The main goal of quantum chemistry is to propose reliable theoretical tools to describe the rich area of chemistry.
The accurate computation of the electronic structure of molecular systems plays a central role in the development of methods in quantum chemistry,
but despite intense developments, no definitive solution to that problem have been found.
The theoretical challenge to be overcome falls back in the category of the quantum many-body problem due the intrinsic quantum nature
of the electrons and the coulomb repulsion between them, inducing the so-called electronic correlation problem.
Tackling this problem translate to solving the Schroedinger equation for a $N$~-~electron system, and two roads have emerged to approximate the solution to this formidably complex mathematical problem: the wave function theory (WFT) and density functional theory (DFT).
Although both WFT and DFT spring from the same problem, their formalisms are very different as the former deals with the complex
$N$~-~body wave function whereas the latter handles the much simpler one~-~body density.
The computational cost of DFT is very appealing as in its Kohn-Sham (KS) formulation it can be recast in a mean-field procedure.
Therefore, although constant efforts are performed to reduce the computational cost of WFT, DFT remains still the workhorse of quantum chemistry.
From the theoretician point of view, the complexity of description of a given chemical system can be roughly
categorized by the strength of the electronic correlation appearing in its electronic structure.
Weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by the avoidance effects when electron are near the electron coalescence point, which are often called short-range correlation effects,
or far, typically dispersion forces. The theoretical description of weakly correlated systems is one of the more concrete achievement
of quantum chemistry, and the main remaining issue for these systems is to push the limit in terms of the size of the chemical systems that can be treated.
The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibits
a much more exotic electronic structure.
Transition metals containing systems, low-spin open shell systems, covalent bond breaking or excited states
have all in common that they cannot be even qualitatively described by a single electronic configuration.
It is now clear that the usual approximations in KS-DFT fails in giving an accurate description of these situations and WFT has become
the standard for the treatment of strongly correlated systems.
From the theoretical point of view, the complexity of the strong correlation problem is, at least, two-fold:
i) the presence of near degeneracies and/or strong interactions among a primary set of electronic configurations
ii) the quantitative description of the systems must take into account weak correlation effects which requires to take into account many
other electronic configurations with typically much smaller weights in the wave function.
Fulfilling these two objectives is a rather complicated task, specially if one adds the requirement of size-extensivity and additivity of the computed energy in the case of non interacting fragments, which is a very desirable property for any approximated method.
To tackle this problem, many WFT methods have emerged which can be categorized in two branches: the single-reference (SR)
The advantage of these approaches rely on the rather straightforward way to improve the level of accuracy,
which consists in increasing the rank of the excitation operators used to generate the CC wave function.
Despite its appealing elegant simplicity, the computational cost of the CC methods increase drastically with the rank of the excitation
operators, even if alternative approaches have been proposed using stochastic techniques\cite{alex_thom,piotr} or symmetry-broken approaches\cite{scuseria}.
In the MR approaches, the zeroth order wave function consists in a linear combination of Slater determinants which are supposed to concentrate most of strong interactions and near degeneracies.
On top of this zeroth-order wave function, weak correlation is introduced by the addition of other configurations
A sensible advantage of WFT is its systematically improvable character to tend to the exact solution, which is the so-called full configuration interaction (FCI) in a complete basis set (CBS).
Such a path can be expressed in two ways which are quite independent one from another: i) improving the description of the wave function in terms of multiple excitations expansion ii) improving the quality of the one-particle basis set.
The theoretical framework of the basis set correction have been derived in details in \cite{GinPraFerAssSavTou-JCP-18}, so we recall briefly the main equations involved for the present study.
