Efficient Procedure to Compute the Microcanonical Volume of Initial Conditions that Lead to Escape Trajectories from a Multidimensional Potential Well

Abstract

A procedure is presented for computing the phase space volume of initial conditions for trajectories that escape or "react" from a multidimensional potential well. The procedure combines a phase space transition state theory, which allows one to construct dividing surfaces that are free of local recrossing and that minimize the directional flux, and a classical spectral theorem. The procedure gives the volume of reactive initial conditions in terms of a sum over each entrance channel of the well of the product of the phase space flux across the dividing surface associated with the channel and the mean residence time in the well of trajectories which enter through the channel. This approach is illustrated for HCN isomerization in three dimensions, for which the method is several orders of magnitude more efficient than standard Monte Carlo sampling.

Microcanonical rates, gap times, and phase space dividing surfaces

“…III we consider the dynamics of the HCN isomerization reaction. We analyze previously computed classical trajectory results 155,156 in light of the discussion given in Sec. II.…”

“…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.…”

“…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.…”

“…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.…”

“…The particular reaction chosen for study is the isomerization HCN CNH, 155,156,[168][169][170][171][172][173][174][175][176] for which several relevant theoretical quantities have recently been computed for a ͑classical͒ model of the HCN molecule at fixed energy. 155,156 We are concerned with the rate of reaction, at fixed energy, for a system described by a time-independent, n DoF classical Hamiltonian. One measure of the rate at which trajectories leave a region of the energy surface is given by the ͑magnitude of the͒ flux of trajectories leaving that region ͑units of energy surface volume/time͒ divided by the energy surface volume of initial conditions in that region corresponding to trajectories that will eventually leave the region.…”