Phase space geometry and reaction dynamics near index two saddles

TL;DR

This paper explores the complex phase space structures near index 2 saddles in Hamiltonian systems, revealing new invariant manifolds that influence reaction pathways and introduce roaming trajectories beyond classical transition state theory.

Contribution

It extends the understanding of phase space geometry from index 1 to index 2 saddles, identifying new invariant manifolds that act as barriers and influence reaction dynamics.

Findings

Invariant manifolds near index 2 saddles differ from index 1 cases.

Existence of barriers dividing the energy surface around index 2 saddles.

Identification of roaming trajectories not aligned with classical pathways.

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. 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.