Reaction rate coefficient trajectory calculation

Let us consider first the in vacuo cases. Dynamical aspects of the reaction in vacuo may be recovered by resorting to calculations of semiclassical trajectories. A cluster of independent representative points, with accurately selected classical initial conditions, are allowed to perform trajectories according to classical mechanics. The reaction path, which is a static semiclassical concept (the best path for a representative point with infinitely slow motion), is replaced by descriptions of the density of trajectories. A widely employed approach to obtain dynamical information (reaction rate coefficients) is based on modern versions of the Transition State Theory (TST) whose original formulation dates back to 1935. Much work has been done to extend and refine the original TST. 

The key idea that supplements RRK theory is the transition state assumption. The transition state is assumed to be a point of no return. In other words, any trajectory that passes through the transition state in the forward direction will proceed to products without recrossing in the reverse direction. This assumption permits the identification of the reaction rate with the rate at which classical trajectories pass through the transition state. In combination with the ergodic approximation this means that the reaction rate coefficient can be calculated from the rate at which trajectories, sampled from a microcanonical ensemble in the reactants, cross the barrier, divided by the total number of states in the ensemble at the required energy. This quantity is conveniently formulated using the idea of phase space. 

In Ref. Esposito, Capitelli, Kustova Nagnibeda (2000), the dissociation rate coefficients diss calculated within the framework of the Treanor-Marrone model are compared with those obtained from trajectory calculations Esposito, Capitelli Gorse (2000), some recommendations for the optimum choice for the parameter U for the specific reactions are given. Eigure 1 presents the temperature dependence of the state-dependent dissociation rate coefficients in an (N2, N) mixture. The coefficients are calculated for different... 

Table 2.3 shows a comparison of the calculated and experimental rate coefficients for several other compounds. The comments made for the H3O+ reaction with acetone apply also to this larger group of reactions. Where supplied, the error margins for the experimental rate coefficients are about 20-30% of the mean value. In every reaction listed in Table 2.3, the capture rate coefficient lies within the stated error margins and in most cases is close to the quoted experimental mean value. From this comparison it seems fairly safe to assume that the calculated heap rate coefficient is likely to be every bit as reliable as any experimentally determined value and thus, as mentioned earlier, in the absence of an experimental rate coefficient, trajectory methods provide the best option for determining a reaction rate coefficient for subsequent use in analysing PTR-MS data. 

We can calculate the thermal rate constants at low temperatures with the cross-sections for the HD and OH rotationally excited states, using Eqs. (34) and (35), and with the assumption that simultaneous OH and HD rotational excitation does not have a strong correlated effect on the dynamics as found in the previous quantum and classical trajectory calculations for the OH + H2 reaction on the WDSE PES.69,78 In Fig. 13, we compare the theoretical thermal rate coefficient with the experimental values from 248 to 418 K of Ravishankara et al.7A On average, the theoretical result... 

When potential surfaces are available, quasiclassical trajectory calculations (first introduced by Karplus, et al.496) become possible. Such calculations are the theorist s analogue of experiments and have been quite successful in simulating molecular reactive collisions.497 Opacity functions, excitation functions, and thermally averaged rate coefficients may be computed using such treatments. Since initial conditions may be varied in these calculations, state-to-state cross sections can be obtained, and problems such as vibrational specificity in the energy release of an exoergic reaction and vibrational selectivity in the energy requirement of an endo-... 

Recent advances have resulted from the development of more powerful experimental methods and because the classical collision dynamics can now be calculated fully using high-speed computers. By applying Monte Carlo techniques to the selection of starting conditions for trajectory calculations, a reaction can be simulated with a sample very much smaller than the number of reactive encounters that must necessarily occur in any kinetic experiment, and models for reaction can therefore be tested. The remainder of this introduction is devoted to a simple explanation of the classical dynamics of collisions, a description of the parameters needed to define them, and the relationship between these and the rate coefficient for a reaction [9]. 

Recently, Monte Carlo trajectory studies have been performed [325, 326] on systems with appreciable energy barriers, with a view to discovering whether excitation of particular degrees of freedom in the reagents promotes reaction. For three-atom reactions, if the barrier lies in the exit valley, vibrational (rather than translational or rotational) excitation can be used most effectively for surmounting the barrier. Conversely, if the barrier is in the entry valley, it is most easily surmounted if energy is located in the relative translation of the products rather than in vibration. For appreciably endothermic reactions, the barrier is very likely to be in the exit valley [323], and the conclusion that vibrational excitation will considerably assist the occurrence of such reactions is supported by calculations based on the applications of microscopic reversibility to the detailed rate coefficients for several exothermic reactions [327, 227]. It appears that similar criteria apply to four-center reactions of the AB + CD - AC + BD type [317], and the effect of vibrational excitation on the rate of such reactions has been investigated [316,317]. 

Similarly, the rate coefficient for a thermal reaction occurring with the influence of a spherically symmetric potential V(r) can be calculated from equation (63) by relating the cross-section to the potential. A useful relationship from classical scattering dynamics [16] is found in terms of the impact parameter, b. The impact parameter is the distance of closest approach between two particles in the absence of an interparticle force. At large separation, the collision trajectories of two particles will be parallel straight lines, and the impact parameter is the perpendicular distance between the trajectories. The cross-section is given by equation (64),... 

The development of the collision dynamics approach to bimolecular reactions has for the most part departed from models that seek analytical expressions for rate coefficients, and has centered on trajectory calculations, a method made possible by the development of high speed computers. 

It is evident from these equations that trajectory calculations give microscopic, or state-to-state, information for a reaction. Finally, the thermal rate coefficient for the elementary reaction can be obtained by summing the k(v, J) over each weighted level... 

Esposito, R, Capitelli, M., Kustova, E. Nagnibeda, E. (2000). Rate coefficients for the reaction N2(i)+N=3N a comparison of trajectory calculations and the Treanor-Marrone model., Chem. Phys. Lett. 330 207-211. 

In Table II we compare rate coefficients calculated [20] for the He + HCt reaction using three different theories - the ACCSA, the statistical adiabatic channel model (SACM) of Troe [14] and classical trajectory calculations [16]. The trajectory calculations have been parameterized to give the simple formula... 

While the results of trajectory calculations provide an accurate testing ground for more approximate theories, and, in the parameterised form developed by Su, Chesnavich and Bowers [25,26], a widely applied means of calculating capture rate coefficients for these more complex interactions, they provide less insight into reaction mechanisms and rate coefficient determinants than more analytic approaches. The simplest approach is provided by phase space theory (PST) which assumes an isotropic potential between the reactants [31]. The centrifugal term in the effective potential in (3.2) can be expressed in terms of the orbital angular momentum quantum number, , for the collision, so that the equation for Vejf (Rab) becomes ... 

Theoretical analyses of the reaction have been performed by Bettens et al. [39] and by Klippenstein et al. [ 15]. The former used classical trajectories on a calculated potential energy surface. Klippenstein et al. used high level electronic structure methods to calculate the potential energy surface, coupled with the analytical solutions outlined above [36], detailed transition state theory and trajectory calculations. Their paper contains a great deal of insight into the mechanism and the temperarnre dependence of the overall rate coefficient. 

Instead of seeking an analytical expression for the rate coefficient on the basis of classical expressions for the ion-molecule interaction potential, an alternative approach is to model the reaction process through a series of classical trajectory calculations. The details are somewhat complicated and so the interested reader should consult the original references. Importantly, the results of such trajectory calculations have been parameterized by Su and Chesnavich to give expressions which allow calculation of the rate coefficient [27], The resulting rate coefficient, termed the capture rate coefficient, heap, is given by... 

These studies seem to indicate that, for structureless particles, it is most important to understand the dependence of nucleation rate coefficients on cluster size for very small clusters. At the low temperatures appropriate for argon nucleation, the decay rate coefficient for excited clusters for clusters larger than seven or eight monomers becomes essentially zero, and the capture cross section for this size cluster apparently increases very slowly with n. These facts should make it very easy to compute steady-state nucleation rates for argon, provided similar information is available for the rate coefficients for the "quenching" reactions of equation (2), since it may not be necessary to use trajectories to calculate any of these rate coefficients for clusters larger than ten or twelve monomers in size. 

These calculations show that classical trajectory techniques as usually applied to chemical reactions are a useful tool for the study of cluster dynamics. We have made direct calculations of microscopic rate coefficients for some of the elementary steps of the early stages of nucleation. We have focused our attention on the formation and dissociation of quasibound clusters, however, this same approach would provide useful, fundamental information if applied to other mechanistic steps, particularly the stabliziation step, equation (2). 

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