  Good many-body methods for medium and heavy nuclei are important. Here we
combine ideas from standard generator-coordinate methods (GCM) and the
so-called Monte Carlo shell model, and set forth a novel approach: starting
from a mean-field solution (Hartree-Fock-Bogoliubov), we create a set of
non-orthogonal basis states, by using low-lying vibrational
quasiparticleTamm-Dancoff modes, and then project onto states of good angular
momentum and particle number. The results we benchmark against full shell model
calculations. Even with just a few such modes we find improvement over standard
GCM calculations in excitation spectra. We also find significant improvement in
$0\nu\beta\beta$ nuclear matrix elements.
