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We study the ability of variational approaches based on self-consistentmean-field and beyond-mean-field methods to reproduce exact energies andelectromagnetic properties of the nuclei defined within the $sd$-shell valencespace 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 exactsolutions {obtained by} the full diagonalization of the Hamiltonian. We analyzethe role played by the proton-neutron ($pn$) mixing as well as the quadrupoleand pairing degrees of freedom (including both isoscalar and isovectorchannels) in the description of the spectra of even-even, even-odd and odd-oddnuclei in the whole $sd$-shell.

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