Reaction Probabilities and Cross Sections

We have obtained converged reaction probabilities and cross sections according to Eqs. (5.7) and (5.5), respectively. There are 13 convergence parameters to optimize. These fall into four roughly independent groups ABC parameters (A , A , Zq ) for defining the absorbing potential [cf. Eq. (5.39)], parameters 

We demonstrate the convergence of selected reaction probabilities with respect to max) which Control the spatial extent of the grid and the strength 

Reaction probabiHties which give dominant contributions to their respective rate constants at T = 300/tT are Pv ij=o i-o(E = 1.0 eV), = 1.014 eV), 

The partial cross sections required to construct or = i,j=o(E) are shown in Fig. 5.6 for various total energies. These were computed using Kmax — 2, based on the convergence behavior seen in Fig. 5.5. The linear increase for small 7 results 

The D+H2(t = l,i) reaction cross sections are shown in Fig. 5.7, as a function of total energy. The thick lines show the present calculations for j = (0,1,2,3), and the thin dotted line is the j = 0 result of Zhang and Miller obtained from the SMKVP [16]. We see complete agreement for j = 0 between the two methods over the entire energy range. The discrepancies between the SMKVP and the ABC-DVR-Newton method for J = 0 and E 1 eV seem to have averaged out in the sum over orbital and total angular momentum. 

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In this section, we shall consider how the solution of the classical equations of motion for more than two atoms may be used to find reaction probabilities and cross-sections for chemical reactions. Although the treatment is based on classical mechanics, it is termed quasi-classical because quantization of vibrational and rotational energy levels is accounted for. 

Because of the very large well in the i)otential energy surface, very few accurate quantum dynamical results were available five years ago, either by tiirK dcpendcnt or by time-independent methods. Indeed, a very largo nunit)cr of channels or of grid points are necessary to converge reaction probabilities and cross sections. 

The theory presented here is for calculating the initial state-specific total reaction probabilities and cross sections for a diatom-diatom reaction AB + CD - A + BCD in full dimensions. The Hamiltonian expressed in the reactant Jacobi coordinates shown in Fig. 1 for a given total angular momentum J can be written as... 

In the previous section we have introduced a semiclassical methodology which is capable of giving total reaction probabilities and cross sections. If one is interested in state-to-state resolved quantities, it is necessary to introduce a coordinate system in which reactant and product channels are described in the same manner, i.e., so that the separation into a quantum and a classical part does not discriminate between the channels. The reason is that it is not possible to construct a proper wave function for the product channel from a quantum-classical method in which a mixture of classical and quantum coordinates is... 

Fig. 4.2 Schematic illustration of the threshold resonances (a) and queintum dynamical resonance (b), adapted from [67]. In each panel, the left figure illustrates the effective dynamical potential along the reaction coordinate R, and the middle and the right ones tire the c-dependence of the reaction probability Pj Ef) and the reaction cross section a(Ec). In the case of threshold resonance, the non-zero values of the reaction probability and cross section start at a smaller collision energy than the height of the barrier, which manifests the effect of zero-point eneigy...

An important recent theoretical development is the direct approaches for calculating rate constants. These approaches express the rate constant in terms of a so-called flux operator and bypass the necessity for calculating the complete state-to-state reaction probabilities or cross-sections prior to the evaluation of the rate constant [1-3]. This is the theme of this chapter. 

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. 

CSDW technique included [28] Partial wave reaction probabilities, integral cross sections and product rotational distributions. In addition differential cross sections were obtained from the reaction probabilities by applying Herschbach s optical model. 

These and similar qualitative considerations have been taken as the basis for the formulation of the kinematic models of the exchange reactions for which only a part of the potential surface is used in the calculation of the probability and cross section of the reaction. The dynamic problem can then be subdivided into several more simple problems which can be treated more readily. In this connection we mention the direct interaction model involving repulsion of products (DIRP) [372], various simulations of vibrational and translational energy redistribution by forced oscillators (FOTO) [371] and the model of sudden transformation of the reactant state into the product states, also referred to as the Franck-Condon model [415, 416]. 

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. 

This is the mcvcimumpossible rate ojbimolecular reaction, the collision rate of the molecules that can react. We must multiply this by a probability of reaction in the collision so actual rates must be less than this. We know that the units of k should be Uters/mole time, and, since velocity is in lengthAime and cross section is in area/time, the units are correct if we make sure that we use volume in liters, and compute the area of a mole of molecules. If the molecular weight is 28 (air) and the temperature is 300 K, then we have... 

The results presented here are in snpport of earlier stndies which have correlated structure in the partial wave reaction probabilities and structure in the differential cross sections. Especially noteworthy among these earlier stndies are the ones of Redmon and Wyatt(21) who first suggested this correlation based on their j -conserving quantum calculations of the F- H2 -> HF-i-H reaction. He applied the CEQ version of the reduced dimensionality method to a study of that reaction( ) as well as the F- HD -> BF+D, DF+H reac-tions(9) and found similar correlations. More recently the BCRLN was used in an extensive study of differential cross sections in the... 

Most of the classical, semiclassical and quantal calculations of transition probabilities (or cross sections) refer to colinear atom-diatom gas phase reactions. However, a consideration of the nonlinear collisions seems to be very important for an adequate description of the chemical elementary processes in physical space. Quite recently, encouraging progress in this direction has been made / /. 

In light nuclei it is often possible to follow the decay of compound nuclei in resonance reactions for the spacing of levels can be wide compared with the resolving power of existing apparatus. For heavy nuclei, discrete levels in the compound nucleus usually can be detected at the present time only by the interaction of slow neutrons with nuclei. This subject is discussed elsewhere in this series. We are concerned here with compound nuclei in which a great many levels are excited. The theoretical discussion therefore necessitates a. statistical approach. Thus if the probability of decay of the compound nucleus C in the direction (7.1) is represented hy F B,b), and if Fq is the sum of the probabilities of all possible reactions, then the cross section for the reaction aA->bB is ... 

Additional theoretical refinement is needed before we can quantitatively compare BCRLM differential cross sections with 3D. The shape of the BCRLM differential cross section contains information about the impact parameter dependence of the reaction probability, and whenever the 3D angular distribution retains only this level of dynamical detail, we would expect the BCRLM differential cross section to compare nicely. Consequently, it is likely that BCRLM will fare best when compared to 3D differential cross sections from the ground state of reactants to all product rotational states, since this type of cross section retains the least amount of rotational Information. 

Then, with the following formulae, the thermally averaged dynamical quantities, the total reaction probability and the total cross-section can be drawn (F is the flux operator) ... 

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