We study the formation of highly excited neutral atoms during the break-up of strongly-driven molecules. Past work on this significant phenomenon has shown that during the formation of highly excited neutral atoms ($mathrm{H^{*}}$) during the break-up of H$_{2}$ in a linear laser field the electron that escapes does so either very quickly or after remaining bound for a few periods of the laser field. Here, we address the electron-nuclear dynamics in $mathrm{H^{*}}$ formation in elliptical laser fields, through Coulomb explosion. We show that with increasing ellipticity two-electron effects are effectively switched-off. We perform these studies using a toolkit we have developed for semiclassical computations for strongly-driven multi-center molecules. This toolkit includes the formulation of the probabilities of strong-field phenomena in a transparent way. This allows us to identify the shortcomings of currently used initial phase space distributions for the electronic degrees of freedom. In addition, it includes a 3-dimensional method for time-propagation that fully accounts for the Coulomb singularity. This technique has been previously developed in the context of celestial mechanics and we currently adopt it to strongly-driven systems. Moreover, we allow for tunneling during the time-propagation. We find that this is necessary in order to accurately describe the fragmentation of strongly-driven molecules.
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