Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed

Abstract

The fundamental assumption of conventional transition state theory is the existence of a dividing surface having the property that trajectories originating in reactants must cross the surface only once and then proceed to products. Recently it has been shown [1,2] how to construct a dividing surface in phase space for Hamiltonian systems with an arbitrary (but finite) number of degrees of freedom having the property that trajectories only cross once locally. In this letter we provide an argument showing that the flux across this dividing surface is a minimum with respect to certain types of variations of the dividing surface. In this sense, conventional transition state theory is shown to be equivalent to variational transition state theory. PACS: 82.20.Db, 82.20.Nk,

Microcanonical rates, gap times, and phase space dividing surfaces

“…148,149 Some early attempts were made to compute and visualize such manifolds in a three DoF system describing surface diffusion of atoms 150 and for a fourdimensional symplectic mapping modeling the dissociation of a van der Waals complex. 151,152 As discussed in more detail below, based on the notion of the NHIM and on the development of efficient algorithms for computing normal forms ͑NFs͒ at saddles, there has been significant recent progress in the development and implementation of phase space transition state theory 144,148,149,[153][154][155][156][157][158][159] ͑see also Refs. 130 and 160-167͒.…”

“…99,121 In the present paper we show how the expression for the microcanonical rate of reaction described above can be evaluated using the phase space approach to reaction dynamics developed in a recent series of papers. 148,149,[153][154][155][156][157][158][159] Moreover, we analyze in detail the properties of the gap time distribution previously obtained for HCN isomerization using the phase space reaction rate theory. 156 Our approach explicitly focuses on the gap time distribution for an ensemble of trajectories with initial conditions distributed uniformly on the constant energy dividing surface; this then implies that we consider the decay characteristics of an ensemble of phase points that fills the reactant region of phase space uniformly at constant energy, with nonreactive regions automatically excluded.…”

“…32 We demonstrate that various dynamical quantities related to the unimolecular reaction rate defined within Thiele's approach can be rigorously defined and computed within the recently developed theoretical framework for phase space transition state theory. 148,149,[153][154][155][156][157][158][159] Special emphasis is placed on the properties of the gap time distribution, and the relation between the inverse gap time and the rate of reaction. In Sec.…”

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).…”

Periodic-orbit formula for quantum reactions through transition states

“…This shows that the transition state at energy E is formed by a normally hyperbolic invariant manifold (NHIM) (see [4]), which in this case is an invariant sphere of dimension 2d−3, where d is the number of degrees of freedoms, and normal hyperbolicity means that the contraction and expansion rates associated with the directions normal to the sphere dominate those of the directions tangential to the sphere. For d = 2, this simply is the unstable periodic orbit of the PODS [5]. In fact, the NHIM spans another sphere which is of dimension 2d − 2 and hence has one dimension less than the energy surface and can be taken as a dividing surface.…”