Phonons in Molecular Quantum Dots: Density Functional Calculation of Franck-Condon Emission Rates in External Fields

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.