PCET as an overlap problem Vibronic coupling and nonadiabatic proton-coupled electron transfer

Rate

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Donor–acceptor distance

Driving force and environment

Preset

Rate expression

k = \sum_{μ} P_{μ} \sum_{ν} \frac{{| V_{μ ν} |}^{2}}{ℏ} \sqrt{\frac{π}{λ k_{B} T}} \exp [\frac{- {({ΔG}_{μ ν} + λ)}^{2}}{4 λ k_{B} T}]

term	meaning	set by
$V_{μ ν}$	$V_{el} \cdot S_{μ ν}$	Distance R
${ΔG}_{μ ν}$	$ΔG ° + (E_{ν} - E_{μ})$	ΔG°
$λ$	reorganization energy	λ
$P_{μ}$	Boltzmann population	T

The double sum runs over the lowest four vibrational states on each side. Only the overlap S_μν — the shaded area above — depends on distance; pulling the donor and acceptor apart by a few tenths of an Ångström thins it toward zero and the rate falls by orders of magnitude. Deuterium, being more compact, loses its overlap faster: that is where the kinetic isotope effect comes from.

⟨ k ⟩ = \int k (R') G_{σ} (R' - R) d R'

Gating stands for thermal motion in the donor–acceptor distance itself — a slow heavy-atom vibration that stretches and compresses the hydrogen bond. Rather than evaluate k at one fixed R, this averages it over a Gaussian spread of R centred on the slider value, with standard deviation σ set by Gating amplitude. Because k(R) falls off so steeply, that average is dominated by the shortest, fastest instants in the spread — not by the mean R — which is why gating raises the rate and, by partly erasing the very R-dependence that separates the isotopes, collapses the KIE.

Illustrative parameters, not fitted to a specific system. The proton wells are analytic Morse potentials of fixed shape; the donor–acceptor distance only moves them apart. Electronic coupling V_el is held fixed at .