Modeling voltammetry curves for proton coupled electron transfer: The importance of nuclear quantum effects

Abstract

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.

Nonadiabatic Dynamics at Metal Surfaces: Fewest Switches Surface Hopping with Electronic Relaxation

“…Finally, in Figure , we compare the electron transfer rate predicted by FSSH-ER as a function of Γ with three other methods: (1) Marcus theory, which is valid in the small Γ limit; (2) TST, which is valid in the large Γ limit; and (3) BCME, which interpolates between both limits. These results have been previously reported in ref . To test the FSSH-ER method above, we perform a simulation with all trajectories initialized as the ground state of the left well and subject to an external nuclear friction γ n = 2 m ω (and the corresponding random force that obeys the fluctuation–dissipation theorem).…”

“…In order to go beyond this limitation, the BCME extrapolates the motion from raw diabatic PESs to “broadened” diabatic PESs so that, altogether, nuclear motion recovers the proper PMF (for Γ < kT or Γ > kT ). A schematic diagram of the broadened diabats is shown in Figure for the noninteracting Anderson–Holstein model, for which the BCME has been shown to yield accurate and efficient results. , Recently, the BCME has also been applied to model electrochemical problems . Now, at bottom, because the BCME is formulated in a modified molecular diabatic picture, such a construction has natural upsides and downsides.…”

“…By performing CME-like dynamics on broadened diabats, the BCME is able to work in the Γ > kT regime and recover the correctly broadened barrier between the well on the left and the well on the right; note that the correctly broadened PMF can be much lower than the unbroadened PMF at the crossing point. More details about the BCME can be found in refs ,,, .…”

“…88,89 Recently, the BCME has also been applied to model electrochemical problems. 90 Now, at bottom, because the BCME is formulated in a modified molecular diabatic picture, such a construction has natural upsides and downsides. The upside is that, by construction, the BCME is very inexpensive because it does not need to treat a large number of electronic states explicitly (as opposed to IESH where all one-electron eigenstates are explicitly involved).…”

Nonadiabatic dynamics at metal surfaces: fewest switches surface hopping with electronic relaxation

“…The BCME yields accurate and efficient results for the Anderson-Holstein model 90,91 and has been recently applied to electrochemical model problems. 92 However, at bottom, the BCME method is formulated in a (modified) molecular diabatic picture, which has both upsides and downsides. The upside is that, by construction, the BCME is very inexpensive because it does not need to treat a large number of electronic states explicitly (as opposed to IESH where all one-electron eigenstates are explicitly involved).…”