Phase-space geometry and reaction dynamics near index 2 saddles

Abstract

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. In recent years, similar studies have been carried out for index 1 saddles of potential energy surfaces, and the phase-space geometry associated with classical transition state theory has been elucidated. In this case, 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 1 in the energy surface and have the structure of spherical cylinders, and thus act as the conduits for reacting trajectories in phase space. The situation for index 2 saddles is quite different, and their relevance for reaction dynamics has not previously been fully recognized. 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 in order to constrain reactions. Rather, in the index 2 case there are different types of invariant manifolds, containing the NHIM and its stable and unstable manifolds, that act as co-dimension 1 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. The first involves isomerization on a potential energy surface with multiple (four) symmetry equivalent minima; the dynamics in the vicinity of the saddle enables a rigorous distinction to be made between stepwise (sequential) and concerted (hilltop crossing) isomerization pathways. The second example involves two potential minima connected by two distinct transition states associated with conventional index 1 saddles, and an index 2 saddle that sits between the two index 1 saddles. For the case of non-equivalent index 1 saddles, our analysis suggests a rigorous dynamical definition of ‘non-MEP’ or ‘roaming’ reactive events.