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

The union of rotational and vibrational modes in generator-coordinate-type calculations, with application to neutrinoless double-beta decay