket*{_{i}^{\alpha}}$. %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.~$\mel{0}{\hH}{_{i}^{\alpha}} \neq 0$. 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} %\begin{gather}
% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j} % \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} \begin{equation}
f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la}, f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la},
\end{equation} \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} \begin{equation}
\exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} a_\nu \exp( - \la_\nu r_{12}^2 ). \exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} a_\nu \exp( - \la_\nu r_{12}^2 ).
\end{equation} \end{equation}
The coefficients $a_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study. 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 %%% %%% TABLE 1 %%%
\begin{table} \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. 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.} The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lcccd} \begin{tabular}{lcdd}
Atom & $N$ & FCI-F12 & CIPSI & \text{FCI} \\ Atom & $N$ & \mcc{FCI-F12} & \mcc{FCI} \\
\hline \hline
\ce{He} & D & & & -2.887\,595 \footnotemark[1] \\ \ce{He} & D & & -2.887\,595 \footnotemark[1] \\
(cc-pV$N$Z) & T & & & -2.900\,232 \footnotemark[1] \\ (cc-pV$N$Z) & T & & -2.900\,232 \footnotemark[1] \\
& Q & & & -2.902\,411 \footnotemark[1] \\ & Q & & -2.902\,411 \footnotemark[1] \\
& 5 & & & -2.903\,152 \footnotemark[1] \\ & 5 & & -2.903\,152 \footnotemark[1] \\
& 6 & & & -2.903\,432 \footnotemark[1] \\ & 6 & & -2.903\,432 \footnotemark[1] \\
& $\infty$ & & & -2.903\,724 \footnotemark[2] \\ & $\infty$ & & -2.903\,724 \footnotemark[2] \\
\hline \hline
\ce{Li} & D & & & -7.466\,025 (FCI) \\ \ce{Li} & D & & -7.466\,025 (FCI) \\
(cc-pCV$N$Z) & T & & & -7.474\,251 (FCI) \\ (cc-pCV$N$Z) & T & & -7.474\,251 (FCI) \\
& Q & & & -7.476\,373 (FCI) \\ & Q & & -7.476\,373 (FCI) \\
& $\infty$ & & & -7.478\,060 \footnotemark[3] \\ & $\infty$ & & -7.478\,060 \footnotemark[3] \\
\hline \hline
\ce{Be} & D & & & -14.651\,833 (FCI) \\ \ce{Be} & D & & -14.651\,833 (FCI) \\
(cc-pCV$N$Z) & T & & & -14.662\,368 (FCI) \\ (cc-pCV$N$Z) & T & & -14.662\,368 (FCI) \\
& Q & & & -14.665\,566 (CIPSI) \\ & Q & & -14.665\,566 (CIPSI) \\
& $\infty$ & & & -14.667\,356 \footnotemark[4] \\ & $\infty$ & & -14.667\,356 \footnotemark[4] \\
& $\infty$ & & & -14.667\,39 (TOTO) \\ & $\infty$ & & -14.667\,39 (TOTO) \\
\hline \hline
\ce{B} & D & & & -24.619\,101 (FCI) \\ \ce{B} & D & & -24.619\,101 (FCI) \\
(cc-pwCV$N$Z) & T & & & -24.643\,222 (CIPSI) \\ (cc-pwCV$N$Z) & T & & -24.643\,222 (CIPSI) \\
& Q & & & -24.650\,331 (CIPSI) \\ & Q & & -24.650\,331 (CIPSI) \\
& 5 & & & -24.652\,309 (CIPSI) \\ & 5 & & -24.652\,309 (CIPSI) \\
& $\infty$ & & & -24.653\,866 \footnotemark[5] \\ & $\infty$ & & -24.653\,866 \footnotemark[5] \\
& $\infty$ & & & -24.653\,90 (TOTO) \\ & $\infty$ & & -24.653\,90 (TOTO) \\
\hline \hline
\ce{C} & D & & & -37.792\,469 (FCI) \\ \ce{C} & D & & -37.792\,469 (FCI) \\
(cc-pwCV$N$Z) & T & & & -37.829\,847 (CIPSI) \\ (cc-pwCV$N$Z) & T & & -37.829\,847 (CIPSI) \\
& Q & & & -37.839\,816 (CIPSI) \\ & Q & & -37.839\,816 (CIPSI) \\
& 5 & & & -37.842\,731 (CIPSI) \\ & 5 & & -37.842\,731 (CIPSI) \\
& $\infty$ & & & -37.840\,129 6 \\ & $\infty$ & & -37.840\,129 6 \\
& $\infty$ & & & -37.845\,0 (TOTO) \\ & $\infty$ & & -37.845\,0 (TOTO) \\
\hline \hline
\ce{N} & D & & & -54.517\,650 (FCI) \\ \ce{N} & D & & -54.517\,650 (FCI) \\
(cc-pwCV$N$Z) & T & & & \\ (cc-pwCV$N$Z) & T & & \\
& Q & & & \\ & Q & & \\
& 5 & & & -54.585\,926 (CIPSI) \\ & 5 & & -54.585\,926 (CIPSI) \\
& $\infty$ & & & -54.588\,917 \footnotemark[7] \\ & $\infty$ & & -54.588\,917 \footnotemark[7] \\
& $\infty$ & & & -54.589\,3 (TOTO) \\ & $\infty$ & & -54.589\,3 (TOTO) \\
\hline \hline
\ce{O} & D & & & \\ \ce{O} & D & & \\
(cc-pwCV$N$Z) & T & & & \\ (cc-pwCV$N$Z) & T & & \\
& Q & & & -75.054\,737 (CIPSI) \\ & Q & & -75.054\,737 (CIPSI) \\
& 5 & & & -75.062\,002 (CIPSI) \\ & 5 & & -75.062\,002 (CIPSI) \\
& $\infty$ & & & -75.066\,892 \footnotemark[7] \\ & $\infty$ & & -75.066\,892 \footnotemark[7] \\
& $\infty$ & & & -75.067\,4 (TOTO) \\ & $\infty$ & & -75.067\,4 (TOTO) \\
\hline \hline
\ce{F} & D & & & -99.566\,902 (CIPSI) \\ \ce{F} & D & & -99.566\,902 (CIPSI) \\
(cc-pwCV$N$Z) & T & & & -99.682\,616 (CIPSI) \\ (cc-pwCV$N$Z) & T & & -99.682\,616 (CIPSI) \\
& Q & & & -99.715\,563 (CIPSI) \\ & Q & & -99.715\,563 (CIPSI) \\
& 5 & & & -99.726\,249 (CIPSI) \\ & 5 & & -99.726\,249 (CIPSI) \\
& $\infty$ & & & -99.733\,424 \footnotemark[7] \\ & $\infty$ & & -99.733\,424 \footnotemark[7] \\
& $\infty$ & & & -99.734\,1 (TOTO) \\ & $\infty$ & & -99.734\,1 (TOTO) \\
\hline \hline
\ce{Ne} & D & & & \\ \ce{Ne} & D & & \\
(cc-pwCV$N$Z) & T & & & \\ (cc-pwCV$N$Z) & T & & \\
& Q & & & \\ & Q & & \\
& 5 & & & \\ & 5 & & \\
& $\infty$ & & & -128.937\,274 \footnotemark[7] \\ & $\infty$ & & -128.937\,274 \footnotemark[7] \\
& $\infty$ & & & -128.938\,3 (TOTO) \\ & $\infty$ & & -128.938\,3 (TOTO) \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\footnotetext[1]{Reference \onlinecite{Kong12}} \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 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 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. 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. 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.---} \begin{acknowledgments}
This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2016-0510 and from GENCI-TGCC (Grant 2016-08s015). 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} \bibliography{CI-F12}