  We report the calculation of various phonon overlaps and their corresponding
phonon emission probabilities for the problem of an electron tunneling onto and
off of the buckyball-dimer molecular quantum dot $C_{72}$, both with and
without the influence of an external field. We show that the stretch mode of
the two balls of the dumbbell couples most strongly to the electronic
transition, and in turn that a field in the direction of the bond between the
two $C_{36}$ balls is most effective at further increasing the phonon emission
into the stretch mode. As the field is increased, phonon emission increases in
probability with an accompanying decrease in probability of the dot remaining
in the ground vibrational state. We also present a simple model to gauge the
effect of molecular size on the phonon emission of molecules similar to our
$C_{72}$ molecule, including the experimentally tested $C_{140}$. In our
approach we do not assume that the hessians of the molecule are identical for
different charge states. Our treatment is hence a generalization of the
traditional phonon overlap calculations for coupled electron-photon transition
in solids.
