  We have developed a method to simulate multiple electron scattering in a
vacuum barrier using real-space single-electron wavefunctions of the separate
surfaces. The tunnelling current is calculated to first order in the Dyson
series. In zero order, we find a result which differs from the usual Bardeen
approach by the decay constants of the wavefunctions, entering the description
as individual weights of tunnelling transitions. To first order we find
multiple electron scattering, which can be formulated in terms of Bardeen
matrices. Here, we also derive a first-principles formulation for the
interaction energy between the two surfaces. With this method the tunnelling
current can in principle be computed to any order in the Dyson expansion.

Materials Science