Dynamical hierarchy in transition states: Why and how does a system climb over the mountain?

Abstract

How a reacting system climbs through a transition state during the course of a reaction has been an intriguing subject for decades. Here we present and quantify a technique to identify and characterize local invariances about the transition state of an N-particle Hamiltonian system, using Lie canonical perturbation theory combined with microcanonical molecular dynamics simulation. We show that at least three distinct energy regimes of dynamical behavior occur in the region of the transition state, distinguished by the extent of their local dynamical invariance and regularity. Isomerization of a six-atom Lennard-Jones cluster illustrates this: up to energies high enough to make the system manifestly chaotic, approximate invariants of motion associated with a reaction coordinate in phase space imply a many-body dividing hypersurface in phase space that is free of recrossings even in a sea of chaos. The method makes it possible to visualize the stable and unstable invariant manifolds leading to and from the transition state, i.e., the reaction path in phase space, and how this regularity turns to chaos with increasing total energy of the system. This, in turn, illuminates a new type of phase space bottleneck in the region of a transition state that emerges as the total energy and mode coupling increase, which keeps a reacting system increasingly trapped in that region.T he pervasive concept of the mechanism of the most common class of chemical reactions is that of a system moving on a single effective potential surface, typically in a space of 3N-6 independent variables, for an N-body system, from one local minimum, the state of the reactants, across a saddle or transition state, to a second local minimum, that of the products. The concept of transition state or dividing surface was introduced by Eyring (1) and Wigner (2). If the system begins in thermal equilibrium, and if conditions justify assuming a quasiequilibrium between the reactants and systems crossing the transition state in the forward direction (i.e., toward the products) along the reaction coordinate q 1 , then the apparatus of ''transitionstate theory'' in any of several forms (1-12) can be invoked to evaluate the rate coefficient of the reaction. The greater part of the effort in using any of the specific methods is establishing the hypersurface dividing reactant from product so that it is as free as possible from the ''recrossing problem.'' Such dividing surfaces have usually been defined in configurational space (1-8); Davis and Gray (13) first showed that in any Hamiltonian system with two degrees of freedom (dof), the transition state defined as the separatrix in phase space is always free from barrier recrossings. This analysis, however, has been limited to systems with only two dof; there has been no general theory yet for systems of higher dimensionality (14-17). Several recent developments, theoretical (18-23) and experimental (24, 25), have shed light on mechanics of passage through reaction bottlenecks. Indicative symptoms of a local regul...

Nonlinear dynamical effects on reaction rates in thermally fluctuating environments

“…The blue dotted line shows the reaction probability calculated by the Gaussian approximation for y 1 , that is, only the terms in the first bracket in Eq. (25). The red solid line is calculated by the Gram-Charlier A series up to the cubic term [n = 3 in Eq.…”

“…45 The validity of the usage of perturbation theory to take into account such nonlinearity in the region of rank-one saddle has been ensured by several studies in experiments 46,47 and theories [48][49][50][51][52][53][54][55][56][57] on the regularity of crossing dynamics over the saddle and the corresponding phase space geometrical structure (e.g., a no-return TS) in a wide class of Hamiltonian systems. [4][5][6][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] One can naturally adopt this perturbation theory without loss of generality as far as the total energy of the system is not so very high that any perturbation treatment is invalidated. These developments, however, are all based on the Hamiltonian formalism, which corresponds to isolated systems (i.e., gas phase).…”

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

“…LCPT has been applied to seeking for such local descriptions in a perturbative manner from integrable solutions, and shown to be versatile in various types of Hamiltonian in the research fields such as celestial mechanics [26,27], atomic physics [28,29], cluster physics [30][31][32][33][34][35][36]. For example, in the context of chemical reaction dynamics, LCPT has been applied to seeking (locally-)no-return transition state and the associated reaction coordinate buried in the phase space for many-degrees of freedom Hamiltonian systems such as intramolecular proton transfer in malonaldehyde [37,38], argon cluster isomerization [30][31][32][33][34][35][36], O( 1 D) + N 2 O → NO + NO [39], a hydrogen atom in crossed electric and magnetic fields [29,40], HCN isomerization [41,42,1,2], and so forth. LCPT was generalized to dissipative systems such as multidimensional (generalized) Langevin formulation to describe reactions under thermal fluctuation, in which no-return transition state can be obtained by incorporating nonlinearity of the system and interactions with heat bath [43][44][45][46][47][48][49][50].…”

Robust Existence of a Reaction Boundary to Separate the Fate of a Chemical Reaction

“…While H 0 can take any kind of functional form, recent studies [7][8][9][22][23][24][25][26][27] have found that it is possible to introduce a coordinate transformation ðx; $; q 2 ; . .…”

“…(4) is called ''partial normal form'' (PNF) in the sense that only the action of the reactive mode is transformed as an invariant, which survives robustly even at a moderately high energy regime (because resonance does not meet between the reactive and nonreactive modes [7][8][9][22][23][24][25][26][27]). It is also possible to construct a ''full normal form'' that makes all the actions transform as invariants of motion [7][8][9][10][22][23][24][25].…”