where ∇ is the average origin shift across all oscillators. To investigate the validity of the expected linearily of Equation , Cynwat & Frank fitted experimental internal conversion measurements for various carotenoids species, and found a high fidelity exponential trend (linear in the case of ln (W )), with a negative gradient of 0.755 cm -1 s -1 . Further, this trend was able to predict the rates of numerous other carotenoids. This was further experimentally validated by Maciejewski and co-workers . Further investigation by Andersson and coworkers , who examined the rate constants for various carotenes, found that multiple terms in Equation 30 could be considered as constants. In addition to the pre-exponential terms, E λ was found to depend weakly on the length of carotene chain. Indeed, the rate of internal conversion was found to increase by a factor of ∼300 when the chain length was increased from five to eleven, implying some degree of invariance with respect to the former exponential term. Further it was highlighted that coupling for the ployacene series that the coupling was similar, and therefore insentive to the size of the chromophore . In the limit of the natural logarithm, it is the former exponential term that is dominant. Following, a simplified form of Equation 30 can be cast :