Cylindrical manifolds in phase space as mediators of chemical reaction dynamics and kinetics. I. Theory

Chemical reaction rates must increasingly be determined in systems that evolve under the control of external stimuli. In these systems, when a reactant population is induced to cross an energy barrier through forcing from a temporally varying external field, the transition state that the reaction must pass through during the transformation from reactant to product is no longer a fixed geometric structure, but is instead timedependent. For a periodically forced model reaction, we develop a recrossing-free dividing surface that is attached to a transition state trajectory [T. Bartsch, R. Hernandez, and T. Uzer, Phys. Rev. Lett. 95, 058301 (2005)]. We have previously shown that for single-mode sinusoidal driving, the stability of the timevarying transition state directly determines the reaction rate [G. T. Craven, T. Bartsch, and R. Hernandez, J. Chem. Phys. 141, 041106 (2014)]. Here, we extend our previous work to the case of multi-mode driving waveforms. Excellent agreement is observed between the rates predicted by stability analysis and rates obtained through numerical calculation of the reactive flux. We also show that the optimal dividing surface and the resulting reaction rate for a reactive system driven by weak thermal noise can be approximated well using the transition state geometry of the underlying deterministic system. This agreement persists as long as the thermal driving strength is less than the order of that of the periodic driving. The power of this result is its simplicity. The surprising accuracy of the time-dependent noise-free geometry for obtaining transition state theory rates in chemical reactions driven by periodic fields reveals the dynamics without requiring the cost of brute-force calculations.

Section: Introductionmentioning

confidence: 99%

Chemical reactions induced by oscillating external fields in weak thermal environments

Craven

Bartsch

Hernandez

2015

“…97,98 Important theoretical advances were subsequently made relating nonstatistical behavior to molecular phase space structure, 99 in particular, the existence of intramolecular bottlenecks to energy transfer, [100][101][102] "vague tori," 103,104 broken separatrices, [105][106][107] and reactive islands and cylinders. [108][109][110][111][112][113][114] The multimode case remains less thoroughly explored and understood. Methods for defining approximate intramolecular bottlenecks and reactive dividing surfaces have been devised.…”

Section: Introductionmentioning

confidence: 99%

Microcanonical rates, gap times, and phase space dividing surfaces

Ezra

Waalkens

Wiggins

2009

112

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 isomerization reaction HCN CNH, previously studied using the methods of phase space transition state theory. Both algebraic ͑power law͒ 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 t Շ 0.5 ps, after which the product branching ratio is 1:1, the "statistical" value. At intermediate times, this statistical product ratio is accompanied by nonexponential ͑nonstatistical͒ 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.

“…5 can be obtained by noting its similarity with the reactive islands observed by De Leon et al [11][12][13] These authors interpreted the reactive islands in a two-dimensional autonomous system as the imprints of invariant "cylindrical manifolds" that separate reactive from nonreactive trajectories and that intertwine in a complicated manner as they follow the intricacies of a homoclinic tangle.…”

Section: The Driven Hénon-heiles Systemmentioning

confidence: 99%

“…Through numerical trajectory calculations we find that the set of trajectories that lead into any given channel possesses a structure reminiscent of the "reactive islands" known in autonomous systems. 4,6,[11][12][13] The time-dependent invariant manifolds introduced here form the boundaries of the reactive islands and thus provides a geometric interpretation of the island structure. Time-dependent normal form theory will prove to be an effective tool for their calculation.…”

Section: The Driven Hénon-heiles Systemmentioning

confidence: 99%

See 1 more Smart Citation

Transition state theory for laser-driven reactions

Kawai

Bandrauk

Jaffé

et al. 2007

Recent developments in Transition State Theory brought about by dynamical systems theory are extended to time-dependent systems such as laser-driven reactions. Using time-dependent normal form theory, we construct a reaction coordinate with regular dynamics inside the transition region. The conservation of the associated action enables one to extract time-dependent invariant manifolds that act as separatrices between reactive and non-reactive trajectories and thus make it possible to predict the ultimate fate of a trajectory. We illustrate the power of our approach on a driven Hénon-Heiles system, which serves as a simple example of a reactive system with several open channels. The present generalization of Transition State Theory to driven systems will allow one to study processes such as the control of chemical reactions through laser pulses.