where ρ (E mr -E ns ) is the density of states. For more complex systems, a Boltzmann weighting factor P r can be employed, however for many cases, the lack of is fine . Due to the lack of computational chemistry during early years of the methods' development, vibronic coupling could not be calculated directly, however, at low temperatures, the rate is maximised when the energy gap and rigidity parameters are nearly equal . Instead, it was approximated from the promoting mode (strong participating mode) frequency . Lin took the derivation further; assuming a promotingaccepting mode paradigm, Equation 12 can be recast in terms of the overlap between these two modes K j and G j ′ :