Calculation of Reaction Probabilities

Zhang D H and Zhang J Z H 1995 Quantum calculations of reaction probabilities for HO + CO and bound states of HOCO J. Chem. Phys. 103 6512... 

Among those TDQM studies, exact quantum dynamical calculations were usually limited to the total angular momentum / = 0. For / > 0, most of the authors used a capture model (or L-shift model) [77] to estimate the reaction probability from the / = 0 results. Even the direct calculations of reaction probabilities for / > 0 were performed using the centrifugal sudden (CS) approximation. Carroll... 

Agrawal. P.M. and Raff. L.M. (1981) Calculation of reaction probabilities and rate cocfficicnrs for collincar three-body exchange reactions using time-dependent wave packet methods J. Chem. Phys. 74. 5076-5081. 

Siace the discovery of quantum mechanics,more than fifty years ago,the theory of chemical reactivity has taken the first steps of its development. The knowledge of the electronic structure and the properties of atoms and molecules is the basis for an understanding of their interactions in the elementary act of any chemical process. The increasing information in this field during the last decades has stimulated the elaboration of the methods for evaluating the potential energy of the reacting systems as well as the creation of new methods for calculation of reaction probabilities (or cross sections) and rate constants. An exact solution to these fundamental problems of theoretical chemistry based on quan-tvm. mechanics and statistical physics, however, is still impossible even for the simplest chemical reactions. Therefore,different approximations have to be used in order to sii lify one or the other side of the problem. 

Before ending this section we mention a novel application of periodic orbits towards understanding the structure of resonances in reactive scattering. Exact quantal calculations of reaction probabilities are available for example, for the H+H2 exchange reaction on two different potential energy surfaces - the PK(II) and the LSTH surfaces.One finds, that qualitatively the quantal reaction probabilities are identical on the two surfaces. It came therefore as quite a surprise to find that for the MU+D2 reaction, the LSTH surface exhibits many narrow resonances which simply don t exist on the PKII surface. On the other hand, for MU+H2 the reaction probabilities on the two surfaces are again qualitatively similar - no resonances are found for either surface. 

We begin our brief review of recent research on the a+HQ reaction with Ref. [4], which reported the first accurate ID quantum calculation of reaction probabilities for Q+XQ->QX+Q (X=Mu,H,D), using the BCMR surface. Reference [4] also contains many citations to pre-1983 research on the Q+HQ reactioa... 

There are several avenues for future study suggested by the present results. In this Section we consider the following first, possible improvements to the iterative calculation of reaction probabilities by an ABC Green s function on a grid and second, a generalization of Makri s effective free particle propagator. 

Here, we report the results of quantum mechanical close coupling calculations of reaction probabilities on the ah initio surface of Walch et at. and the DIM one of Whitlock et The integration... 

The curve marked ion-dipole is based on the classical cross-section corresponding to trajectories which lead to intimate encounters (9, 13). The measured cross-sections differ more dramatically from the predictions of this theory than previously measured cross-sections for exothermic reactions (7). The fast fall-off of the cross-section at high energy is quite close to the theoretical prediction (E 5 5) (2) based on the assumption of a direct, impulsive collision and calculation of the probability that two particles out of three will stick together. The meaning of this is not clear, however, since neither the relative masses of the particles nor the energy is consistent with this theoretical assumption. This behavior is, however, probably understandable in terms of competition of different exit channels on the basis of available phase space (24). 

Free energy profiles can also be evaluated within the partial path transition interface sampling method (PPTIS), a path sampling technique designed for the calculation of reaction rate constant in systems with diffusive barrier-crossing events [31,32], In this approach, the reaction rate is expressed in terms of transitions probabilities between a series of nonintersecting interfaces located between regions. c/ and... 

For determination of reaction probability and reaction cross section, a large number of collision trajectories have to be considered and appropriate averages over the initial conditions performed. The reaction probability is calculated for a specified initial relative velocity vR (i.e. initial relative kinetic energy), rotational state /, and impact parameter b. The reaction probability is the ratio of number of reactive trajectories to the total number trajectories, i.e. 

Fig. 6.4 Plot of reaction probability vs. initial translational energy for the H + HH = HH + H reaction for a certain empirical potential energy surface (the Porter-Karplus surface). Curves (reading down) are shown for the path shown as PP in Fig. 6.3a. (marked Marcus-Coltrin), the exact quantum mechanical result for the Porter-Karplus surface (marked Exact QM), the usual TST result calculated for the MEP, QQ (Fig. 6.3a) (The data are from Marcus, R. A. and Coltrin, M. E., J. Chem. Phys. 67, 2609 (1977))...

It is possible to solve Eq. (1.10) numerically for the nuclear motion associated with chemical reactions and to calculate the reaction probability including detailed state-to-state reaction probabilities (see Section 4.2). However, with the present computer technology such an approach is in practice limited to systems with a small number of degrees of freedom. 

In order to obtain agreement with experimental data, a realistic interaction potential must be used and it is necessary to calculate the reaction probability from the basic equations of motion. 

This paper draws a parallel between the (full) six-dimensional H + H2O —> H2 -I- OH and the (reduced) seven-dimensional H -l- CH4 —> H2 + CH3 abstraction reactions. In Sec. 2, we briefly present the initial state TD quantum wave packet approach for the A -I- BCD and X + YCZ3 reactions. The Hamiltonians, body-fixed (BE) parity-adapted rotational basis functions, initial state construction and wave packet propagation, and extraction of reaction probabilities, reaction cross sections, and thermal rate coefficients from the propagated wave packet to compare with experiments are discussed. In Sec. 3 we briefly outline the potential energy surfaces used in the calculations. Some... 

One of the most important and elusive quantities in molecular djmamics calculations of reacting system is the intermolecular energy transfer probability density function P(E ,E) which is used in master equation calculations of reaction rate coefficients [1]. 

The derivations of this example are detailed and serve as a completion to chapter 3.2 and in particular to section 3.2-4. Of special importance is the calculation of the probabilities P33 and P34 elaborated below. The following reactions are considered showing at some conditions a complicated mixed-mode behavior [69]. ... 

Statistical models for the analysis of NMR data are used in two complementary approaches (Fig. 2) an analytical (model fitting) approach and a synthetic (computer simulation) approach. In the analytical approach, assigned NMR resonance intensities are fit to expected intensities based on statistical models. In the synthetic approach, spectral intensities are first calculated using reaction probabilities predicted by theoretical models these theoretical intensities are matched with those observed in the NMR spectrum. The calculation is based on theoretical probability expressions or Monte Carlo simulation. In an integrated approach, both methods are used for more complex systems. 

In the reduction of NO by CO at low temperature, on Pd/MgO(100) model catalysts, the reaction rate is independent of CO pressure but increases with NO pressure [64,65], then the TON of the reaction is modified by the reverse-spillover effect and, in particular, depends on particle size. In that case, it is no longer possible to compare the TON value measured on different particle sizes, it is more appropriated to calculate the reaction probability [64], which takes into account for the real flux of molecules that reach the clusters which can be measured by molecular beam experiments. Reverse-spillover effects have also been recently observed for the CO oxidation on size-selected Pd clusters soft-landed on MgO epitaxial films [66]. 

The product function retrieves the partial atomic sigma charge of reaction center 3 and assigns it to the variable qsigp3. Finally, the reactivity is calculated using the partial sigma charges of both reaction centers. The rule returns OK if no error occurred and provides the reactivity value for further evaluation of reaction probability. 

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