  In this paper we present an evolution of our derivation of the shell-model
effective Hamiltonian, namely introducing effects of three-body contributions.
More precisely, we consider a three-body potential at next-to-next-to-leading
order in chiral perturbation theory, and the induced three-body forces that
arise from many-body correlations among valence nucleons. The first one is
included, in the derivation of the effective Hamiltonian for one- and
two-valence nucleon-systems, at first order in the many-body perturbation
theory. Namely, we include only the three-body interaction between one or two
valence nucleons and those belonging to the core. For nuclei with more than two
valence particles, both induced - turned on by the two-body potential - and
genuine three-body forces come into play. Since it is difficult to perform
shell-model calculations with three-body forces, these contributions are
estimated for the ground-state energy only. In order to establish the
reliability of our approximations, we focus attention on nuclei belonging to
the p shell, aiming to benchmark our calculations against those performed with
the ab initio no-core shell-model. The obtained results are satisfactory, and
pave the way to the application of our approach to nuclear systems with heavier
masses.
