Effective quantum reorganization energy for electron transfer

1. Problem Statement

The paper addresses a significant discrepancy in the field of condensed-phase electron transfer (ET), particularly in the context of the Carbon Dioxide Reduction Reaction (CO2RR):

• The Discrepancy: Experimental kinetic data for CO2RR suggests a reorganization energy ($\lambda_{\text{eff}}$) of approximately 0.75 eV. However, theoretical calculations based on standard Marcus theory (using ab initio methods) predict a classical reorganization energy ($\lambda$) of approximately 6.3 eV.
• The Paradox: Despite this order-of-magnitude difference, experimental reaction rates follow Marcus-like rate expressions with remarkable accuracy.
• The Core Issue: Standard Marcus theory defines reorganization energy as a purely classical quantity derived from the intersection of diabatic potential energy surfaces, assuming vanishing electronic coupling ($V \to 0$). The authors argue that in systems with strong electronic coupling (common in inner-sphere reactions like CO2RR), the classical definition fails to capture the true activation barrier, yet the form of the Marcus equation remains valid. The question is: What is the physical quantity that replaces the classical $\lambda$ in the presence of strong coupling?

2. Methodology

The authors employ a rigorous mathematical treatment of a harmonic two-level quantum system to resolve this discrepancy.

• Model System: They utilize a Hamiltonian describing two diabatic electronic states ($|\phi_a\rangle$ and $|\phi_b\rangle$) coupled by an electronic interaction $V(q)$. The energies of the diabatic states are harmonic functions of a collective nuclear coordinate $q$:
  • $E_a(q) = \lambda q^2$
  • $E_b(q) = \lambda(1-q)^2 + \Delta E$
  • Where $\lambda$ is the classical reorganization energy and $\Delta E$ is the driving force.
• Adiabatic Treatment: Instead of treating the system non-adiabatically (where nuclei move, then electrons hop), they solve the secular equation for the full Hamiltonian to obtain the exact adiabatic ground state energy ($E_-$).
• Perturbative Analysis: They perform a quadratic expansion of the exact activation barrier ($E^*$) derived from the adiabatic surface. They compare this expansion to the standard Marcus barrier form:
  $$E^* = \frac{(\lambda_{\text{eff}} + \Delta E)^2}{4\lambda_{\text{eff}}}$$
• Derivation: By matching terms in the expansion, they derive a closed-form expression for an effective reorganization energy ($\lambda_{\text{eff}}$) that depends on both the classical $\lambda$ and the electronic coupling $V$.
• Application to Electrochemistry: They extend this derivation to heterogeneous Faradaic reactions at electrochemical interfaces, integrating over the continuum of electronic states in a metal electrode (similar to Marcus-Hush-Chidsey kinetics) but substituting the classical parameters with the derived quantum effective parameters.

3. Key Contributions

The paper makes three primary theoretical contributions:

1. Definition of Effective Quantum Reorganization Energy ($\lambda_{\text{eff}}$):
   The authors prove that the parameter governing the activation barrier in the "normal region" is not the classical $\lambda$, but a quantum mechanical object dependent on electronic coupling. Under the Condon approximation (constant coupling $V$), the relationship is:
   $$\lambda_{\text{eff}} = \lambda \left( 1 - \frac{2V}{\lambda} \right)^2$$

   • This recovers the classical $\lambda$ only when $V \to 0$.
   • As $V$ increases, $\lambda_{\text{eff}}$ decreases significantly, explaining why experimental fits yield much lower values than ab initio calculations of $\lambda$.

2. Unification of Adiabatic and Non-Adiabatic Regimes:
   The work demonstrates that Marcus-like rate expressions remain accurate even in the strongly adiabatic regime (strong coupling), provided one uses $\lambda_{\text{eff}}$ instead of $\lambda$. This resolves the paradox of why CO2RR follows Marcus kinetics despite having strong coupling.

3. Closed-Form Rate Expression for Electrochemical Interfaces:
   The authors derive a new rate equation for ET-limited Faradaic reactions (Eq. 15) that is valid from the non-adiabatic limit to the strongly adiabatic limit.

   • Unlike traditional adiabatic corrections (which simply lower the barrier by a constant $-V$), this expression correctly accounts for the change in the curvature of the potential energy surface.
   • The prefactor transitions from the Fermi-Golden Rule form (non-adiabatic) to a classical attempt frequency (adiabatic) naturally within the framework.

4. Results

• Resolution of the CO2RR Discrepancy: By applying the derived formula to the CO2RR data ($\lambda = 6.3$ eV, $\lambda_{\text{eff}} = 0.75$ eV), the authors calculate an electronic coupling of $V \approx 2$ eV. This value is consistent with independent ab initio calculations, confirming that the "missing" energy is due to the quantum mechanical reduction of the barrier caused by strong coupling.
• Accuracy of Approximations: Numerical comparisons (Fig. 2) show that:
  • Standard non-adiabatic Marcus theory significantly overestimates the barrier at high coupling.
  • Traditional "constant shift" corrections ($-V$) underestimate the barrier at high coupling.
  • The new $\lambda_{\text{eff}}$ model reproduces the exact numerical solution of the adiabatic barrier with excellent agreement across a wide range of driving forces ($\Delta E$) and coupling strengths.
• Current-Overpotential Relations: The derived rate equation (Eq. 15) accurately predicts reduction currents and Arrhenius behavior (Fig. 4) for systems with strong coupling, matching exact numerical integrations where standard theories fail.
• Barrierless Transition: The model predicts that as $V \to \lambda/2$, $\lambda_{\text{eff}} \to 0$, leading to a barrierless transition, consistent with known physical limits for adiabatic ET.

5. Significance

• Theoretical Unification: This work provides a unified quantum mechanical description of electron transfer activation barriers, bridging the gap between the non-adiabatic (hopping) and adiabatic (concerted) regimes. It corrects a "category error" in the literature where the classical $\lambda$ was incorrectly assumed to be the sole parameter in Marcus-like expressions for all coupling strengths.
• Practical Impact on Catalysis: It explains why Marcus theory works so well for complex inner-sphere reactions (like CO2RR, hydrogen evolution, etc.) where strong coupling is expected. It allows researchers to extract physically meaningful coupling constants from experimental kinetic data.
• First-Principles Prediction: The framework enables the prediction of Faradaic reaction kinetics using ab initio calculations of $\lambda$ and $V$ without needing empirical fitting parameters for the activation barrier.
• Broader Applicability: While motivated by electrochemistry, the findings are relevant to other fields involving coupled harmonic states, such as disordered organic semiconductors, mixed-valence complexes, and charge transfer in DNA $\pi$-stacks, where electronic coupling is often comparable to reorganization energy.

In summary, the paper redefines the reorganization energy as a quantum mechanical, coupling-dependent quantity, thereby extending the validity of Marcus-like kinetics into the strongly adiabatic regime and resolving long-standing discrepancies between theory and experiment in electrocatalysis.