A formula to compute the microcanonical volume of reactive initial conditions in transition state theory

Abstract

We present the formal proof of a procedure to compute the phase-space volume of initial conditions for trajectories that, for a constant energy, escape or 'react' from a multi-dimensional potential well with one or several exit/entrance channels. The procedure relies on a phase-space formulation of transition state theory. It gives the volume of reactive initial conditions as the sum over the exit/entrance channels where each channel contributes by the product of the phase-space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel. An example is given to demonstrate the computational efficiency of the procedure.

Microcanonical rates, gap times, and phase space dividing surfaces

“…178 and 179͒, have only recently become amenable to direct computation via the use of NF approaches. 148,149,[153][154][155][156][157][158][159][160][161][162] A. Phase space dividing surfaces: definition and properties…”

“…For multidimensional systems such as polyatomic molecules ͑N Ն 3 DoF͒, it is in general no longer possible to define or compute a dividing surface with desirable dynamical attributes such as the no-recrossing property by working in configuration space alone, and a phase space perspective is necessary. 148,149,[153][154][155][156][157][158][159][160][161][162] While the phase space approach to reaction dynamics reviewed briefly below does not give a complete solution to the problem ͑questions of transition state bifurcations and nonintegrability remain to be addressed, for example͒, it does provide an important component, which, when coupled with knowledge of certain properties of the system Hamiltonian function, does provide the necessary information to identify the reactant region in some situations of interest. Saddle points of the potential energy surface still play a role-in particular, rank one saddles.…”

“…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.…”