Variational approximations to exact solutions in shell-model valence spaces: systematic calculations in the $sd$-shell

Abstract: 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 non-trivial 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.