Multiple scattering in a vacuum barrier obtained from real-space wavefunctions

We have developed a method for simulating multiple electron scattering in a vacuum barrier using real-space single-electron wavefunctions for the separate surfaces. The Green functions in the vacuum barriers are calculated to first order in the Dyson series. We find that the zero-order current is equal to the usual Bardeen approach only in the limit of zero bias and derive the modifications in the finite bias regime. We also derive a first-principles formulation for the energy of interaction between the two surfaces, and show that it is proportional to the tunnelling current. With this method the tunnelling current can in principle be computed to any order in the Dyson expansion.