Variational approximations to exact solutions in shell-model valence spaces: Systematic calculations in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:math> shell

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