Density functional theory based direct comparison of coherent tunneling and electron hopping in redox-active single-molecule junctions

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To define the conductance of single-molecule junctions with a redox functionality in an electrochemical cell, two conceptually different electron transport mechanisms, namely, coherent tunneling and vibrationally induced hopping, compete with each other, where implicit parameters of the setup such as the length of the molecule and the applied gate voltage decide which mechanism is the dominant one. Although coherent tunneling is most efficiently described within Landauer theory and the common theoretical treatment of electron hopping is based on Marcus theory, both theories are adequate for the processes they describe without introducing accuracy-limiting approximations. For a direct comparison, however, it has to be ensured that the crucial quantities obtained from electronic structure calculations, i.e., the transmission function $T(E)$ in Landauer theory and the transfer integral $V$, the reorganization energy $\ensuremath{\lambda}$, and the driving force $\ensuremath{\Delta}{G}^{0}$ in Marcus theory, are derived from similar grounds, as pointed out by Nitzan and coworkers in a series of publications. In this paper our framework is a single-particle picture, for which we perform density functional theory calculations for the conductance corresponding to both transport mechanisms for junctions with the central molecule containing one, two, or three Ruthenium centers, from which we extrapolate our results in order to define the critical length of the transition point of the two regimes which we identify at 5.76nm for this type of molecular wire. We also discuss trends in the dependence on an electrochemically induced gate potential.