  We use the generator-coordinate method with realistic shell-model
interactions to closely approximate full shell-model calculations of the matrix
elements for the neutrinoless double-beta decay of $^{48}$Ca, $^{76}$Ge, and
$^{82}$Se. We work in one major shell for the first isotope, in the
$f_{5/2}pg_{9/2}$ space for the second and third, and finally in two major
shells for all three. Our coordinates include not only the usual axial
deformation parameter $\beta$, but also the triaxiality angle $\gamma$ and
neutron-proton pairing amplitudes. In the smaller model spaces our matrix
elements agree well with those of full shell-model diagonalization, suggesting
that our Hamiltonian-based GCM captures most of the important valence-space
correlations. In two major shells, where exact diagonalization is not currently
possible, our matrix elements are only slightly different from those in a
single shell.
