Reduced Effective Reorganization Energy for Adiabatic Electron Transfer

Abstract

We predict that in the adiabatic limit of the Marcus normal regime, Marcus kinetics will be observed with a reduced effective reorganization energy that is a function of the standard reorganization energy and the coupling strength. This result enables the derivation of a closed-form Marcus-Hush-Chidsey type rate expression for heterogeneous electron transfer in the adiabatic limit, which also involves a different prefactor than in the non-adiabatic case.

Figures (5)

• Figure 1: (a) Energy $E$ plotted against the reaction coordinate $q$ for the two-level Marcus system under study with parameter values taken to be $\lambda=4.0$ eV, $\Delta G^{\circ}=0.3$ eV. Diabats with harmonic energies $E_\text{a}$ (blue small dashes) and $E_\text{b}$ (green large dashes) are mixed by large coupling $V$=1.0 eV. (b) A proposed mechanism for heterogeneous, adiabatic electron transfer in which a continuum of metalic states, each with Fermi-Dirac occupancy $n(\epsilon)$, couple adiabatically to the accepting state with curvature governed by the reduced effective reorganization energy.
• Figure 2: (a)-(b) Activation barrier $E^*$ plotted against the thermodynamic driving force $\Delta G^{\circ}$ for $\lambda=4.0$ eV and (a) $V=1.0$ eV or (b) $V=0.5$ eV. (c)-(d) Activation barrier $E^*$ plotted against the coupling value $V$ for $\lambda=4.0$ eV and (c) $\Delta G^{\circ}$=0.3 eV or (d) $\Delta G^{\circ}=-0.5$ eV. (e)-(f) Activation barrier $E^*$ plotted against the reorganization energy $\lambda$ for $V=0.4$ eV and (e) $\Delta G^{\circ}$=0.3 eV or (f) $\Delta G^{\circ}=-0.5$ eV. In all panels results are shown for non-adiabatic Marcus theory with no mixing (blue solid line), the exact ground state adiabat (black dotted line), the traditional adiabatic correction to Marcus theory obtained by lowering the barrier by $V$ (green dashes) and the correction considered in this work (orange dashes).
• Figure 3: (a) Activation barrier $E^*$ plotted against the thermodynamic driving force $\Delta G^{\circ}$ for $\lambda=4.0$ eV and linear coupling with $V(0)=0.6$ eV and $V(1)=1.0$ eV. (b) Activation barrier $E^*$ plotted against the coupling value of the reactant minimum $V(0)$ for linear coupling with $V(1)=1$ eV and $\lambda=4.0$ eV, $\Delta G^{\circ}=-0.5$ eV. For the constant shift method, $V(q=1/2)$ is used.
• Figure 4: (a) Tafel plot showing the (base-10) logarithm of the (scaled) reaction rate $k$ plotted against the formal overpotential $\eta_\text{f}$ at $T=300$ K. (b) Arrhenius plot showing the dependence of the rate on the inverse temperature $1/T$ for fixed formal overpotential $\eta_\text{f}=-0.3$ V. In both panels the reorganization energy and coupling are set to $\lambda=4.0$ eV and $V=0.5$ eV respectively. In both panels results are shown for non-adiabatic Marcus theory with no mixing (blue solid line), the exact ground state adiabat (black dotted line), the traditional adiabatic correction to Marcus theory obtained by lowering the barrier by $V$ (green dashes), and the correction proposed in this Letter (orange dashes).
• Figure 5: Tafel plots showing the (base-10) logarithm of the (scaled) reaction rate $k$ plotted against the formal overpotential $\eta_\text{f}$ for $\lambda=4.0$ eV at $T=300$ K and (a) $V(0)=0.1$ eV, $V(1)=0.5$ eV, (b) $V(0)=0.2$ eV, $V(1)=1.0$ eV, and (c) $V(0)=0.6$ eV, $V(1)=1.0$ eV. For the constant shift method, $V(q=1/2)$ is used.