It has often been observed in the light-harvesting antenna system of bacterial photosynthesis that excitation-energy transfer (EET) takes place very rapidly to or from a nearly (or completely) optically forbidden state. So far, the rate constant of EET has usually been calculated by Forster's formula, which regards EET as arising from the overlap integral between the luminescence spectrum of the excitation donor and the absorption spectrum of its acceptor. The observed EETs are much faster than expected from this formula, since the transition dipole as a whole nearly (or completely) vanishes in this state, giving a very small (or completely vanishing) Forster's overlap integral. We note in these EETs that at least one of the donor and acceptor is an aggregate of chromophores. Individual chromophores comprising the aggregate retain nonvanishing transition dipoles, even if their sum nearly (or completely) vanishes at an exciton state which is nearly (or completely) optically forbidden therein. EET to (or from) the exciton state can arise due to interactions to (or from) individual transition dipoles in the aggregate when the distance between the donor and the acceptor is not much larger than the physical size of the aggregate. A new formula is proposed for calculating the rate constant of such an EET. It fills well the gap between theory and experiment mentioned above.