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Although self-consistent multi-configuration methods have been used fordecades to address the description of atomic and molecular many-body systems,only a few trials have been made in the context of nuclear structure. This workaims at the development of such an approach to describe in a unified wayvarious types of correlations in nuclei, in a self-consistent manner where themean-field is improved as correlations are introduced. The goal is to reconcilethe usually set apart Shell-Model and Self-Consistent Mean-Field methods. Thisapproach is referred as "variational multiparticle-multihole configurationmixing method". It is based on a double variational principle which yields aset of two coupled equations that determine at the same time the expansioncoefficients of the many-body wave function and the single particle states. Theformalism is derived and discussed in a general context, starting from athree-body Hamiltonian. Links to existing many-body techniques such as theformalism of Green's functions are established. First applications are doneusing the two-body D1S Gogny effective force. The numerical procedure is testedon the $^{12}$C nucleus in order to study the convergence features of thealgorithm in different contexts. Ground state properties as well assingle-particle quantities are analyzed, and the description of the first $2^+$state is examined. This study allows to validate our numerical algorithm andleads to encouraging results. In order to test the method further, we willrealize in the second article of this series, a systematic description of morenuclei and observables obtained by applying the newly-developed numericalprocedure with the same Gogny force. As raised in the present work,applications of the variational multiparticle-multihole configuration mixingmethod will however ultimately require the use of an extended and moreconstrained Gogny force.

physical review. c

Description of nuclear systems with a self-consistent  configuration-mixing approach. I: Theory, algorithm, and application to the  $^{12}$C test nucleus