4) The important properties of coalescence points and various forms of electron-photon interaction. While the analysis, in points 1) and 2), of the importance of the coalescence points does not depend on the choice of the form of the photoionization matrix element, in different forms the needed information about these coalescences is differently positioned among the interaction, the final state, and the initial state wave functions. We find that if the A-form is used the important information about the singularities is contained in the electron-photon interaction. Only the proper value of the initial state wave function at the coalescence points is needed to evaluate the leading contributions. No information on coalescences from the final state wave function is needed. This analysis agrees with that of [5,11], who discussed the SO region.
In the V form, but only when s-singlet states are considered, the coalescences which give the leading contribution to any region of the spectrum are simply the initial state coalescences. Here information on the slope of the wave function at the coalescence points is needed to evaluate the leading contributions. This information is contained in Schrödinger equation and is connected to the value of the wave function at coalescences. This connection is usually described in terms of (Kato) cusp conditions. No information on coalescences from the final state wave function is needed. This is known for the SO region [4,11]. However, it should be noted that, if non-s-states are considered, the final state wave function coalescences are important and must be treated correctly[12]
In the L form the needed information about the
coalescence is contained in both initial and final state, even when
ionization of the ground state is considered. Inconsistent treatment
of coalescence points in initial and final states can lead to a
spurious
dependence of the double ionization total
cross section, rather than the correct
(in dipole
approximation). This is an example of the general experience that,
unless improvements are introduced consistently, they may not lead to
improvements in prediction. According to Åberg the spurious
contribution appears when using approximate initial state wave
functions which do not satisfy the e-N Kato cusp condition, together
with final state wave functions which do. This is correct neglecting
e-e coalescence in both initial and final state (as Åberg does).
More generally, if approximate wave functions are used, the
coalescences in both initial and final states must be treated in the
same way, to ensure cancelation of spurious terms. If the initial
state wave function is accurately described at coalescence points, so
the final state wave function must also be accurately described. For
example, if the C3 function (which properly treats coalescence points
in the final state) is used, the initial state wave function must
satisfy Kato conditions at both coalescence points in order to cancel
the spurious
contribution.
5) The treatment of states other than the ground state. Although we are mainly concerned with the photoionization of the He ground state our approach allows us to address problems in treating other states. In particular we wish to address the issue of how appropriate it is to neglect various final state interactions (as is often done at high energies), considering the various forms of the photoionization matrix element (L, V, A) which may be utilized. We know that in single electron problems [or in independent particle approximation (IPA)] the final state interaction can not be neglected in L form for any initial state. In the case of V-form the final state interaction can be neglected only for s-state ionization. In A-form the final state interaction can be neglected for any state.
However, if one goes beyond the IPA picture, the situation changes. Recently deviations from the IPA predictions were observed for 2p-state single ionization of Ne [12]. This was explained as due to final state interaction (beyond IPA) between outgoing electron and the ion; there is also a correction to the initial state normalization due to initial state correlation interactions. It can be shown (the most convenient way is in the A-form), using our approach of expansion around the singular point (only the e-N singular point contributes in single ionization), that the important part of the final state interaction beyond IPA, which may persist (depending on the configuration) even in the asymptotic limit of 2p ionization, is the interaction of the outgoing continuum electron at the electron-nucleus coalescence.
Most of the discussion regarding various forms (including the discussion of the treatment of non-s states) can be presented using the example of single photoionization in independent particle approximation, which involve only one (electron-nucleus) coalescence. We may start with such an example, in order to explain our approach in calculating the asymptotic behavior of a Fourier transform of a function with singularities, using expansion around the singularity. We can analyze the A, V and L forms of the matrix element and for each identify the needed quality of approximations for the initial and final state wave functions to get the proper high energy result. This can be illustrated analytically in the special case of the point Coulomb potential, demonstrating how the same result is obtained in all these forms when singularities are treated properly in both initial and final states.