The variational multiparticle-multihole configuration mixing approach (MPMH) to nuclei has been proposed about a decade ago. While the first applications followed rapidly, the implementation of the full formalism of this method has only been recently completed and applied in [C. Robin, N. Pillet, D. Pe~na Arteaga and J.-F. Berger, Phys. Rev. C 93, 024302 (2016)] to $^{12}$C as a test-case. The main objective of the present paper is to carry on the study that was initiated in that reference, in order to put the MPMH method to more stringent tests. To that aim we perform a systematic study of even-even sd-shell nuclei. The wave function of these nuclei is taken as a configuration mixing built on orbitals of the sd-shell, and both the mixing coefficients of the nuclear state and the single-particle wave functions are determined consistently from the same variational principle. The calculations are done using the D1S Gogny force. Various ground-state properties are analyzed. In particular, the correlation content and composition of the wave function as well as the single-particle orbitals and energies are examined. Binding energies and charge radii are also calculated and compared to experiment. The description of the first excited state is also examined and the corresponding transition densities are used as input for the calculation of inelastic electron and proton scattering. Special attention is paid to the effect of the optimization of the single-particle states consistently with the correlations of the system. Globally, the results are satisfying and encouraging. In particular, charge radii and excitation energies are nicely reproduced. However, the chosen valence-space truncation scheme precludes achieving maximum collectivity in the studied nuclei. Further refinement of the method and a better-suited interaction are necessary to remedy this situation.
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