Distance dependence of heterogeneous electron transfer through the nonadiabatic and adiabatic regimes

The Landau-Zener formalism, which is strictly valid for a two-state system, is extended to multistate systems by assuming that the electronic interaction between the redox moiety and a given energy level in the electrode is independent of the energy of the level and of the neighboring levels. The resultant electron transmission coefficient, κel,m, over the full range (nonadiabatic to adiabatic regimes) is defined by κel,m=2(1-exp[-(νel,m0/2νn)exp[-β(r-r0)]])/(2-exp[-(νel,m0/2νn)exp[-β(r-r0)]]) where r (cm) is the distance between the electrode and the redox moiety, r0 (cm) is the distance between the electrode and the plane of closest approach for the redox moieties, νn (s−1) is the effective nuclear vibration frequency, νel,m0(s-1) is the energy-independent electron-hopping frequency when r = r0 and when the reactants and products have the same nuclear configurations and energies, and β (cm−1) is the decay constant for electronic coupling. This relationship is shown to be an adequate approximation of the more rigorously derived results of Kuznetsov et al. [J. Electroanal. Chem. 532 (2002) 171] which is valid from weakly coupled (nonadiabatic, νel,m0/νn≪1) to strongly coupled (adiabatic, νel,m0/νn≫1) regimes. We also show that the distance dependence of κel,m is consistent with the experimental observations of Smalley et al. [J. Am. Chem. Soc. 125 (2003) 2004]. The expression for κel,m also leads to a remarkably simple description of khet (units: cm s−1), the rate constant for heterogeneous electron transfer between an electrode and redox species in solution: khet=(νnκn,m/β)ln(1+νel,m0/νn) where κn,m is the nuclear reorganization factor.