A free energy relationship (FER) between the activation free energy Δ G ⧧ and the reaction asymmetry Δ G 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-bond 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.

A quadratic free energy relationship (FER) between the kinetic activation free energy Δ G ⧧ and the thermodynamic reaction asymmetry Δ G RXN is derived for acid-base ionization proton-transfer reactions AH···B→ A-···HB+ in a polar environment in the proton adiabatic regime, in which the proton is treated quantum mechanically, but does not tunnel. The description differs from traditional treatments in both the proton quantization and the identification of a solvent coordinate as the reaction coordinate. The key coefficients in the FER are analyzed analytically for the simplified case, where the proton donor−acceptor distance is held fixed (a restriction removed in the following paper). In particular, the intrinsic barrier is shown to be the sum of an intrinsic solvent barrier, largely determined by solvent reorganization, and the zero point energy difference of the proton between the reactant and the transition state in a solvent coordinate. The Brønsted coefficient is related to the quantum proton-averaged solute electronic structure at, and the position of, this transition state along this reaction coordinate. Similarities and differences of the FER with the well-known Marcus relation are discussed.