Quantum rate theory and electron-transfer dynamics: A theoretical and experimental approach for quantum electrochemistry

Quantum rate theory is based on a first-principle quantum mechanical rate concept that comprises with the Planck–Einstein relationship E = h ν , where ν = e 2 / h C q is a frequency associated with the quantum capacitance C q and E = e 2 / C q is the energy associated with ν . For a single state mode of transmittance, e 2 / C q corresponds to the chemical potential differences Δ μ between donor and acceptor state levels comprising an electrochemical reaction. A statistical mechanic treatment of E is required to compute the contribution of the thermal dynamics at finite temperature. The Arrhenius equation for the temperature dependence of the reaction rate was obtained, as well as Marcus’s Arrhenius-type electron-transfer rate constant as a particular setting of the quantum rate ν . Consequently, this ν concept provides the quantum mechanical foundations for electrochemical reactions at room temperature. The present work also demonstrates that the electron-transfer rate of heterogeneous (diffusionless) reactions can be studied in detail within this theory by measuring C q using time-dependent electrochemical methods. Since the electron transfer follows a statistical mechanics version of the Planck–Einstein E = h ν relationship, the electrochemical reaction dynamics cannot be appropriately modeled using non-relativistic Schrödinger wave mechanics, which is the ongoing quantum approach to electrochemistry. Accordingly, a relativistic analysis that takes into account the spin dynamics of the electron is more appropriate. The latter assumption implies quantum electrodynamics within a particular quantum transport mode intrinsically coupled to the electron-transfer rate of electrochemical reactions that have not been considered thus far. Here it is demonstrated that the consideration of this inherent quantum transport is key to obtaining an in-depth understanding of the electron transfer phenomenon. Finally, the theory is validated through its description of electron transfer, quantum conductance, and capacitance in different electro-active molecular films.