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For defining the conductance of single molecule junctions with a redoxfunctionality in an electrochemical cell, two conceptually different electrontransport mechanisms, namely coherent tunnelling and vibrationally inducedhopping compete with each other, where implicit parameters of the setup such asthe length of the molecule and the applied gate voltage decide which mechanismis the dominant one. Although coherent tunnelling is most efficiently describedwithin Landauer theory, while the common theoretical treatment of electronhopping is based on Marcus theory, both theories are adequate for the processesthey describe without introducing accuracy limiting approximations. For adirect comparison, however, it has to be ensured that the crucial quantitiesobtained from electronic structure calculations, i.e. the transmission functionT(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 fromsimilar grounds as pointed out by Nitzan and co-workers in a series ofpublications. In this article our framework is a single particle picture, wherewe perform density functional theory calculations for the conductancecorresponding to both transport mechanisms for junctions with the centralmolecule containing one, two or three Ruthenium centers, respectively, where weextrapolate our results in order to define the critical length for thetransition point of the two regimes which we identify at 5.76 nm for this typeof molecular wire. We also discuss trends in dependence on an electrochemicallyinduced gate potential.

A density functional theory based direct comparison of coherent tunnelling and electron hopping in redox-active single molecule junctions