Index k saddles and dividing surfaces in phase space with applications to isomerization dynamics

In this paper, we continue our studies of the phase space geometry and dynamics associated with index k saddles (k > 1) of the potential energy surface. Using Poincaré-Birkhoff normal form (NF) theory, we give an explicit formula for a "dividing surface" in phase space, i.e., a codimension one surface (within the energy shell) through which all trajectories that "cross" the region of the index k saddle must pass. With a generic non-resonance assumption, the normal form provides k (approximate) integrals that describe the saddle dynamics in a neighborhood of the index k saddle. These integrals provide a symbolic description of all trajectories that pass through a neighborhood of the saddle. We give a parametrization of the dividing surface which is used as the basis for a numerical method to sample the dividing surface. Our techniques are applied to isomerization dynamics on a potential energy surface having four minima; two symmetry related pairs of minima are connected by low energy index 1 saddles, with the pairs themselves connected via higher energy index 1 saddles and an index 2 saddle at the origin. We compute and sample the dividing surface and show that our approach enables us to distinguish between concerted crossing ("hilltop crossing") isomerizing trajectories and those trajectories that are not concerted crossing (potentially sequentially isomerizing trajectories). We then consider the effect of additional "bath modes" on the dynamics, by a study of a four degree-of-freedom system. For this system we show that the normal form and dividing surface can be realized and sampled and that, using the approximate integrals of motion and our symbolic description of trajectories, we are able to choose initial conditions corresponding to concerted crossing isomerizing trajectories and (potentially) sequentially isomerizing trajectories.