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We study the performance of self-consistent mean-field and beyond-mean-field approximations in shell-model valence spaces. In particular, Hartree-Fock-Bogolyubov, particle-number variation after projection and projected generator coordinate methods are applied to obtain ground-state and excitation energies for even-even and odd-even Calcium isotopes in the pf -shell. The standard (and non-trivial) KB3G nuclear effective interaction has been used. The comparison with the exact solutions -provided by the full diagonalization of the Hamiltonian-shows an outstanding agreement when particle-number and angular-momentum restorations are performed and both quadrupole and neutron-neutron pairing degrees of freedom are explicitly explored as collective coordinates.

We study the ability of variational approaches based on self-consistent mean-field and beyond-mean-field methods to reproduce exact energies and electromagnetic properties of the nuclei defined within the sd-shell valence space using the nontrivial USD Hamiltonian. In particular, Hartree-Fock-Bogoliubov (HFB), variation after particle-number projection (VAPNP), and projected generator coordinate methods (PGCM) are compared to exact solutions obtained by the full diagonalization of the Hamiltonian. We analyze the role played by the proton-neutron (pn) mixing as well as the quadrupole and pairing degrees of freedom (including both isoscalar and isovector channels) in the description of the spectra of even-even, even-odd, and odd-odd nuclei in the whole sd shell.

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