Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions

Abstract

The three-dimensional hydrogen cyanide/isocyanide isomerization problem is taken as an example to present a general theory for computing the phase space structures which govern classical reaction dynamics in systems with an arbitrary ͑finite͒ number of degrees of freedom. The theory, which is algorithmic in nature, comprises the construction of a dividing surface of minimal flux which is locally a ''surface of no return.'' The theory also allows for the computation of the global phase space transition pathways that trajectories must follow in order to react. The latter are enclosed by the stable and unstable manifolds of a so-called normally hyperbolic invariant manifold ͑NHIM͒. A detailed description of the geometrical structures and the resulting constraints on reaction dynamics is given, with particular emphasis on the three degrees of freedom case. A procedure is given which uses these structures to compute orbits homoclinic to, and heteroclinic between, NHIMs. The role of homoclinic and heteroclinic orbits in global recrossings of dividing surfaces and transport in complex systems is explained. The complete description provided here is inherently one within phase space; it cannot be inferred from a configuration space picture. A complexification of the classical phase space structures to incorporate quantum effects is also discussed. The results presented here call into question certain assumptions routinely made on the global dynamics; this paper provides methods that enable one to understand and quantify the phase space dynamics of reactions without making such assumptions.

Nonlinear dynamical effects on reaction rates in thermally fluctuating environments

“…Numerical values are obtained by trajectory calculations with the initial conditions sampled by the Boltzmann distribution on the surface q 1 = 0 as in Eq. (28). Hereinafter, we mainly focus on reaction dynamics over saddle 2 because they were found to be more subject to nonlinearity than those over Saddle 1.…”

“…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).…”

Transition state theory for laser-driven reactions

“…The stable and unstable manifolds can be continued numerically beyond the region Ω as in the autonomous setting. 4,5 The dimensions of M, W s , W u , and T are 2n − 1, 2n, 2n, and 2n, respectively, in the (2n + 1)-dimensional phase space (including time). These dimensions are increased by 2 compared to the time-independent case because the manifolds are time-dependent and are not confined to an energy shell.…”

“…In this paper, we combine the concept of the TS trajectory with the method of normal form (NF) expansions based on Lie transformations 22 which has been shown to be an effective tool to calculate invariant manifolds in autonomous systems [1][2][3][4][5][6][7][8][9] (see also Refs. 23,24 for a quantum version of Lie transformations).…”

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].…”