^{1,a)}, Holger Waalkens

^{2,b)}and Stephen Wiggins

^{3,c)}

### Abstract

The general approach to classical unimolecular reaction rates due to Thiele is revisited in light of recent advances in the phase space formulation of transition state theory for multidimensional systems. Key concepts, such as the phase space dividing surface separating reactants from products, the average gap time, and the volume of phase space associated with reactive trajectories, are both rigorously defined and readily computed within the phase space approach. We analyze in detail the gap time distribution and associated reactant lifetime distribution for the isomerizationreaction, previously studied using the methods of phase space transition state theory. Both algebraic (power law) and exponential decay regimes have been identified. Statistical estimates of the isomerization rate are compared with the numerically determined decay rate. Correcting the RRKM estimate to account for the measure of the reactant phase space region occupied by trapped trajectories results in a drastic overestimate of the isomerization rate. Compensating but as yet not fully understood trapping mechanisms in the reactant region serve to slow the escape rate sufficiently that the uncorrected RRKM estimate turns out to be reasonably accurate, at least at the particular energy studied. Examination of the decay properties of subensembles of trajectories that exit the HCN well through either of two available symmetry related product channels shows that the complete trajectory ensemble effectively attains the full symmetry of the system phase space on a short time scale , after which the product branching ratio is 1:1, the “statistical” value. At intermediate times, this statistical product ratio is accompanied by nonexponential (nonstatistical) decay. We point out close parallels between the dynamical behavior inferred from the gap time distribution for HCN and nonstatistical behavior recently identified in reactions of some organic molecules.

H.W. acknowledges the EPSRC for support under Grant No. EP/E024629/1. S.W. acknowledges the support of the Office of Naval Research under Grant No. N00014-01-1-0769. G.S.E. and S.W. acknowledge the stimulating environment of the NSF sponsored Institute for Mathematics and its Applications (IMA) where this manuscript was completed.

I. INTRODUCTION

II. GENERAL APPROACH TO UNIMOLECULAR REACTION RATES

A. Phase space dividing surfaces: definition and properties

B. Phase space volumes and gap times

C. Gap time and reactant lifetime distributions

D. Reaction rates

E. Multiple saddles

III. HCN ISOMERIZATIONDYNAMICS

A. Gap time and reactant lifetime distributions

B. Statistical and modified statistical rates

C. Statistical branching ratio accompanied by nonexponential decay

D. Comparison with other calculations

E. Relation to nonstatistical behavior in reactions of organic molecules

IV. SUMMARY AND CONCLUSION

### Key Topics

- Isomerization
- 27.0
- Classical chemical theories
- 20.0
- Surface reactions
- 20.0
- Transition state theory
- 20.0
- Chemical reaction theory
- 19.0

## Figures

Phase space structures for unimolecular reaction (schematic). (a) Definition of reactant region, NHIM, and dividing surface . (b) Definition of gap time and lifetime .

Phase space structures for unimolecular reaction (schematic). (a) Definition of reactant region, NHIM, and dividing surface . (b) Definition of gap time and lifetime .

Isopotential surfaces of the HCN potential energy surface of Ref. 192 in polar representation of the Jacobi coordinates , , and .

Isopotential surfaces of the HCN potential energy surface of Ref. 192 in polar representation of the Jacobi coordinates , , and .

HCN gap time distribution . (a) Complete ensemble. (b) Subensemble reacting via channel 1. (c) Subensemble reacting via channel 2.

HCN gap time distribution . (a) Complete ensemble. (b) Subensemble reacting via channel 1. (c) Subensemble reacting via channel 2.

HCN integrated gap time (reactant lifetime) distribution at short times. is plotted vs for . (a) Total ensemble. (b) Subensemble reacting via channel 1. (c) Subensemble reacting via channel 2.

HCN integrated gap time (reactant lifetime) distribution at short times. is plotted vs for . (a) Total ensemble. (b) Subensemble reacting via channel 1. (c) Subensemble reacting via channel 2.

Integrated gap time (reactant lifetime) distribution for the complete ensemble. (a) A log-log plot shows power law decay at intermediate times. (b) Log plot shows exponential decay .

Integrated gap time (reactant lifetime) distribution for the complete ensemble. (a) A log-log plot shows power law decay at intermediate times. (b) Log plot shows exponential decay .

Integrated gap time (reactant lifetime) distribution for the subensemble reacting via channel 1. (a) A log-log plot shows power law decay at intermediate times. (b) Log plot shows exponential decay .

Integrated gap time (reactant lifetime) distribution for the subensemble reacting via channel 1. (a) A log-log plot shows power law decay at intermediate times. (b) Log plot shows exponential decay .

Integrated gap time (reactant lifetime) distribution for the subensemble reacting via channel 2. (a) A log-log plot shows power law decay at intermediate times. (b) Log plot shows exponential decay .

Integrated gap time (reactant lifetime) distribution for the subensemble reacting via channel 2. (a) A log-log plot shows power law decay at intermediate times. (b) Log plot shows exponential decay .

HCN survival probability . is the fraction of an ensemble of trajectories uniformly distributed throughout the HCN region of phase space at remaining in the well at time . Trajectories are removed from the ensemble once they exit the HCN region by crossing , ; they cannot re-enter the region.

HCN survival probability . is the fraction of an ensemble of trajectories uniformly distributed throughout the HCN region of phase space at remaining in the well at time . Trajectories are removed from the ensemble once they exit the HCN region by crossing , ; they cannot re-enter the region.

The log of the number of trajectories remaining at time versus is plotted for each subensemble: channel 1 (blue), channel 2 (red).

The log of the number of trajectories remaining at time versus is plotted for each subensemble: channel 1 (blue), channel 2 (red).

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