  We investigate the fragmentation dynamics of an atomic chain under tensile
stress. We have classified the location, stability type (indices) and energy of
all equilibria for the general $n$-particle chain, and have highlighted the
importance of saddle points with index $> 1$.
  We show that for an $n=2$-particle chain under tensile stress the index 2
saddle plays a central role in organizing the dynamics. We apply normal form
theory to analyze phase space structure and dynamics in a neighborhood of the
index 2 saddle. We define a phase dividing surface (DS) that enables us to
classify trajectories passing through a neighborhood of the saddle point using
the values of the integrals associated with the normal form. We also generalize
our definition of the dividing surface and define an \emph{extended dividing
surface} (EDS), which is used to sample and classify all trajectories that pass
through a phase space neighborhood of the index 2 saddle at total energies less
than that of the saddle.
  Classical trajectory simulations are used to study single versus double bond
breakage for the $n=2$ chain under tension. Initial conditions for trajectories
are obtained by sampling the EDS at constant energy. We sample trajectories at
fixed energies both above and below the energy of the saddle. The fate of
trajectories (single versus double bond breakage) is explored as a function of
the location of the initial condition on the EDS, and a connection made to the
work of Chesnavich on collision-induced dissociation. A significant finding is
that we can readily identify trajectories that exhibit bond \emph{healing}.
Such trajectories pass outside the nominal (index 1) transition state for
single bond dissociation, but return to the potential well region, possibly
several times, before ultimately dissociating.
