\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.''}
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}
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
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.
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.
Schematic representation of the various orbital spaces and their notation.
The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket*{_{ij}^{\alpha\beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).}
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}
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 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 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}).
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
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}
\alert{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 $H_{IA}\neq0$, $A$ 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}).:
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.
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.}
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}
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 full CI methods.
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.