Controlled golden-rule rate theory for electron transfer at strong electronic coupling in the Marcus inverted regime: rotation-angle optimization, higher-order corrections, and atomistic validation of Optimal Golden Rule theory

Fermi's golden rule (GR) rate theory fails qualitatively when the electronic coupling between diabatic states is large, and in the Marcus inverted regime no established method — except a not-yet-reformulated fourth-order instanton — could treat this limit. Fay's Optimal Golden Rule (OGR) theory (J. Chem. Phys. 161, 014101 (2024)) addresses it by evaluating a golden-rule rate in a globally rotated diabatic basis that minimizes the effective coupling, achieving near-quantitative accuracy on spin-boson models at arbitrary coupling. The paper itself leaves open how to make OGR a controlled, systematically improvable theory: the rotation angle is fixed by a heuristic constrained minimization over $[\theta_{\mathrm{B}},\theta_{\mathrm{A}}]$; recrossing and higher-order-in-coupling corrections at intermediate coupling are neglected (maximum error about a factor of 4); and the theory has not been demonstrated on fully atomistic anharmonic Hamiltonians. The open problem is to complete OGR (or an equivalent golden-rule-based formulation) into a validated theory for the strong-coupling inverted regime.

Background

Electron transfer (ET) is the elementary process in which an electron moves between donor and acceptor species, underlying photovoltaics, photosynthesis, enzymatic catalysis, and radical-pair chemistry. Classical Marcus theory describes ET with two parabolic diabatic free-energy surfaces — nuclear coordinate $q$, reorganization energy $\lambda$ (the energy released when the environment relaxes after the jump), and reaction free energy $\Delta G^{\circ}$ — separated by an electronic coupling $\Delta$. In the normal regime, increasing the driving force $-\Delta G^{\circ}$ lowers the activation barrier; in the Marcus inverted regime ($-\Delta G^{\circ}>\lambda$) it raises it, so rates decrease with driving force. In the inverted regime rates are extremely slow (microsecond lifetimes are typical) and nuclear quantum effects (tunneling, zero-point energy) can enhance rates by many orders of magnitude.

The standard microscopic foundation is Fermi's golden rule (GR): the rate is obtained from second-order perturbation theory in $\Delta$, giving $k_{\mathrm{GR}}\propto\Delta^{2}$ times a Franck–Condon-type weighted density of states; Marcus theory and its quantum-nuclear generalization, Marcus–Levich–Jortner (MLJ) theory, are its classical and single-mode limits. GR assumes $\Delta$ is small. In short-range ET (e.g., charge recombination in tightly stacked chromophores) $\Delta$ can reach many times $k_{\mathrm{B}}T$, and the true rate turns over and decreases with coupling while $k_{\mathrm{GR}}$ grows without bound — GR then overestimates rates by up to four orders of magnitude. In the normal regime, strong coupling makes the reaction adiabatic (it proceeds continuously on a single adiabatic surface), so established single-surface methods — ring-polymer molecular dynamics (a path-integral simulation technique that captures quantum nuclear statistics on a single potential-energy surface), instanton theory (a semiclassical tunneling-path approximation obtained by steepest descent in imaginary time) — apply. In the inverted regime, by contrast, reactants and products remain on different adiabatic states even at strong coupling, so the reaction stays intrinsically non-adiabatic and those methods break down; exact quantum-dynamics methods — hierarchical equations of motion (HEOM, an exact propagation of the reduced density matrix as a convergent hierarchy over the bath) and multilayer multiconfigurational time-dependent Hartree (an exact wavefunction propagation on adaptively contracted basis trees) — work only for simplified spin-boson-type Hamiltonians (a donor/acceptor two-level system linearly coupled to harmonic baths), not for general anharmonic atomistic systems. Before 2024, the strong-coupling toolkit consisted of the interpolation formula of Lawrence, Fletcher, Lindoy and Manolopoulos (which bridges GR and adiabatic limits but fails deep in the inverted regime), the non-adiabatic quantum instanton (NAQI) approximation (a semiclassical rate formula for non-adiabatic reactions built from imaginary-time correlation functions), and higher-order instanton theory — notably the fourth-order instanton of Trenins and Richardson, developed for the normal regime and never reformulated for the inverted regime.

Fay's 2024 Optimal Golden Rule (OGR) theory fills this gap at leading order. The diabatic states $|A\rangle,|B\rangle$ are rotated through a global angle $\theta$ into mixed states $|+\rangle,|-\rangle$ in which the effective coupling $\Delta_{+-}(\theta)$ is reduced; the GR rate is then evaluated in the rotated basis (fully compatible with existing golden-rule path-integral and instanton machinery for anharmonic potentials). Because the coupling cannot be minimized simultaneously for both states, $\theta$ is constrained to $[\theta_{\mathrm{B}},\theta_{\mathrm{A}}]$ (the angles minimizing the expectation of the coupling in each diabatic state) and chosen to minimize the computed rate — motivated by the observation that the neglected higher-order terms in $\Delta_{+-}$ act mainly as recrossing corrections, making the rotated GR rate an approximate upper bound in the inverted regime; a practical two-bound approximation evaluates only the endpoints. On spin-boson models and an anharmonic model, OGR is accurate to within about a factor of 4 at worst, correctly predicting the rate turnover at large $\Delta$, and it yields analytic modified Marcus/MLJ expressions in which strong coupling effectively renormalizes $\Delta$, $\Delta G^{\circ}$, and $\lambda$. Applied to a BODIPY–anthracene photosensitizer (BODIPY being a boron–dipyrromethene dye widely used in photoredox and triplet-sensitization chemistry), it reduces the computed charge-recombination rate by a factor of four and moves the predicted triplet yield from 0.8 to 0.88, near the experimental 0.93–0.96. The paper is explicit, however, that the theory is a lowest-order approximation with heuristic elements.

Problem Statement

Establish a controlled, systematically improvable golden-rule-based rate theory for electron transfer at arbitrarily strong electronic coupling in the Marcus inverted regime, completing the Optimal Golden Rule (OGR) framework of Fay (2024). The single overarching objective — rate predictions for strong-coupling inverted-regime systems whose accuracy is explained and systematically improved rather than merely observed, compatible with anharmonic and atomistic Hamiltonians and nuclear-quantum-effects methods — comprises three scientifically linked components: (i) a systematic theory (or comparative study with a principled criterion) for the global rotation angle $\theta$, replacing the heuristic constrained minimization of the rate over $[\theta_{\mathrm{B}},\theta_{\mathrm{A}}]$ (the interval bounded by the rotation angles that minimize the effective coupling in each of the two diabatic states) together with its practical two-bound approximation (which evaluates only the endpoints of that interval), and settling whether alternatives such as direct minimization of $\langle\Delta_{+-}^{2}\rangle_{\pm}$, minimization of the initial flux–flux correlation value $\mathrm{Tr}[\hat{F}^{2}e^{-\beta\hat{H}_{\pm}}]$, or independent angles for the flux evaluation and the reactant partition function improve accuracy; (ii) controlled corrections beyond the lowest-order perturbation in the rotated coupling — recrossing and higher-order-in-$\Delta_{+-}$ terms whose neglect may be responsible for residual errors (up to about a factor of 4) at intermediate couplings; and (iii) demonstration of the resulting theory on fully atomistic, anharmonic, electronic-structure-derived Hamiltonians — the route by which, following the source, OGR moves beyond the spin-boson models and model potentials used so far — together with quantitative benchmarking against reliable reference quantum dynamics across the weak-to-strong coupling range wherever such references are obtainable; since numerically exact propagation is unavailable for general anharmonic atomistic systems, the reference standard used for atomistic systems must be explicitly constructed and justified. The source further notes that a completed framework offers a route to generalizing the theory to other non-adiabatic processes (proton-coupled electron transfer, excitation energy transfer, electrochemical electron transfer); these extensions are outlook rather than required deliverables of this problem.

References

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