Factors that control the tunneling signal: (A) A simple tunnel barrier shown as a 1D structure with a gap where the potential (V) exceeds the Fermi energy (EF). This potential barrier, V-EF, can be lowered to a value ΔE by the presence of an atom in the gap with an eigenstate at EF+ΔE. The extension of this picture to a molecule in the gap is shown in (B) where each atom contributes a level near the gap E1, E2 .... and overlap between the atomic states (Hmn) leads to a delocalized state that connects the left and right electrodes. This mediates a current proportional to the number of available states on the positive electrode (i.e., proportional to Vbias) and the strength of the coupling between the electrodes, as given by the Green's function propagator.