An Extension of the Marcus Equation for Atom Transfer Reactions

The Marcus equation for electron transfer has been widely applied to atom transfer reactions, but the equation does not seem to work well for very endothermic or very exothermic reactions. In this paper, a modified model is proposed. The modified model assumes that the potential energy surface can be written as a sum of the potentials for the individual molecules and an intermolecular potential that keeps the reactants apart. The activation barrier predicted by the model is within 3 kcal/mol of that predicted by the Marcus electron transfer equation when −1 ≤ Δ H r /4 E °≤ 1, where Δ H r is the heat of reaction and E ° is the intrinsic barrier. However, there are significant deviations when Δ H r /4 E ° < −1 and when Δ H r /4 E ° > 1. The modified model predicts that the activation barrier should equal Δ H r /4 E ° in the very endothermic limit, (i.e., Δ H r /4 E ° > 1), while the Marcus electron transfer equation predicts that the activation energy, E a, should diverge from Δ H r . Data shows that E a approaches Δ H r . The modified model predicts that the activation barrier goes to zero for very exothermic reactions, (i.e., Δ H r / E ° < −1) while the Marcus electron transfer equation predicts large barriers. Data shows, though, that the barriers approach zero. We also compare to the Marcus hyperbolic cosine expression and find that the modified model is within 3 kcal/mol of the Marcus hyperbolic cosine expression over the entire energy range. The modified model predicts that the barriers to reaction are associated with Pauli repulsions and not with bond stretching. That prediction agrees with recent ab initio calculations, and the VB model but not with the intersecting parabola model. Overall, the modified model seems to extend the original Marcus equation to very endothermic and very exothermic reactions. Also, it gives predictions similar to the Marcus hyperbolic cosine expression over the entire energy range.