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

For defining the conductance of single molecule junctions with a redox functionality in an electrochemical cell, two conceptually different electron transport mechanisms, namely coherent tunnelling 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 tunnelling is most efficiently described within Landauer theory, while 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 reorganisation energy $\lambda$ and the driving force $\Delta G^0$ in Marcus theory, are derived from similar grounds as pointed out by Nitzan and co-workers in a series of publications. In this article our framework is a single particle picture, where 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, respectively, where we extrapolate our results in order to define the critical length for the transition point of the two regimes which we identify at 5.76 nm for this type of molecular wire. We also discuss trends in dependence on an electrochemically induced gate potential.