Reactant coordinates, reactive scattering

Coupled-channel equations arise in scattering dynamics when all but one of the degrees of freedom of the system are expanded in a square integral basis (of "channels"). The coupled channel equations are then solved numerically and describe motion in the unbound, or scattering coordinate. The principal difficulty of any reactive scattering calculation is that the coordinate system which best describes the asymptotic motions of reactants differs from the coordinate system best suited for products. Consequently, computational methods commonly use different coordinate systems in different parts of configuration space. Boundary conditions are expressed in terms of Jacobi coordinates (sometimes referred to as "cartesian coordinates"), where in the A -BC arrangement... 

Reactive scattering, in which the products of a collision are chemically different from the reactants, is formally similar to inelastic scattering. However, there are complications that arise from the fact that a basis set that efficiently describes the reactants is usually inefficient to describe the products and vice versa. Even the coordinate system to be used requires some care for example, Jacobi coordinates... 

It should be noted that the initial and final wave packets are usually expressed in the Jacobi coordinates of their own arrangements, which results in the difficult coordinate problem in state-to-state reactive scattering as we mentioned previously. One may either choose the product Jacobi coordinate [47,53,65,151], or reactant Jacobi coordinate to propagate the initial wave packet, and there also exist two other methodologies but may be both named as reactant coordinate-based (RGB) method the first one is to employ interpolation schemes for the coordinate transformation [41,89,126,127,156], and the second one is realized by projection of both reactant and product wave packets to an intermediate coordinate [43,127], Alternatively, in the reactant-product decoupling (RPD) method [6,7,96], both the reactant and product coordinates are used, and they are divided and combined by a complex absorbing potential. 

Sun Z, Guo H, Zhang DH (2010) Extraction of state-to-state reactive scattering attributes from wave packet in reactant jacobi coordinates. J Chem Phys 132(8) 084112... 

The single biggest problem in quantum reactive scattering theory is that there are inevitably at least two coordinate systems involved in a calculation. The coordinate system which is best suited to the reactant arrangement is different from that which is best for the product arrangement. In figure 8.1, the two coordinate systems are shown schematically for the reaction A-hBC AB+C. 

The coordinate problem referred to above for reactive scattering is that the Jacobi coordinates (ra,Ra) that are natural for describing the reactants A-hBC are not appropriate for describing the products, AB-hC. There are several ways to deal with this situation, but most of the recent progress in reactive scattering has been based on the formulation [75] in which the Jacobi coordinates for the various arrangements (i.e. A-hBC, AB-hC, AC+B) are all used simultaneously. For the collinear case of Fig. 1.2, for example, the expansion for the wavefunction in this approach is... 

In reactive scattering in principle one needs coordinates which are convenient in describing reactant and product arrangements simultaneously and which have the correct asymptotic behaviour that does not lead to a coupling by the kinetic energy for the separated fragments (e.g. atom and diatom). But there is no such unique coordinate system and so one has to make a compromise between practicability, numerical efficiency, and physical insight and interpretation of the reaction dynamics. 

It should be clear from Fig. 2b (Part I) that either set of mass-scaled Jacobi coordinates alone provides a complete description of the available collinear coordinate space. However, it should be equally clear that while Ra and Va are better suited to describing translational and vibrational motions in the reactant channel, Rc and Tc are more appropriate for a corresponding description of the products. It therefore seems natural to retain both sets of coordinates at once, using each set for convenience as required. Moreover, formulations of quantum reactive scattering based on this idea are quite easy to construct. Indeed a comprehensive account of such a formulation, for the... 

The basic idea behind hyperspherical coordinates is the same as the idea behind natural collision coordinates to find a single set of coordinates that swings naturally from reactants to products. In this case, however, the idea is implemented in a way that is motivated more by mathematical considerations than by physical intuition, with the consequence that hyperspherical coordinate methods currently provide one of the most reliable and widely used solutions to the quantum mechanical reactive scattering problem. 

The Kohn variational method described above for potential scattering extends in a straightforward way to the collinear reactive scattering problem described in Section 2 without the need to introduce any special coordinates (i.e., one can continue to work with the optimum mass-scaled Jacobi coordinates of the reactant and product arrangements). Moreover, the extensions that are required are comparatively minor, and the overall structure of the method remains the same. 

The total reaction probability is typically obtained fiom the reactive flux calculated at the dividing surface placed at a point-of-no-retum.[70,71] This surface is often located in the product channel, but not necessarily at the asymptote where the S-matrix elements are completely converged. Consequently, such calculations can be conveniently carried out in reactant Jacobi coordinates and the computational costs are no more expensive than that for inelastic scattering. Implemented for the Chebyshev propagation, the reaction probability is given as below [72]... 

Stripping, by its very nature, is a large impact parameter process. At small impact parameters the reactants collide head-on thus the Ar+ rebounds and is back scattered in the CM coordinate system either taking a D atom with it or not, depending upon whether the collision is reactive or nonreactive. Thus some back-scattered products are always expected, as is found experimentally (Fig. 8.8). 

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