Regularity in chaotic reaction paths. I. Ar6

Abstract

We scrutinize the saddle crossings of a simple cluster of six atoms to show (a) that it is possible to choose a coordinate system in which the transmission coefficient for the classical reaction path is unity at all energies up to a moderately high energy, above which the transition state is chaotic; (b) that at energies just more than sufficient to allow passage across the saddle, all or almost all the degrees of freedom of the system are essentially regular in the region of the transition state; and (c) that the degree of freedom associated with the reaction coordinate remains essentially regular through the region of the transition state, even to moderately high energies. Microcanonical molecular dynamics simulation of Ar6 bound by pairwise Lennard-Jones potentials reveals the mechanics of passage. We use Lie canonical perturbation theory to construct the nonlinear transformation to a hyperbolic coordinate system which reveals these regularities. This transform “rotates away” the recrossings and nonregular behavior, especially of the motion along the reaction coordinate, leaving a coordinate and a corresponding dividing surface in phase space which minimize recrossings and mode–mode mixing in the transition state region. The action associated with the reactive mode tends to be an approximate invariant of motion through the saddle crossings throughout a relatively wide range of energy. Only at very low energies just above the saddle could any other approximate invariants of motion be found for the other, nonreactive modes. No such local invariants appeared at energies at which the modes are all chaotic and coupled to one another.

Phase transitions in clusters

“…This means that particles of this ensemble do not form a crystal lattice even at zero temperature. The same result follows from the virial theorem for particles interacted through the potential (14) with k > 3 [44], so that the structure of an uniform crystal is unstable.…”

“…These clusters have the PES with many local minima [8][9][10][11][12][13]. In particular, for Lennard-Jones clusters, the number of local minima of the PES rises rapidly with the number n of component atoms, to become of order of a thousand even for n of 13 [8,9,14]. This property of the PES determines a specific cluster behavior at low temperatures [8,15].…”

Nonlinear dynamical effects on reaction rates in thermally fluctuating environments

“…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).…”

Analyzing intramolecular dynamics by fast Lyapunov indicators

“…Also noteworthy are the investigations of Berry and coworkers concerning the nonuniformity of the dynamical properties of Hamiltonian systems representing atomic clusters with up to 13 atoms. In particular, they explored how regular and chaotic behavior may vary locally with the topography of the potential energy surfaces (PES) [15,16,17,18,19,20,21,22,23,24,25]. By analyzing local Lyapunov functions and Kolmogorov entropies, they showed that when systems have just enough energy to go through a saddle in the potential energy surface, the system's trajectories become collimated and regularized through the saddle regions, developing approximate local invariants of the motion different from those in the potential well.…”