Microscopic derivation of the electron-transfer rate connection formula and dephasing scaling for multi-state quantum-dot molecules with off-diagonal electron–phonon couplings

Atomistic simulations of electron transfer in epitaxially fused colloidal quantum dot dimers (Hou, Thoss, Banin and Rabani, Nature Communications 14, 3073, 2023) mapped the full crossover from the incoherent nonadiabatic (Marcus) regime to the coherent adiabatic regime as the donor–acceptor hybridization $J$ is tuned. The simulated rates were summarized with an empirical connection formula $k^{-1}=k_{\rm M}^{-1}+k_{\rm dp}^{-1}$, where the dephasing-limited branch was fitted as $k_{\rm dp}\propto\sqrt{J}$ — a dephasing rate that increases with $J$, opposite to the standard two-level spin-boson result and attributed only qualitatively to off-diagonal electron–phonon couplings and to the multi-state (1Se/1Pe) donor–acceptor manifold. The open problem is to derive this crossover rate law from a microscopic multi-state Hamiltonian with off-diagonal electron–phonon couplings — reproducing the Marcus $J^{2}$ limit at weak coupling, the dephasing-limited adiabatic branch at strong coupling, and the interpolation structure with its apparent $\sqrt{J}$ dephasing scaling — or to establish the correct rate law for the same microscopic model that the empirical fit approximates. No derivation of this rate law has been published; the problem remains open.

Background

Electron transfer (ET) — the quantum jump of an electron from a donor to an acceptor species embedded in a fluctuating environment — is among the most studied elementary processes in chemistry, biology and physics. The classic theory of Marcus (1956) treats the weak-coupling, nonadiabatic regime: the electronic coupling (hybridization) $J$ between localized donor and acceptor states is small, the polarizable environment is represented by a harmonic bath with reorganization energy $\lambda$ (the energy released when the environment relaxes after the jump), and thermal activation over the barrier $\left(\Delta\varepsilon+\lambda\right)^{2}/4\lambda$ (with $\Delta\varepsilon$ the driving force) gives the rate $k_{\rm M}=\frac{|J|^{2}}{\hbar}\sqrt{\frac{\pi}{\lambda k_{\rm B}T}}\,\exp\!\left\{-\frac{(\Delta\varepsilon+\lambda)^{2}}{4\lambda k_{\rm B}T}\right\}.$ Marcus theory is perturbative in $J$ (the rate scales as $J^{2}$). Its opposite is the adiabatic regime, reached when $J$ is large and/or the nuclear motion slow, so that the electron follows a single Born–Oppenheimer surface and the rate is governed by barrier crossing (transition-state theory) rather than by electronic coupling. The two regimes are bridged by the adiabatic parameter $\gamma=\frac{|J|^{2}}{2\hbar\omega_{c}}\sqrt{\frac{\pi}{\lambda k_{\rm B}T}},$ where $\omega_{c}$ is a characteristic nuclear frequency: $\gamma\ll1$ is nonadiabatic, $\gamma\gg1$ adiabatic. Zusman (1980) developed the classic framework for this outer-sphere crossover in polar solvents, where the adiabatic rate is limited by solvent relaxation; experimental crossover studies (e.g. Zhu et al. 2021, on mixed-valence donor–bridge–acceptor complexes — molecules containing the same metal center in two different oxidation states — analyzed with Landau–Zener theory, the standard two-state model of a transition at an avoided crossing of two potential-energy surfaces) remain confined to narrow coupling windows. Colloidal quantum dot (CQD) molecules — two semiconductor nanocrystals (typically CdSe cores with CdS shells) fused epitaxially through a crystalline neck — provide an unusually clean platform for this crossover, because the neck width, core diameter and shell thickness tune $J$ over orders of magnitude while $\lambda$ and $\omega_{c}$ change only mildly. Hou, Thoss, Banin and Rabani (Nature Communications 14, 3073, 2023) built an atomistic multi-state model for such dimers: electronic states from a semi-empirical pseudopotential calculation, vibrations from a Stillinger–Weber force field, and electron–phonon couplings including off-diagonal terms $V_{n\neq m}^{\alpha}$ — couplings in which mode $\alpha$ mixes electronic states $n$ and $m$, rather than merely modulating the energy of state $n$ (the diagonal coupling familiar from the spin-boson model, the standard two-electronic-state ET model coupled to a harmonic bath with spectral density $S(\omega)$, the frequency-resolved distribution of bath-mode coupling strengths). The dynamics were propagated with the Ehrenfest mean-field mixed quantum–classical method (electrons evolve quantum-mechanically in the instantaneous field of classical nuclei, which move on the expectation value of the force), validated against numerically exact MCTDH (multi-configuration time-dependent Hartree) calculations on two-state cuts. Three findings define the problem. First, rates rise by orders of magnitude across the crossover, following $k\propto J^{2}$ as $J\to0$ but increasing only weakly at large $J$. Second, deep in the adiabatic regime the transfer is limited by dephasing (loss of phase coherence between donor and acceptor superposition states), and the effective dephasing rate increases with $J$ — the opposite of the two-level spin-boson result $k_{\rm dp}=2S(2J)\coth(J/k_{\rm B}T)$, which decreases with $J$; the increase was traced qualitatively to the off-diagonal electron–phonon couplings and to population transfer among the several donor and acceptor states (the 1Se-like (s-type) ground and 1Pe-like (p-type) excited electron orbitals are split by only 60–100 meV). Third, the entire rate dataset is captured by a fit to the connection formula $k^{-1}=k_{\rm M}^{-1}+k_{\rm dp}^{-1}\quad\text{with}\quad k_{\rm dp}\propto\sqrt{J},$ in which the total rate is limited by the slower of the Marcus channel and the dephasing channel, like resistances in series. In the source, $k_{\rm dp}$ itself is defined only operationally: the simulated transfer probability is fitted to a two-exponential envelope $p_{\rm env}(t)=A_{\rm M}e^{-k_{\rm M}t}+A_{\rm dp}e^{-k_{\rm dp}t}$, with the total rate taken as either $k_{\rm M}$ or $k_{\rm dp}$ regime by regime (Methods Eq. (20)). Both the interpolation formula and the $\sqrt{J}$ scaling are empirical: the paper presents them as fits (Fig. 5a) and the supplementary note on the physical source of dephasing argues only qualitatively (low-frequency inter-dot breathing and torsional acoustic modes, off-diagonal couplings, 1Se/1Pe transfer). No first-principles theory of the crossover rate law, or of the observed dephasing scaling in multi-state systems, currently exists.

Problem Statement

Derive, from a microscopic Hamiltonian describing several donor and several acceptor electronic states coupled to nuclear vibrations through both diagonal and off-diagonal electron–phonon couplings, a rate theory for electron transfer across the entire nonadiabatic-to-adiabatic crossover that (i) reduces to the Marcus rate $k_{\rm M}\propto J^{2}$ (with its full activation factor) as $J\to0$; (ii) yields, deep in the adiabatic regime $\gamma\gg1$, transfer limited by an effective dephasing rate $k_{\rm dp}$ that increases with the hybridization $J$ — the behavior opposite to the two-level spin-boson formula $k_{\rm dp}=2S(2J)\coth(J/k_{\rm B}T)$, where $S(\omega)$ is the $J$-independent bath spectral density of the two-level model evaluated at the transition frequency $\omega=2J$ — with the scaling of this effective dephasing rate derived microscopically rather than fitted, reproducing the apparent $k_{\rm dp}\propto\sqrt{J}$ dependence over the simulated range of epitaxially fused colloidal quantum dot dimers and identifying the asymptotic behavior of the derived law outside that range; and (iii) accounts for the interpolation structure of the empirical connection formula $k^{-1}=k_{\rm M}^{-1}+k_{\rm dp}^{-1}$ across the crossover — or, alternatively, establishes the correct rate law for the same microscopic model and explains why the empirical fit succeeds locally, if the microscopic analysis shows that this empirical picture (the connection form, the increase of the effective dephasing rate with $J$, or the $\sqrt{J}$ scaling) is not the correct law. In clauses (ii)–(iii), the effective dephasing rate $k_{\rm dp}$ must be defined independently of the connection formula, as in the source (Hou et al. 2023, Methods Eq. (20)): the decay constant of the donor-population envelope $p_{\rm env}(t)=A_{\rm M}e^{-k_{\rm M}t}+A_{\rm dp}e^{-k_{\rm dp}t}$ in the adiabatic (dephasing-dominated) regime, computed from the microscopic multi-state dynamics itself and not back-solved as the residual $[k^{-1}-k_{\rm M}^{-1}]^{-1}$ of a derived total rate. The theory must state its conditions of validity (spectral-density structure, state-manifold spacing, temperature, coupling range) and cover the simulated regime of the CdSe/CdS core–shell dimers of Hou et al. (2023), whose room-temperature rates span roughly 10 ps$^{-1}$ to 100 fs$^{-1}$.

References

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