Proton-Coupled Electron Transfer and the "Linear Approximation" for Coupling to the Donor-Acceptor Distance Fluctuations.

The often-used "linear approximation" for treating the coupling of proton donor-acceptor (D-A) distance fluctuations to proton-coupled electron transfer tunneling reactions is systematically examined. The accuracy of this approximation is found to depend on the potential energy surfaces that are used to describe both the tunneling particle vibrations and the D-A coordinate probability distribution. Harmonic treatment of both the tunneling particle and the D-A coordinates results in a significant breakdown of the linear approximation when the width of the D-A distribution exceeds ∼0.05 Å. When a symmetric back-to-back Morse potential is used to describe the tunneling particle vibrations in the reactant and product states, the D-A distribution width can be expanded up to ∼0.09 Å before the rates calculated using the linear approximation exceed the exact result by an order of magnitude. Incorporation of a more realistic anharmonic D-A potential, based on quantum calculations, includes the important electronic D-A repulsion energy so that the sampling of short D-A distances is reduced. This approach improves the linear approximation as long as the D-A distribution has a width less than ∼0.1 Å. The conditions for the validity of the linear approximation are found to be more fragile when the calculated kinetic isotope effect (KIE) and its temperature dependence are also taken into account.