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 one 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 codimension 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. 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 one saddles, and an index two saddle that sits between the two index one saddles. For the case of non-equivalent index one saddles, our analysis suggests a rigorous dynamical definition of "non-MEP" or "roaming" reactive events.

Phase space barriers and dividing surfaces in the absence of critical points of the potential energy: Application to roaming in ozone

“…35 There is even richer variety of phase space structures and invariant manifolds associated with index two saddles of the potential energy surface, and their implications for reaction dynamics have begun to be explored. 27,31,32 Fundamental theorems assure the existence of these phase space structures and invariant manifolds for a range of energy above that of the saddle. 34 However, the precise extent of this range, as well as the nature and consequences of any bifurcations of the phase space structures and invariant manifolds that might occur as energy is increased, is not known and is a topic of continuing research.…”

“…To show that the vector field is nowhere tangent to the sampled surface, except on the NHIM, we must evaluate the flux form 27 associated with the Hamiltonian vector field through the DS on tangent vectors to the DS. The Hamiltonian vector field is not tangent to the DS if the flux form is nowhere zero on the DS, except at the NHIM.…”

“…[22][23][24][25]. More recently, index two [26][27][28] and higher index 29 saddles have been studied (see also Refs. [30][31][32].…”

Models for Intrinsic Non-RRKM Dynamics. Decomposition of the SN2 Intermediate Cl––CH3Br

“…The phase space for the non-ergodic dynamics of a multi-dimensional polyatomic molecule consists of an Arnold web of corridors of irregular/chaotic motion traversing interconnected regions of regular motion [26,28]. A molecular understanding of intrinsic non-RRKM dynamics desires the identification of the specific mode excitations which lead to irregular or regular atomic-level dynamics [31][32][33][34][35].…”

Reactivity boundaries for chemical reactions associated with higher-index and multiple saddles

“…To reveal the fundamental mechanism of the passage through a saddle with an index greater than 1, the phase-space structure was recently studied on the basis of normal form (NF) theory [68][69][70][71]. For example, the pioneering studies to extend the dynamical reaction theory into higher-index saddles were reported [68] for concerted reactions.…”