The authors propose a regularization technique which allows Lagrange-mesh calculations to retain their accuracy, efficiency and simplicity when the Hamiltonian is singular at finite distance. Compact analytical expressions of kinetic-energy matrix elements are an essential ingredient of the method. The authors present general procedures for deriving them. Their approach is tested on the hydrogen atom and the squared orbital-momentum operator. They demonstrate its accuracy on the non-separable problem of the hydrogen atom in a magnetic field. This method opens the way to numerous new applications of Lagrange-mesh calculations in atomic and molecular physics.

We have combined a laboratory frame independent particle approach with a bodyfixed frame collective coordinates approach to derive new exact parametrizations of the triple differential cross section for double photoionization. The latter disentangle the kinematical from the dynamical effects to the largest possible extent. In addition, they provide an expansion of the so-called correlation factor with respect to the configurations of the electronic pair. The present approach is particularly suitable for the physical analysis of experimental data, as illustrated by the applications reported in the companion paper.

The hyperspherical R-matrix method with semiclassical outgoing waves introduced by us previously [Phys. Rev. A 60, 3667 (1999)] is applied to the one-photon double ionization of the He atom. The absolute differential cross sections obtained are in excellent agreement with experiment for the very challenging dynamical situations considered. This method is the first one to compute the final double continuum state accurately over the entire configuration space, from the vicinity of the nucleus to the asymptotic region.

Experiments using new sources of XUV pulses now tackle the difficult problem of few-photon direct double ionization of atoms. Despite its apparent simplicity, the fundamental process of two-photon direct double ionization of helium is far from being understood. Here, we use a time-dependent approach to study the process. Our results for the electron angular and energy distributions demonstrate that the dominant mechanism for double-electron escape involves a highly correlated electron motion. Angular correlations strongly favour back-to-back electron emission along the polarization axis, while dynamical screening leads to an equipartition of the electron energy for a broad range of field frequencies. These features are reflected in the recoil-ion-momentum distributions that are presently accessible to experiments.

The double photoionization of He is investigated using the hyperspherical R matrix with semiclassical outgoing waves method. Triply, doubly, and singly differential, as well as fully integrated, cross sections are computed in a variety of geometrical and dynamical situations. The results are found to be in excellent agreement with absolute measurements both in shape and, more importantly, in magnitude. This demonstrates the robustness and accuracy of this ab initio method, which also provides a visualization of the formation of the various cross sections during the expansion of the system. This visualization reveals that, for very asymmetric energy sharings, the cross sections take their final form when the electrons are thousands of atomic units away from the ionic core, a distance where no other method is able to describe the double continuum wave function accurately.

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We use the general formalism established in the companion paper I to analyse recent measurements of the triple differential cross sections for double photoionization of He and Ne, for equal energy sharing and for 1 S e symmetry of the residual ion. A dynamical factor, which depends on the energy and on the mutual angle between the two electrons, is extracted from the experiments without relying on any dynamical approximation. This factor is expanded with respect to the one-electron angular momentum , up to a maximum value max , which measures the degree of angular correlation attained by the electron pair. We discuss the physical meaning of max , and the dependence of the dynamical factor on the target, which is observed when comparing helium and neon results.

We demonstrate the feasibility of tackling continuum states with grid techniques. Our method consists of using a Lagrange mesh within the finite R-matrix sphere. We test it on a simple model scattering problem. In contrast with previous proposals, it is capable of handling anisotropic interactions, an essential property to consider realistic applications.