Molecular electronics break-junction experiments are widely used to investigate fundamental physics and chemistry at the nanoscale. Reproducibility in these experiments relies on measuring conductance on thousands of freshly formed molecular junctions, yielding a broad histogram of conductance events. Experiments typically focus on the most probable conductance, while the information content of the conductance histogram has remained unclear. Here, we develop a theory for the conductance histogram, which accurately fits experimental data and augments the information content that can be extracted, by merging the theory of force-spectroscopy with molecular conductance. Specifically, we propose a microscopic model of the junction evolution under the modulation of external mechanical forces and combine it with the non-equilibrium stochastic features of junction rupture and formation. Our formulation captures contributions to the conductance dispersion that emerge due to changes in the conductance during the mechanical elongation inherent to the experiments. The final histogram shape is determined by the statistics of junction rupture and formation. The procedure yields analytical equations for the conductance histogram in terms of parameters that describe the free-energy profile of the junction, its mechanical manipulation, and the ability of the molecule to transport charge. All physical parameters that define our microscopic model can be obtained from separate conductance and rupture force measurements on molecular junctions. Further, the predicted behavior with respect to physical parameters can be used to test the range of validity of the microscopic model, understand the conductance histograms, and design molecular junction experiments with enhanced resolution and molecular devices with more reproducible conductance properties.
