We discuss the systematics of the $2^{+}$ excitation energy and the transition probability from this $2^{+}$ to the ground state for most of the even-even nuclei, from ${}^{16}\mathrm{O}$ up to the actinides, for which data are available. To that aim we calculate their correlated $J=0$ ground state and $J=2$ first excited state by means of the angular-momentum and particle-number projected generator coordinate method, using the axial mass quadrupole moment as the generator coordinate and self-consistent mean-field states only restricted by axial, parity, and time-reversal symmetries. The calculation, which is an extension of a previous systematic calculation of correlations in the ground state, is performed within the framework of a nonrelativistic self-consistent mean-field model using the same Skyrme interaction SLy4 and a density-dependent pairing force to generate the mean-field configurations and mix them. To separate the effects due to angular-momentum projection and those related to configuration mixing, the comparison with the experimental data is performed for the full calculation and also by taking a single configuration for each angular momentum, chosen to minimize the projected energy. The theoretical energies span more than 2 orders of magnitude, ranging below 100 keV in deformed actinide nuclei to a few MeV in doubly-magic nuclei. Both approaches systematically overpredict the experiment excitation energy, by an average factor of about 1.5. The dispersion around the average is significantly better in the configuration mixing approach compared to the single-configuration results, showing the improvement brought by the inclusion of a dispersion on the quadrupole moment in the collective wave function. Both methods do much better for the quadrupole properties; using the configuration mixing approach the mean error on the experimental $B(E2)$ values is only 50%. We discuss possible improvements of the theory that could be made by introducing other constraining fields.

• Received 8 December 2006

DOI:https://doi.org/10.1103/PhysRevC.75.044305