X [formula omitted]: A general quantitative approach to the calculation of ( [formula omitted]) and related processes

Abstract

We present a new variational approach for calculating ( e , 2 e ) and ( γ , 2 e ) processes. We consider the general theory for the electron-impact ionization of atomic hydrogen and we deduce a formula for the scattering amplitude in terms of a surface integral over a five-dimensional hypersurface. We consider configuration space to be divided into three regions separated by five-dimensional hypersurfaces. We calculate the R-matrix on the surface of the innermost hypersurface using a variationally determined two-electron R -operator, a linear operator that relates function values to normal derivatives on the innermost bounding hypersurface inside which the function satisfies the Schrödinger equation. We then use the Light Walker propagation technique, modified by the symmetry condition at r s = r f , to propagate our variationally determined R-matrix through a large volume until we reach the boundary of our second hypersurface. We calculate the S-matrix on this hypersurface and use it to remove the incoming flux and deduce the outgoing wave which is matched to one that has been brought in from some asymptotically large distance. It is shown that it is possible to reduce the five-dimensional integral over the hypersurface to a single numerical integration, times a finite sum over angular terms, thus facilitating the evaluation of the scattering amplitude. The method we propose is computationally efficient. Results will be presented for a number of simple test cases.