A quantum electron-transfer rate theory uniformly valid in the electronic coupling, beyond the Condon, two-state, pathway-separability, and incoherence restrictions

The 2019 interpolation formula of Lawrence, Fletcher, Lindoy, and Manolopoulos combines a Fermi-golden-rule rate and a Born-Oppenheimer rate into a Padé-like approximant that predicts condensed-phase electron-transfer rates at arbitrary electronic coupling with an error no worse than that of its limiting theories. Its authors' conclusion flagged five open issues: the Marcus inverted regime, the Condon approximation, proton-coupled electron transfer, multiple electronic states (superexchange-mediated transfer), and an approach treating both the incoherent rate processes and the coherent electronic dynamics relevant to condensed-phase spectra. The inverted-regime issue has since been resolved by alternative golden-rule-based approaches, most completely Fay's optimal golden-rule theory of 2024, and the dividing-surface validity condition of the source's Eqs. (39)-(40) has been substantially reworked within nonadiabatic ring-polymer instanton theory. The surviving open problem is the remaining core: construct and validate a quantum-mechanical electron-transfer rate theory, uniformly valid in the electronic coupling, that removes the Condon approximation (coordinate-dependent diabatic coupling), the restriction to two electronic states (superexchange-mediated transfer), the separability of reaction pathways (proton-coupled electron transfer with competing pathways), and the restriction to incoherent rate processes (coherent electronic dynamics of condensed-phase spectra).

Background

Electron transfer — the movement of an electron between two chemical species or sites — is conventionally described in a two-state diabatic representation: nuclear motion is governed by potential-energy surfaces $V_0(\mathbf{q})$ and $V_1(\mathbf{q})$ on which the electronic state is held fixed (diabatic surfaces, as opposed to adiabatic surfaces on which the electronic state follows the nuclei), and the two states are coupled by an electronic coupling $\Delta$. Marcus theory, the classical limit of this picture, predicts rates from the crossing of parabolic free-energy surfaces and distinguishes a normal regime from an inverted regime, in which a large negative free-energy change raises rather than lowers the barrier. The quantum-mechanical rate has two well-understood limits. For weak coupling the reaction is nonadiabatic and the rate obeys Fermi's golden rule, $k_{\mathrm{GR}} \propto \Delta^2$; for strong coupling the Born-Oppenheimer approximation holds and the reaction proceeds as barrier crossing on the lower adiabatic surface with rate $k_{\mathrm{BO}}$. Between these limits no simple exact theory exists for condensed-phase reactions.

Nuclear quantum effects (tunnelling, zero-point energy) and solvent friction are captured in the golden-rule limit by Wolynes theory, an imaginary-time path-integral steepest-descent approximation to the golden-rule rate, and in the adiabatic limit by ring-polymer molecular dynamics (RPMD) rate theory, which combines discretized imaginary-time path-integral (ring-polymer) quantum statistics with classical real-time dynamics and gives a rate independent of the choice of dividing surface separating reactants from products. The standard benchmark is the spin-boson model — a two-state system linearly coupled to a harmonic solvent bath — for which the hierarchical equations of motion (HEOM) provide numerically exact reference rates.

Lawrence, Fletcher, Lindoy, and Manolopoulos (2019) showed that the exact rate at arbitrary coupling strength is well approximated by combining only the two limiting rates through the Padé-like interpolation formula

$$k_{\mathrm{IF}}(\Delta)=\frac{k_{\mathrm{GR}}(\Delta)\,k_{\mathrm{BO}}(\Delta)}{k_{\mathrm{GR}}(\Delta)+k_{\mathrm{BO}}(\Delta=0)}$$

which reduces to the golden-rule rate as $\beta\Delta\to 0$ and to the Born-Oppenheimer rate as $\beta\Delta\to\infty$ ($\beta$ is the inverse temperature), satisfies detailed balance, and — combined with Wolynes theory and RPMD — reproduced exact HEOM rates for a demanding set of spin-boson models with an error comparable to that of the limiting theories themselves. The formula's validity, however, rests on restrictive conditions: the diabatic coupling is taken constant (the Condon approximation); only two electronic states are treated; reactants and products must be equally well defined by a position-space dividing surface on the lower adiabatic surface and by diabatic state projectors, a condition tested only on spin-boson and one-dimensional scattering models; and only incoherent, Markovian rate processes are described. In the Marcus inverted regime no Born-Oppenheimer rate exists at all, so the formula cannot be applied there — an issue the authors flagged for alternative approaches, which have since been provided (most completely Fay's optimal golden-rule theory of 2024). The source paper closes by asking for the theoretical work needed to remove the remaining restrictions, and that request defines the problem stated below.

Problem Statement

The problem, as posed by the concluding section of Lawrence, Fletcher, Lindoy, and Manolopoulos (2019), is to develop and validate a quantum-mechanical electron-transfer rate theory that is uniformly valid in the electronic coupling strength — reducing to the correct Fermi-golden-rule rate for weak coupling and the correct Born-Oppenheimer rate for strong coupling — while removing its remaining restrictions:

1. Non-Condon coupling: the theory must remain accurate when the electronic coupling $\Delta(\mathbf{q})$ depends strongly on the nuclear coordinates, rather than assuming a constant $\Delta$.
2. Multiple electronic states: it must treat reactions involving more than two diabatic states, including superexchange (bridge-mediated) electron transfer, in which the perturbative structure of the rate changes — the leading term is fourth rather than second order in the coupling for a three-state bridge.
3. Non-separable pathways: it must treat reactions such as proton-coupled electron transfer, in which concerted and sequential pathways compete and cannot be decomposed into independent golden-rule and Born-Oppenheimer components.
4. Coherence: it must describe, within one framework, both the incoherent rate processes treated by current theories and the coherent electronic dynamics relevant to condensed-phase electronic spectra.

The source additionally notes that the interpolation formula's accuracy rests on the equivalence of the position-space dividing-surface and diabatic-projection definitions of reactants and products, a condition tested only on spin-boson and one-dimensional scattering models and not guaranteed on multidimensional and atomistic systems; its conclusion correspondingly describes applying the formula to more realistic models as an interesting direction for future work. The source presents the four restrictions above as several distinct open issues warranting further theoretical work toward a practically usable rate theory for charge-transfer reactions at arbitrary coupling. The four restrictions are independent sub-problems of this single objective — resolving one of them advances the objective without resolving it — and the objective is met only when one theory removes all four jointly while retaining uniform validity in the electronic coupling, so any such theory must be validated on systems that exercise each restriction (coordinate-dependent coupling, three or more electronic states, competing proton-coupled pathways, and coherent electronic dynamics) rather than on the two-state spin-boson benchmark alone. The Marcus inverted regime, flagged as open in the source, has been addressed since 2018-2024 by alternative golden-rule-based theories and is not part of the surviving core, although a fully general theory would ideally remain applicable there as well.

References

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