Trajectory calculations, classical

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. 

Here J and v represent the initial rotational and vibrational conditions, respectively. Nr is the number of trajectories resulting in reaction, and TOTAL the total number of trajectories. The reactive cross-section is obtained by integrating over the impact parameters. 

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 

6 Kinetic Isotope Effects Continued Variational Transition State Theory and Tunneling 

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At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

Mayne H R 1991 Classical trajectory calculations on gas-phase reactive collisions/of. Rev. Phys. Chem. 10 107-21... 

Grigoleit U, Lenzer T and Luther K 2000 Temperature dependence of collisional energy transfer in highly excited aromatics studied by classical trajectory calculations Z. Phys. Chem., A/F214 1065-85... 

Quasiclassical calculations are similar to classical trajectory calculations with the addition of terms to account for quantum effects. The inclusion of tunneling and quantized energy levels improves the accuracy of results for light atoms, such as hydrogen transfer, and lower-temperature reactions. 

Fig. 5.7. Comparison of experimental data from [191] (1), [221] (2), [220] (3) and ultrasonic point [216] (4) with SCS calculation [191] (solid line) and classical trajectories calculations from [222] (dotted line), [223] (broken line) and [216] (4 ).

The important fact is that the number of collisions Zr increases with temperature. It may be attributed to the effect of attraction forces. They accelerate the molecule motion along the classical trajectories favouring more effective R-T relaxation. This effect becomes relatively weaker with increase of temperature. As a result the effective cross-section decreases monotonically [199], as was predicted for the quantum J-diffusion model in [186] (solid line) but by classical trajectory calculations (dotted and broken lines) as well. At temperatures above 300 K both theoretical approaches are in satisfactory mutual agreement whereas some other approaches used in [224, 225] as well as SCS with attraction forces neglected [191] were shown to have the opposite temperature dependence for Zr [191]. Thus SCS results with a... 

Russel J. D., Bernstein R. B., Curtiss C. F. Transport properties of a gas of diatomic molecules. VI. Classical trajectory calculations of the rotational relaxation time of the Ar-N2 system, J. Chem. Phys. 57, 3304-7 (1972). 

The second channel, producing CO, was first observed by Seakins and Leone [64], who estimated 40% branching to this channel. Later measurements by Lockenberg et al. [65] and Preses et al. [66] concluded the branching to CO is 18%. Note that decomposition of formaldehyde formed in reaction (26a) is not a possible source of CO due to the large barrier for formaldehyde decomposition. Marcy et al. [67] recently combined time-resolved Lourier spectroscopy experiments with direct dynamics classical trajectory calculations to examine the mechanism of the CO product channel. They observed two pathways for CO formation, neither of which involve crossing a TS. 

Fig. 1. Potential energy surface and classical trajectory calculations on the H + H2 hydrogen exchange reaction. Note the orbiting trajectory in the vicinity of Lake Eyring . Despite the unrealistic nature of a well near the transition state of this reaction, many of the modern ideas of chemical reaction theory can be seen in action already in this work. (See Ref. 1.)...

From the point of view of associative desorption, this reaction is an early barrier reaction. That is, the transition state resembles the reactants.46 Early barrier reactions are well known to channel large amounts of the reaction exoergicity into product vibration. For example, the famous chemical-laser reaction, F + H2 — HF(u) + H, is such a reaction producing a highly inverted HF vibrational distribution.47-50 Luntz and co-workers carried out classical trajectory calculation on the Born-Oppenheimer potential energy surface of Fig. 3(c) and found indeed that the properties of this early barrier reaction do include an inverted N2 vibrational distribution that peaks near v = 6 and extends to v = 11 (see Fig. 3(a)). In marked contrast to these theoretical predictions, the experimentally observed N2 vibrational distribution shown in Fig. 3(d) is skewed towards low values of v. The authors of Ref. 44 also employed the electronic friction theory of Tully and Head-Gordon35 in an attempt to model electronically nonadiabatic influences to the reaction. The results of these calculations are shown in... 

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... 

Classical trajectory calculations, performed on the PES1 and PESl(Br) potential energy surfaces described above, have provided a detailed picture of the microscopic dynamics of the Cl- + CH3Clb and Cl" + CH3Br SN2 nucleophilic substitution reactions.6,8,35-38 In the sections below, different aspects of these trajectory studies and their relation to experimental results and statistical theories are reviewed. 

The criterion for association in the classical trajectory calculations is the presence of multiple inner turning points between Cl and the CH3Clb center of mass. Two types of Cl -CH3Clb complexes, short-range and long-range, were observed in the trajectories. The two different complexes were identified by the Cl -C distance... 

A simple model for the dynamics of nonresonant laser-induced desorption of adsorbates from surfaces has been formulated by Lucchese and Tully (LT). LT present the result of stochastic, classical trajectory calculations for thermal and laser-induced desorption of NO from LiF(100). For the LID simulations the initial temperature was set at 0 K and temperature Jumps of several thousand degrees were driven in a few picoseconds through nonspecific heating of the substrate. The interaction potential for these calculations... 

The exchange between the gas-phase and chemisorbed states of small molecules plays a vital role in such technologically important fields as heterogeneous catalysis and corrosion. The dynamics involved in these processes, however, are not currently well understood. Molecular-beam studies combined with classical trajectory calculations have proven to be a successful tool for understanding the underlying features of atomic-scale motion in the gas phase. The extension of these techniques to surfaces has also helped in elucidating the details of gas-surface reactions. 

Figure 7. Dissociation rates k as extracted from the quantum mechanical calculations (open circles). The statistical rates are represented by the step functions and the filled circles represent the classical rate constants as obtained from elaborate classical trajectory calculations. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)...

J. Troe Let me point out that the ab initio calculations of the HO2 surface still must be in error in their long-range part. High-pressure H + O2 recombination experiments clearly show that the potential is completely loose and not semirigid like the potential you used for illustration. For loose potentials SACM and classical trajectory calculations of product distributions (in the classical range) nicely agree. 

Of particular interest is the relation between the various forms of VTST and SACM. The charge-dipole system is particularly well suited to investigate this aspect in a quantitative way. In the following we show that, for potentials without pronounced energy barriers, VTST and SACM in general are not equivalent, the numerical differences depending on the chosen variant of VTST. Because SACM agrees well with classical trajectory calculations, the comparison of VTST with SACM may help to identify artifacts of the VTST treatment. 

Below we shall compare this result with that from the corresponding classical trajectory calculations as well as from variational transition-state treatments. 

Comparisons of classical trajectory calculations with various versions of VTST have also been performed for quite a series of other reaction systems such as neutral radical association/dissociation and radical-surface association processes [27-30], In these studies, the various treatments employed... 

Such a method has recently been developed by Miller. et. al. (28). It uses short lengths of classical trajectory, calculated on an upside-down potential energy surface, to obtain a nonlocal correction to the classical (canonical) equilibrium probability density Peq(p, ) at each point then uses this corrected density to evaluate the rate constant via eq. 4. The method appears to handle the anharmonic tunneling in the reactions H+HH and D+HH fairly well (28), and can... 

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