Using molecules as electronic components is a powerful new direction in the science and technology of nanometre-scale systems. Experiments to date have examined a multitude of molecules conducting in parallel, or, in some cases, transport through single molecules. The latter includes molecules probed in a two-terminal geometry using mechanically controlled break junctions or scanning probes as well as three-terminal single-molecule transistors made from carbon nanotubes, C(60) molecules, and conjugated molecules diluted in a less-conducting molecular layer. The ultimate limit would be a device where electrons hop on to, and off from, a single atom between two contacts. Here we describe transistors incorporating a transition-metal complex designed so that electron transport occurs through well-defined charge states of a single atom. We examine two related molecules containing a Co ion bonded to polypyridyl ligands, attached to insulating tethers of different lengths. Changing the length of the insulating tether alters the coupling of the ion to the electrodes, enabling the fabrication of devices that exhibit either single-electron phenomena, such as Coulomb blockade, or the Kondo effect.

We measure electron tunneling in transistors made from C 140 , a molecule with a mass−spring−mass geometry chosen as a model system to study electron-vibration coupling. We observe vibration-assisted tunneling at an energy corresponding to the stretching mode of C 140 . Molecular modeling provides explanations for why this mode couples more strongly to electron tunneling than to the other internal modes of the molecule. We make comparisons between the observed tunneling rates and those expected from the Franck−Condon model. When electrons travel through molecules, vibrational modes of the molecules can affect current flow. Molecular-vibrationassisted tunneling was first measured in the 1960s using devices whose tunnel barriers contained many molecules. 1 Recently, effects of vibrations in single molecules have been measured using scanning tunneling microscopes, 2 singlemolecule transistors, 3,4 and mechanical break junctions. 5 Theoretical considerations suggest that different regimes may exist depending on whether tunneling electrons occupy resonant energy levels on the molecule, and also on the relative magnitudes of the rate of electron flow, the vibrational frequency, and the damping rate of vibrational energy. [6][7][8][9][10][11][12][13][14] A quantitative analysis of electron-vibration interactions has been difficult to achieve in previous molecular-transistor experiments. In transistors made from cobalt coordination complexes, 4 neither the precise nature of the vibrational modes nor their energies was determined independently of transport measurements. In transistors made from C 60 , 3 the "bouncing-ball" mode of a single C 60 molecule against a gold surface was observed, a mode not intrinsic to the molecule itself. In this letter we study single-molecule transistors made using a molecule, C 140 , with low-energy internal vibrational modes that are well understood. We observe clear signatures

Density functional calculations for the electronic conductance of single molecules are now common. We examine the methodology from a rigorous point of view, discussing where it can be expected to work, and where it should fail. When molecules are weakly coupled to leads, local and gradientcorrected approximations fail, as the Kohn-Sham levels are misaligned. In the weak bias regime, XC corrections to the current are missed by the standard methodology. For finite bias, a new methodology for performing calculations can be rigorously derived using an extension of time-dependent current density functional theory from the Schrödinger equation to a Master equation. Contents

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