Improvement in the First Born Theory of Electron Scattering by Molecular Systems. I. Theory

The asymptotic or Bethe expansion of the first Born cross section, for large magnitude of the initial collision velocity $V$, is considered for a particular electronic excitation of a molecular target. The sum of the Bethe cross section over the complete set of wave functions for nuclear motion in the final electronic state is shown to be asymptotic to the analogous sum of the Born cross section over states allowed by energy conservation. The closure relation is then used to simplify the matrix elements which define the Bethe cross section without further approximation. The advantages gained by this approach are demonstrated in the following paper by the treatment of a particular example.