  Background: Time-dependent techniques in nuclear theory often rely on
mean-field or Hartree-Fock descriptions. Beyond mean-field dynamical
calculations within the time-dependent density matrix (TDDM) theory have often
invoked symmetry restrictions and ignored the connection between the mean-field
and the induced interaction. Purpose: We study the ground states obtained in a
TDDM approach for nuclei from $A=12$ to $A=24$, including examples of even and
odd-even nuclei with and without intrinsic deformation. We overcome previous
limitations using three-dimensional simulations and employ density-independent
Skyrme interactions self-consistently. Methods: The correlated ground states
are found starting from the Hartree-Fock solution, by adiabatically including
the beyond-mean-field terms in real time. Results: We find that, within this
approach, correlations are responsible for $\approx 4-5 \%$ of the total
energy. Radii are generally unaffected by the introduction of beyond mean-field
correlations. Large nuclear correlation entropies are associated to large
correlation energies. By all measures, $^{12}$C is the most correlated isotope
in the mass region considered. Conclusions: Our work is the starting point of a
consistent implementation of the TDDM technique for applications into nuclear
reactions. Our results indicate that correlation effects in structure are
small, but beyond-mean-field dynamical simulations could provide new insight
into several issues of interest.
