A Hamiltonian model is formulated for electrochemical electron-transfer reactions involving frequency changes of the inner-sphere modes. The adiabatic free energy surface is calculated and it is shown that this nonlinear coupling leads to a transfer coefficient different from , and that the transfer coefficient can no longer be identified with the electron occupation probability in the transition state saddle point, as in the linear coupling case.
Schmickler, W., & Koper, M. T. M. (1999). Adiabatic electrochemical electron-transfer reactions involving frequency changes of inner-sphere modes. Electrochemistry Communications, 1(9), 402-405. https://doi.org/10.1016/S1388-2481(99)00087-9