Cross-sections and rate constants

We begin by establishing the relation between the so-called reaction cross-section jr and the bimolecular rate constant. Let us consider an elementary gas-phase reaction, 

These requirements can be met in a so-called crossed molecular-beam experiment, which is sketched in Fig. 2.1.1. Here we can generate beams of molecules with well-defined velocities and it is possible to determine the speed of the product molecules, e.g., vc = vc, by the so-called time-of-Sight technique. The elimination of multiple scattering in the reaction zone and collisions in the beams are obtained by doing the experiments in high vacuum, that is, at very low pressures. 

In an experiment, we can monitor the number of product molecules, C or D, emerging in a space angle dFl around the direction Q Kl is given by the physical design of the detector and Q by its position (fl is conveniently specified by the two polar angles 9 and j ). This is the simplest analysis of a scattering process, where we just count the number of product molecules independent of their internal state 

1The word collision should not be taken too literally, since molecules are not, say, hard spheres where it is straightforward to count the hits . Thus, a collision should really be interpreted as the broader term a scattering event . 

At sufficiently low pressure (as in the beam experiment) where an A molecule only collides with one single B molecule in the reaction zone, it will hold that the number of product molecules is proportional to the number of collisions between A and B molecules. Clearly, that number depends on the relative speed of the two molecules, v = i a — vB, the time interval dt, and the number of B molecules. Therefore, if we assume that the number density (number/m3) of B molecules in quantum state j and with velocity vb is b(j b) and that the flux density of A molecules relative to the B molecules is JA(i,v) (number/(m2 s)), then the number of collisions between A and B in the time interval dt is proportional to nB(j,va)V JA(i,v) Adt. Here V is the volume of the reaction zone (see Fig. 2.1.1), and A the cross-sectional area of the beam of A molecules. 

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Zhang D H and Zhang J Z H 1994 Accurate quantum calculations for H2+OH probabilities, cross sections and rate constants J. Chem. Phys. 100 2697... 

The evaluation of the action of the Hamiltonian matrix on a vector is the central computational bottleneck. (The action of the absorption matrix, A, is generally a simple diagonal damping operation near the relevant grid edges.) Section IIIA discusses a useful representation for four-atom systems. Section IIIB outlines one aspect of how the action of the kinetic energy operator is evaluated that may prove of general interest and also is of relevance for problems that require parallelization. Section IIIC discusses initial conditions and hnal state analysis and Section HID outlines some relevant equations for the construction of cross sections and rate constants for four-atom problems of the type AB + CD ABC + D. 

Reaction Probability, Cross-Section, and Rate Constant 420... 

A.I.Maergoiz, E.E.Nikitin, and J.Troe, Calculation of cross sections and rate constants for capture of two identical linear dipole molecules, Khim. Fiz. 12, 841... 

Gray, S.K.. Goldfield, E.M.. Schatz, G.C. and Balint-Kurti, G.G. (1999) Helidty decoupled quantum dynamics and capture model cross sections and rate constants... 

REACTION PROBABILITY, CROSS SECTION, AND RATE CONSTANT... 

For a review of the use of hyperspherical harmonics as orbitals in quantum chemistry, (see [97]). Applications to bound state problems have mainly regarded nuclear physics, and are outside the scope of this article. The hyperquantization algorithm had been successfully applied to the prototype ion-molecule reaction He + IlJ HeH+ + H [98,99] and atom-molecule reaction F + H2 HF + H [100,101]. For the latter, resonances were characterized [102,103] and benchmark state-to-state differential cross sections and rate constants [104,105] were given. 

Langhoff, S.R., R.L. Jaffe, and J.O. Arnold, Effective cross sections and rate constants for predissociation of CIO in the earth s atmosphere. J Quant Spectrosc Radiat... 

Some of the earliest potentials computed by the SRS variant of SAPT were for Ar-H2 [149] and for He-HF [150,151]. An application of the latter potential in a calculation of differential scattering cross sections [152] and comparison with experiment shows that this potential is very accurate, also in the repulsive region. Some other SAPT results are for Ar-HF [153], Ne-HCN [154], CO2 dimer [155], and for the water dimer [129,156]. The accuracy of the water pair potential was tested [130,131] by a calculation of the various tunnehng splittings caused by hydrogen bond rearrangement processes in the water dimer and comparison with high resolution spectroscopic data [132,133]. Other complexes studied are He-CO [157,158], and Ne-CO [159]. The pair potentials of He-CO and Ne-CO were applied in calculations of the rotationally resolved infrared spectra of these complexes measured in Refs. [160,161]. They were employed [162-165] in theoretical and experimental studies of the state-to-state rotationally inelastic He-CO and Ne-CO collision cross sections and rate constants. It was reaffirmed that both potentials are accurate, especially the one for He-CO. 

Observables, such as cross sections and rate constants, can be extracted from the output of the trajectory calculations. Again, the event of interest must be clearly defined. For instance, in an AB + CD collision, the event of interest may be reaction to ABC + D whether this has occurred is relatively easily determined. As was outlined in Sec. III.C, each trajectory which satisfies the condition is weighted with 1, else 0. The event of interest, however, could be more detailed. If, for instance, one wished to compare with quantum (or experimental) results in which the process AB(v = 0, j = 0) + CD(v = 0,... 

The preceding presentation describes how the collision impact parameter and the relative translational energy are sampled to calculate reaction cross sections and rate constants. In the following, Monte Carlo sampling of the reactant s Cartesian coordinates and momenta is described for atom + diatom collisions and polyatomic + polyatomic collisions. Initial energies are chosen for the reactants, which corresponds to quantum mechanical vibrational-rotational energy levels. This is the quasi-classical model [2-4]. 

The cross section and rate constant expressions for an A + reaction, where both reactants are polyatomics, are the same as those above for an atom + diatom reaction, except for polyatomics there are more vibrational and rotational quantum numbers to consider. If the polyatomics are symmetric tops with rotational quantum numbers J and K, the state specific cross section becomes a, (urel, vA, JA, KA, vB, JB, KB), where vA represents the vibrational quantum numbers for A. If the polyatomic reactants have Boltzmann distributions of vibrational-rotational energies, the reactive cross section becomes a function of viel and T = TA = TB [i.e., ffr(vTCl T)] and is determined by summing over the quantum numbers as is done in Eq. 

Chernyi, G.G., Losev, S.A., Macheret, S.O., Potapkin, B.V (2002), Physical and Chemical Processes and Gas Dynamics Cross- Sections and Rate Constants, Progress in Astronautics and Aeronautics, vol. 196, American Institute of Aeronautics Astronautics, Reston, VA. 

The essential validity of the above model for many ion-molecule reactions at thermal energies is demonstrated by the magnitude of the reaction cross-sections and rate constants, and the variation of these quantities with collision energy and temperature, respectively. At low collision energies this model can then serve as a basis for calculating the rate of formation of collision complexes. In order to include the prediction of detailed kinetic data, however, the model must be... 

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. 

We list here and discuss the various conceivable stages of refinement in an idealized experiment to examine the rate of an ion-molecule reaction. Anticipating the discussion in Section 2, which relates cross sections and rate constants, we recognize that the fundamental information is summarized in cross-section information. In Sections 1.4.1-1.4.3, this ideal measurement would yield a cross section, but in Sections 1.4.4 and 1.4.5, this is further reduced to a reaction probability or collision efficiency. Very little of this is possible at the present time, but it is nevertheless stated for the sake of completeness to indicate just how far a rate measurement can be taken. A strong case may be made that such a complete investigation... 

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