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

Abstract

Nonlinear dynamics around a rank-one saddle is investigated in a high energy regime above the reaction threshold. The transition state (TS) is considered as a surface of a ''point of no return'' through which all reactive trajectories pass only once in the process of climbing over the saddle before being captured in the product state. A no-return TS ceases to exist above a certain high energy regime. However, even at high energies where the no-return TS can no longer exist, it is shown that ''an impenetrable barrier'' in the phase space robustly persists, which acts as a boundary between reactive and nonreactive trajectories. This implies that we can yet predict the fate of reactions even when the no-return TS may not exist. As an example, we show the analysis of dynamical systems theory for a hydrogen atom in crossed electric and magnetic fields.

Phase space geometry of dynamics passing through saddle coupled with spatial rotation

“…19,70,71 Further relaxation of the normalization is also possible, by which we can still extract the stable/unstable invariant manifolds (W s /W u ) with yet better convergence even when no-return TS (T ) may not exist. 30,49 Note also that the generating function, and therefore also the new Hamiltonian, are rational functions, rather than polynomials, of the dynamical variable J a as in Eq. (37).…”

“…30,70 We compare the value ofH truncated at μth order perturbation with true Hamiltonian H . The difference is denoted by H (μ) .…”

“…The potential of the theories has been demonstrated not only in chemical reactions with 17,22 and without [23][24][25][26][27] time-dependent external field but also in ionization of a hydrogen atom in crossed electric and magnetic fields, [28][29][30] isomerization of clusters, [31][32][33][34][35][36] and the escape of asteroids from Mars 37,38 [Just recently the theory was also generalized to quantum Hamiltonian systems [39][40][41] and dissipative (generalized) Langevin systems. [42][43][44][45][46][47][48][49][50][51] The dimension of the phase space of an N -particle nonrigid system is (6N − 10) in the upper limit.…”

“…Thanks to normal form (NF) theories (a classical analog of quantum Van Vleck perturbation theory), it was revealed in classical Hamiltonian systems with many degrees of freedom, 9,10,17,19,[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] that one can robustly extract the NHIM, no-return TS, and the stable/unstable invariant manifolds up to a moderately high energy regime above the saddle point energy, even under the existence of chaos arising from nonlinear couplings. The potential of the theories has been demonstrated not only in chemical reactions with 17,22 and without [23][24][25][26][27] time-dependent external field but also in ionization of a hydrogen atom in crossed electric and magnetic fields, [28][29][30] isomerization of clusters, [31][32][33][34][35][36] and the escape of asteroids from Mars 37,38 [Just recently the theory was also generalized to quantum Hamiltonian systems [39][40][41] and dissipative (generalized) Langevin systems.…”

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].…”