Quantum Collision Theory

Joachain C J 1975 Quantum Collision Theory (Amsterdam North-Holland) p 383... 

I tried to consistently distinguish between the quantum mechanics of atomic and molecular properties on the one hand and quantum collision theory on the other. The present discussion follows the same pattern. 

Joachain, C.J. (1975). Quantum Collision Theory, Vol.l (North-Holland, Amsterdam). 

C. J. Joachain (1975) Quantum Collision Theory, North-Holland Physics Publishing, Chapter 20... 

Pig.24a Temperatiare dependence of the rate constants of reactions H2 + H (Curves a and a ) and D2 + D (curves b and b ) according to quantum collision theory (curves a and b) and semiclassical collision theory (curves a and b ) based on Weston-Mortensen potential surface. Simbols represent experimental results. 

H2 + H reaction according to quantum collision theory based on Porter-Karplus potential surface. Curves 1, 1 and correspond to colinear, complanar and... 

To set the stage for further dispositions, Jeremy Hutson describes in Chapter 1 (entitled Theory of Cold Atomic and Molecular Collisions ) the basic theory of atomic and molecular scattering at low temperatures. This chapter introduces the concepts of the cross-section, the rate coefficient, and the scattering length, and describes the main ingredients of quantum collision theory. It is shown that the... 

L. Eno and H. Rabitz, "Generalized sensitivity analysis in quantum collision theory", J. Chem. Phys. 71, 4824 (1979)... 

Simple collision theories neglect the internal quantum state dependence of a. The rate constant as a function of temperature T results as a thennal average over the Maxwell-Boltzmaim velocity distribution p Ef. 

Fig. 3. The normalized excitation functions in A2 versus collision energy for the two isotopic channels for the F+HD reaction. The solid line is the result of quantum scattering theory using the SW-PES. The QCT simulations from Ref. 71 are plotted for comparison. The experiment, shown with points, is normalized to theory by a single scaling factor for both channels. Also shown in (a) is the theoretical decomposition of the excitation function into direct and resonant contributions using the J-shifting procedure.

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

The Arrhenius relation means that the rate constant or the diffusivity increases with temperature. Typically, at low temperatures (0-60°C), a 10-degree increase in temperature results in a doubling of reaction rates. In this section, two theories are introduced to account for the Arrhenius relation and reaction rate laws. Collision theory is a classical theory, whereas transition state theory is related to quantum chemistry and is often referred to as one of the most significant advances in chemistry. 

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

Many books and reviews give exemplary treatments of two-body collision theory, especially for our purposes those treating molecular collisions.41 5 Therefore, we confine ourselves to brief discussion and statement of results essential to understanding the differential scattering, giving both quantum and classical mechanical formulas where possible. 

Of the above reactions, only (15) is consistent with the experimental observations. Calculations from collision theory based on the endother-micity of reaction (13) indicate that only one collision in about 104 could yield oxygen atoms. Since the quantum yield of ozone formation has been estimated at 0.03 (91) and as high as 0.14 (18), and since long chains are not possible, this reaction cannot be important. In reaction (14) one mercury atom is removed for each oxygen atom formed. Since 60 or more ozone molecules may be formed for each mercury atom removed, this reaction cannot be important unless mercury is regenerated. The postulated reaction (61)... 

At low temperature the classical approximation fails, but a quantum generalization of the long-range-force-law collision theories has been provided by Clary (1984,1985,1990). His capture-rate approximation (called adiabatic capture centrifugal sudden approximation or ACCSA) is closely related to the statistical adiabatic channel model of Quack and Troe (1975). Both theories calculate the capture rate from vibrationally and rotationally adiabatic potentials, but these are obtained by interpolation in the earlier work (Quack and Troe 1975) and by quantum mechanical sudden approximations in the later work (Clary 1984, 1985). 

Raymond Daudel, Alberte Pullman, and Lionel Salem, Collision Theory, Reaction Path, Static Indexes, in Quantum Theory of Chemical Reactions, Vol. 1, Reidel, Dordrecht,... 

Lindemann s suggestion was amplified in 1926 by Hinshel-wood8 and by Fowler and Rideal.9 Independently and practically simultaneously Rice and Ramsperger10 proposed the fairly complete mathematical treatment for the collision theory which accounted for all the facts and predicted quantitatively the decrease in reaction velocity constant at decreased pressures. Kassel11 amplified these theories still further and introduced a refinement based on the quantum theory. 

In cases where quantum calculations are too demanding because of the large number of reactant and product states that are coupled together, semi-classical methods might provide a tractable alternative. Semi-classical collision theory exploits the ease of solving Hamilton s classical equations for motion over a reaction potential energy surface to approximate the... 

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