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We are comparing the performance of various numerical methods that can be used to solve the Schrodinger's Equation for the Hydrogen Atom interacting with a strong laser pulse (too strong to use perturbation methods). When using discretization schemes for the radial part, it seems like most (all) people put the atom in a box, just chopping the radius off at some large value and solving for those basis sets. How does this compare with mapping the radial variable to a finite domain, and then discretizing that domain (in the process, throwing out most of the basis sets available)? Is there a reason nobody seems to do that?

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    $\begingroup$ The reason is probably that taking the box large enough will not influence the results at all for the given numerical accuracy, so nobody bothers with mapping the variable. However, a simple google search revealed for example this publication: dx.doi.org/10.1137/S1064827596301418 that deals with mapping the infinite domain to a finite interval. $\endgroup$ – Ondřej Čertík Aug 17 '12 at 18:10
  • $\begingroup$ What is the functional form of the pulse? I don't see why this can't be solved nearly analytically. $\endgroup$ – Jeff Oct 5 '12 at 1:41
  • $\begingroup$ @Jeff: The pulse is likely too short to use Flouquet methods, and even if they could be used I suspect that the OP is interested in other species besides the H-atom. $\endgroup$ – Dan Oct 6 '12 at 1:05
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Baker et al. proposed such a mapping for a radial grid for atomic & molecular electronic structure computations in 1994. It is still used in modern electronic structure codes, e.g. FHI-AIMS uses them, as described in a recent paper.

Even with such a mapping, the same problems still remains: if something interesting should happen beyound the outermost grid point, you will miss it. However, these mappings do have the advantage that the grid can be systematically improved toward the inclusion of distant grid points. (This is explained in section 4.1 of the recent FHI-AIMS paper).

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