Nonlinear free energy relations for adiabatic proton transfer reactions in a polar environment. II. Inclusion of the hydrogen bond vibration

A free energy relationship (FER) between the activation free energy DeltaG(double dagger) and the reaction asymmetry DeltaG(RXN) was derived in the preceding paper for acid ionization proton transfer (PT) reactions in a polar environment, in which the proton is treated quantum mechanically, but does not tunnel. In the present paper, the inclusion of the proton donor-acceptor vibration-the vibration of the hydrogen (H-) bond-and its impact on the FER are analyzed. The structure of the resulting FER, which includes quantization of both the proton and the H-bond coordinates, is found to be identical to that for the fixed donor-acceptor case, but with a re-interpretation for certain components, which reflects a significant coupling that exists between the H-bond vibration and the solvent reaction coordinate. This coupling derives from the increased mixing of the reactant and product valence bond electronic structures as the transition state is reached. Analytical expressions for the FER ingredients including these features are obtained. The present description of PT in an H-bund is compared with that of a bond energy-bond order characterization, which is sometimes employed in characterizing condensed phase PT systems. A comparison of the derived FER for PT is also made with the empirical Marcus FER and with other FERs in the literature.