Calculation of ionization within the close-coupling formalism.

The CCC method was developed initially [1,2] to describe scattering of electrons from one-electron targets. The method made use of Laguerre basis states that are are L functions to discretize the target continuum. The method provided convergent amplitudes for scattering to low–lying discrete states and total–ionization cross– sections. The theory is fully symmetrized since it is based on a symmetrized expansion of the two–electron wave function. Bray and Fursa [3] used the fact that the positive–energy pseudostates were approximations to true continuum functions to propose a method for calculating energy differential cross sections for ionization. They did this in a way that preserved “two–particle” unitarity implicit in the CCC formalism with the L target states. This had the consequence of yielding different magnitude amplitudes C( a, b) and C( b, a) for the same physically indistinguishable process of the two ionized electrons emerging with energies a, b, depending on which electron, a or b, is represented by a Coulomb wave. Bray and Fursa argued that this asymmetry could be interpreted in terms of “distinguishable” electrons. This has been criticized by Bencze and Chandler [4] (BC) who argue that the symmetrization property of the amplitudes is a fundamental tenet of scattering theory and that the CCC amplitudes must satisfy this property counter to the numerical evidence. They conclude that “The numerical CCC amplitudes have not, therefore, converged to accurate approximations of the exact amplitudes.” It is the dichotomy between a method of calculating ionization processes that seems very impressive in the quality of agreement achieved between theory and experiment, and the criticism of the model on a very fundamental level, that provides the motivation for the present work. To make the discussion as transparent as possible we choose the Temkin–Poet model (TP) [5,6] of electron–hydrogen scattering. This model is the solution of the scattering problem assuming spherical averaging over both electrons and solving for the total angular momentum zero two–electron wave function. The Schrödinger equation becomes