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 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 \cite{Huron73, Evangelisti83}.
This method, involving effective Hamiltonian theory, has been shown to be successful in other scenarios \cite{Heully92}.
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}.
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.
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
As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various authors \cite{Pack66, Morgan93}), for small $r_{12}$, the two-electron correlation factor $f_{12}$ in Eq.~\eqref{eq:Ja} must behave as
It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F}=0$, which demonstrates that our dressed CI method becomes exact in the limit of a complete one-electron basis.
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{This yields the following linear system of equations
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}.
% 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).}
Compared to a conventional CI calculation, new matrix elements are required.
The simplest of them $f_{IJ}$ --- required in Eqs.~\eqref{eq:IHF} and \eqref{eq:tI} --- can be easily computed by applying Condon-Slater rules \cite{SzaboBook}.
They involve two-electron integrals over the 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}.
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}.
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}).
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{PT2}.
%The set $\mC$ is defined by two simple rules.
%First, because $f$ is a two-electron operator (and thanks to the matrix element $f_{AJ}$ in \eqref{eq:IHF-RI}), we know that the sum over $A$ is restricted to the singly- or doubly-excited determinants with respect to the determinant $\kJ$.
%Second, to ensure that $H_{IA} \neq 0$, $A$ must be connected to $\kI$, i.e.~differs from $\kI$ by no more than two spin orbitals.
%Three types of determinants have these two properties:
%i) the pure doubles $\ket*{_{ij}^{\alpha \beta}}$;
%ii) the mixed doubles $\ket*{_{ij}^{a \beta}}$;
%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.~$\mel{0}{\hH}{_{i}^{\alpha}}\neq0$.
Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero \cite{Kutzelnigg91}.
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:atoms}, we report the total atomic energy of the neutral atoms from $Z =2$ to $10$ for various Dunning's basis sets.
In all calculations, the associated OPTRI basis is used as CABS \cite{Yousaf08, Yousaf09}.
In Table \ref{tab:molecules}, we report the total energy of the \ce{H2}, \ce{F2} and \ce{H2O} molecules at experimental geometry \cite{Giner13, Giner15, Caffarel16}.
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.
