  We study the phase space geometry associated with index 2 saddles of a
potential energy surface and its influence on reaction dynamics for $n$
degree-of-freedom (DoF) Hamiltonian systems. For index 1 saddles of potential
energy surfaces (the case of classical transition state theory), the existence
of a normally hyperbolic invariant manifold (NHIM) of saddle stability type has
been shown, where the NHIM serves as the "anchor" for the construction of
dividing surfaces having the no-recrossing property and minimal flux. For the
index 1 saddle case the stable and unstable manifolds of the NHIM are
co-dimension one in the energy surface, and act as conduits for reacting
trajectories in phase space. The situation for index 2 saddles is quite
different. We show that NHIMs with their stable and unstable manifolds still
exist, but that these manifolds by themselves lack sufficient dimension to act
as barriers in the energy surface. Rather, there are different types of
invariant manifolds, containing the NHIM and its stable and unstable manifolds,
that act as co-dimension one barriers in the energy surface. These barriers
divide the energy surface in the vicinity of the index 2 saddle into regions of
qualitatively different trajectories exhibiting a wider variety of dynamical
behavior than for the case of index 1 saddles. In particular, we can identify a
class of trajectories, which we refer to as "roaming trajectories", which are
not associated with reaction along the classical minimum energy path (MEP). We
illustrate the significance of our analysis of the index 2 saddle for reaction
dynamics with two examples.
