Wigner's dynamical transition state theory in phase space: classical and quantum

Abstract

We develop Wigner's approach to a dynamical transition state theory in phase space in both the classical and quantum mechanical settings. The key to our development is the construction of a normal form for describing the dynamics in the neighborhood of a specific type of saddle point that governs the evolution from reactants to products in high dimensional systems. In the classical case this is the standard Poincaré-Birkhoff normal form. In the quantum case we develop a normal form based on the Weyl calculus and an explicit algorithm for computing this quantum normal form. The classical normal form allows us to discover and compute the phase space structures that govern classical reaction dynamics. From this knowledge we are able to provide a direct construction of an energy dependent dividing surface in phase space having the properties that trajectories do not locally "re-cross" the surface and the directional flux across the surface is minimal. Using this, we are able to give a formula for the directional flux through the dividing surface that goes beyond the harmonic approximation. We relate this construction to the fluxflux autocorrelation function which is a standard ingredient in the expression for the reaction rate in the chemistry community. We also give a classical mechanical interpretation of the activated complex as a normally hyperbolic invariant manifold (NHIM), and further describe the structure of the NHIM. The quantum normal form provides us with an efficient algorithm to compute quantum reaction rates and we relate this algorithm to the quantum version of the flux-flux autocorrelation function formalism. The significance of the classical phase space structures for the quantum mechanics of reactions is elucidated by studying the phase space distribution of scattering states. The quantum normal form also provides an efficient way of computing Gamov-Siegert resonances. We relate these resonances to the lifetimes of the quantum activated complex. We consider several one, two, and three degree-of-freedom systems and show explicitly how calculations of the above quantities can be carried out. Our theoretical framework is valid for Hamiltonian systems with an arbitrary number of degrees of freedom and we demonstrate that in several situations it gives rise to algorithms that are computationally more efficient than existing methods.

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.…”

Using scattering theory to compute invariant manifolds and numerical results for the laser-driven Hénon-Heiles system

“…The methods of this paper are of particular interest in chemistry, since invariant manifolds has proved to be useful to compute reaction rates in transition state theory (see, for example, Bartsch et al (2012); Waalkens et al (2008)). The computation of physically relevant quantities from transition state theory such as the branching ratio has been improved using invariant manifolds.…”

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

“…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.…”