Cross-sections and Rates

Detailed Cross-sections and Rates.—The RRKM version of transition-state theory for unimolecular reactions, as developed 25 years ago and sununarized in its useful practical form in recent books, has continued to find wide applications in unimolecular rate theory. As has been pointed out by Marcus in the 1973 Faraday Discussion on molecular beams, it is both a weakness and a strength of transition-state theory that it does not make very detailed statements on specific cross-sections and rates. With such information becoming accessible experimentally, more detailed statistical dynamical theories were to come. We have now four such detailed statistical approaches  

Since (Q may be considered to be die most general one in the seme that the other three modds, including the normal and extmded RRKM treatment, can be obtained from it by simplifying assumptions, we shall hri y devdop diis iqiproach here. 

Fmr the construction of adiabatic charnels, one sedEs a s nration of the molecular Hamiltonian 3 = Jfo + such that the solutions of the Schrodinger equation = EaVo are given by the product functions Yo = bJiq,FS) 

The (m(q) are one-dimensional continuum functions in the reaction co-ordinate q, and m(q,R) are channel functions which depend on all other co-ordinates R and parametrically on q. The multidimensional problem is then reduced to the one-dimensional motion of a particle (reduced mass ( ) in an effective adiabatic channel potential K.(q) defined as shown in equation (40). Suitable approximate 

Hamiltonians and co-ordinate systems have been discussed. One may further 

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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... 

Collisional ionization can play an important role in plasmas, flames and atmospheric and interstellar physics and chemistry. Models of these phenomena depend critically on the accurate detennination of absolute cross sections and rate coefficients. The rate coefficient is the quantity closest to what an experiment actually measures and can be regarded as the cross section averaged over the collision velocity distribution. 

The END equations are integrated to yield the time evolution of the wave function parameters for reactive processes from an initial state of the system. The solution is propagated until such a time that the system has clearly reached the final products. Then, the evolved state vector may be projected against a number of different possible final product states to yield coiresponding transition probability amplitudes. Details of the END dynamics can be depicted and cross-section cross-sections and rate coefficients calculated. 

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 new R-matrix approach for calculating cross-sections and rate coefficients for electron-impact excitation of complex atoms and ions is reviewed in [307]. It is found that accurate electron scattering calculations involving complex targets, such as the astrophysically important low ionization stages of iron-peak elements, are possible within this method. 

Cross-sections and rate coefficients for electron impact recombination (25 785 sets). 

Reactive scattering and quantum dynamics (RSQD) methods are important to both scientific and technological development endeavors. Because the behavior of chemical species (molecule—molecule, atom-molecule, electron scattering, etc.) is rigorously described by quantum mechanics, which is built into the RSQD theoretical methods, accurate and converged solutions are achievable. Pursuant to a central goal of theoretical chemistry, these methods determine the cross sections and rates of chemical reactions. There are three basic methods ... 

In order to evaluate DSMC chemistry models, we require experimental and/or detailed theoretical results. Data of interest that can be measured experimentally include reaction cross sections, and rate coefficients. The most useful type of theoretical data are generated by detailed analysis of the collision and reaction dynamics using potential surfaces obtained from high level quantum chemical methods. 

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... 

Exact reaction cross sections and rates by the hyperquantization algorithm Auzinsh M. 

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... 

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