  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.
