Breakup of the aligned H$_2$ molecule by xuv laser pulses: A time-dependent treatment in prolate spheroidal coordinates

We have carried out calculations of the triple-differential cross section for one-photon double ionization of molecular hydrogen for a central photon energy of $75$~eV, using a fully {it ab initio}, nonperturbative approach to solve the time-dependent Schro equation in prolate spheroidal coordinates. The spatial coordinates $xi$ and $eta$ are discretized in a finite-element discrete-variable representation. The wave packet of the laser-driven two-electron system is propagated in time through an effective short iterative Lanczos method to simulate the double ionization of the hydrogen molecule. For both symmetric and asymmetric energy sharing, the present results agree to a satisfactory level with most earlier predictions for the absolute magnitude and the shape of the angular distributions. A notable exception, however, concerns the predictions of the recent time-independent calculations based on the exterior complex scaling method in prolate spheroidal coordinates [Phys.~Rev.~A~{bf 82}, 023423 (2010)]. Extensive tests of the numerical implementation were performed, including the effect of truncating the Neumann expansion for the dielectronic interaction on the description of the initial bound state and the predicted cross sections. We observe that the dominant escape mode of the two photoelectrons dramatically depends upon the energy sharing. In the parallel geometry, when the ejected electrons are collected along the direction of the laser polarization axis, back-to-back escape is the dominant channel for strongly asymmetric energy sharing, while it is completely forbidden if the two electrons share the excess energy equally.