Atoms collinear collisions

Let us continue with the atom-diatom collinear collision model, this time allowing for the possibility of the reaction A -r BC —> AB -i- C. We first introduce mass-scaled coordinates, as these are especially convenient to describe rearrangements, using... 

Baer M 1975 Adiabatic and diabatic representations for atom-molecule collisions treatment of the collinear arrangement Chem. Rhys. Lett. 35 112... 

We note that the coordinate X directly reflects the distance between atoms one and two, whereas the coordinate X2 reflects a combination of both distances. Therefore, a knowledge of the two coordinates does not directly tell us what the distances are between the involved atoms. Also, for the potential energy function in a collinear collision, the natural variables will be the distances between atoms A and B and atoms B and C. These variables appear as the components along a new set of coordinate axes, if instead of a rectangular coordinate system we use a mass-weighted skewed angle coordinate system. 

Computer modelling has provided a wealth of numerical results. Many accurate cross sections are now available for collinear collisions of atoms with diatoms, where motion is constrained to a line at all times. These models are clearly not for immediate application to physical situations, but they are useful in other respects. They provide exact results with which approximations may be tested. This testing may prevent waste of effort while developing approximations for physical (spatial motion) situations. Results on coplanar motion are available and others are likely soon on spatial motion,... 

Paulsen et al. (1972) developed an optical model for vibrational relaxation in reactive systems. Only collinear atom-diatom collisions were considered, i.e. impact parameter dependencies were omitted. The model was applied to vibrational relaxation of electronically excited I2 in inert gases, in which case dissociation of I2 is responsible for flux loss. Olson (1972) used an absorbing-sphere model for calculating integral cross sections of ion-ion recombination processes A++B ->A + B + AE, with A or B atoms or molecules. He employed the Landau-Zener formula to obtain a critical crossing distance Rc, and assumed the opacity to be unity for distances... 

Shin and Keizer considered a simple model for a triple collision, in which a diatomic molecule undergoes collinear collisions with two atoms, one from each side. The initial velocities of the two atoms were assumed to be uncorrelated, and the time between the two collisions was taken to be random (a Poisson distribution). The interference effect was explicitly calculated and shown to change the relaxation rate for a low frequency mode of CSj in He by as much as 30% the effect was smaller for high frequency modes or heavier collision partners. The Keizer-Shin model probably underestimates the true interference effect for two reasons first, more than two collisions can interfere with each other, and second, hard collisions will tend to be grouped together. 

Equation (2.2) does not appear to be very helpful, since we would have to know the answer, E(R), to begin with. However, keeping Eq. (2.2) in mind, let s look at what we know qualitatively about E(R) in a simple case the collinear collision of an atom. A, with a diatomic, BC ... 

The majority of VTST calculations performed to date have been for atom-diatom collisions.For that kind of collision, reasonably accurate calculations of the vibrational energy levels are possible without excessive labor. For example, for a collinear minimum-energy path the vibrations orthogonal to the path consist of one stretch and a twofold degenerate bend. Use of a curvilinear bend coordinated 57,65 reduces the bend-stretch coupling, and principal anharmonicity can be included accurately in the bend by the harmonic-quartic approximation described above or by the WKB approximation. The stretch can also be treated accurately by the WKB approximation. 5 xt is also possible to estimate the effect of bend-rotational coupling,57 and in particularly... 

Turning back to practical application of the generalized Landau-Teller model, one can assert that with the help of the improved semiclassical approximation for the transition probability and the asymptotic method for calculation of the exchange interaction it is possible to get a reliable estimate of the adiabatic Ehrenfest exponent for a collinear atom-diatom collision. This implies of course, a non-empirical prediction, within the exponential accuracy, of the temperature dependence of the average transition probability. It remains to be seen how well these ideas can be extended for a three-dimensional collision. 

The technique of solving equations of motion by constructing a mechanical system that is moving under the same equations as die real problem is known as analog (as opposed to digital) computation. Note, however, fliat here this is only useful for collinear collisions and cannot describe, for example, migratory dynamics such as in H + ICl where the attack is on one atom, iodine, as discussed in Section 5.1.5.1, but the final product is HCl. 

Figure 2 shows the result of a two wavepacket calculation for the model system, i.e., a collinear collision between an atom A and a Morse-oscillator BC. The wavefunctions are expanded in 18 oscillator states. Initially we have u (to) = 5 o and = 0, i.e., the second GWP is not activated. The two wavepackets are initially centered around Ro = 4 k and 4.2 A, respectively, with the same initial momentum Pq and width parameters. As they approach the BC molecule the second wavepacket is activated. Finally most (85%) of the oscillator amplitude is associated with the first GWP. Thus the interaction between the classical particle represented by the first wavepacket and the quantum oscillator creates a second classical particle also represented by a GWP with an amplitude of about 15%. 

In order to decrease the number of coordinates, we fix the p value, P = 7t, that is, we will consider the collinear collision (Fig. 2.5, b). In this case, the potential energy depends only on two coordinates Rbc and Ra- Anotho coordinate can be chosen instead of Ra, for exanq)Ie, the distance (Rab) between the atoms A and B. 

J. W. Duff and P. Brumer, Exponentiating trajectories and statistical behavior in collinear atom-diatom collisions,... 

I. H. Zimmermann and T. F. George, Quantum mechanical study of electronic transitions in collinear atom-molecule collisions, Chem. Phys. 7 323 (1975). 

D. C. Clary, Quantum-dynamical study of translational-vibrational energy transfer in the collinear collisions of atoms with triatomic molecules. Mol. Phys. 39 1295 (1980). 

Baer, M. (1975) Adiabatic and diabatic representations for atom-molecule collisions treatment of the collinear arrangement. Chem. Phys. Lett., 35, 112 Baer, M. (1976) Adiabatic and diabatic representations for atom-diatom collisions treatment of the ihree-dimensional. Chem. Phys., 15, 49. 

Kaye J A and Kuppermann A 1988 Mass effect in quantum-mechanical collision-induced dissociation in collinear reactive atom diatomic molecule collisions Chem. Phys. 125 279-91... 

By the introduction of the (x, y) coordinate system, one has reduced the problem to the motion of a particle of mass (i in a two-dimensional rectilinear space (x, y). Thus, the problem of the collision between an atom and a diatomic molecule in a collinear geometry has been converted into a problem of a single particle on the potential energy surface expressed in terms of the coordinates x and y rather than the coordinates rAB and rBc The coordinates x and y which transform the kinetic energy to diagonal form in such way that the kinetic energy contains only one (effective) mass are referred to as mass scaled Jacobi coordinates. 

For simplicity, we assume that the collision is collinear, say along the rr-axis. The atoms A, B, and C are numbered 1, 2, and 3, respectively, to harmonize the notation with the previous section. Then, according to the analysis above, the following internal Jacobi coordinates are consistent with a diagonal kinetic energy ... 

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