We investigate rates of proton-coupled electron transfer (PCET) in potential sweep experiments for a generalized Anderson–Holstein model with the inclusion of a quantized proton coordinate. To model this system, we utilize a quantum classical Liouville equation embedded inside of a classical master equation, which can be solved approximately with a recently developed algorithm combining diffusional effects and surface hopping between electronic states. We find that the addition of nuclear quantum effects through the proton coordinate can yield quantitatively (but not qualitatively) different IV curves under a potential sweep compared to electron transfer (ET). Additionally, we find that kinetic isotope effects give rise to a shift in the peak potential, but not the peak current, which would allow for quantification of whether an electrochemical ET event is proton-coupled or not. These findings suggest that it will be very difficult to completely understand coupled nuclear–electronic effects in electrochemical voltammetry experiments using only IV curves, and new experimental techniques will be needed to draw inferences about the nature of electrochemical PCET.
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Hsys can also be expressed in second quantized notation as , where are annihilation (creation) operators for the proton vibrational state μ and is the identity operator.
Of course, Wk must depend strongly on x, and it is for this reason that we allow hopping only at the x = 0 grid point. Moreover, in Ref. 25, it was shown that, as long as the coupling decays quickly with distance from the electrode (e.g. exponentially), we can incorporate an x-dependence in the coupling by integrating over the relevant hopping regime and redefining the hopping rate at the first grid point in x. In practice, for any reasonably spaced uniform grid (which is usually on the order of microns), all relevant hopping should be only at the first grid point.
We note that recent work by Matyushov and Newton49 has shown that nonergodic effects due to a slowly relaxing medium (of comparable order to the reaction time) can play a significant role in electrochemical ET by reducing the solvent reorganization energy below its thermodynamic limit from Marcus theory. This effect is mainly noticeable in electrochemical ET due to the multiple orders of magnitude in rates covered over the course of the entire potential sweep. While we do not account for this effect in our framework, this observation provides an additional way that one could fine tune the above model by incorporating an overpotential dependence into λ.
For a more rigorous justification for this choice of electrode boundary condition, see Ref. 25.
