Natural collision coordinates

Feshbach resonances is purely model dependent since trapping well may exist on one type of adiabatic potential, say in hyperspherical coordinates, while only a barrier may exist on another type, say in natural collision coordinates. However, this is not correct since there are fundamental differences between QBS and Feshbach states. First, the pole structure of the S-matrix is intrinsically different in the two cases. A Feshbach resonance corresponds to a single isolated pole of the scattering matrix (S-matrix) below the real axis of the complex energy plane, see the discussion below. On the other hand, the barrier resonance corresponds to an infinite sequence of poles extending into the lower half plane. For a parabolic barrier, it is easy to show that the pole positions are given by... 

Fig. 1.10. Description of "natural collision coordinates" for a reaction AB + C — A + BC, s and n. for the collinear case. (Those for the three-dimensional problem are described in ref. [53].) The s is the reaction coordinate, measured from any fixed point O on C to the foot P of the perpendicular from the point P. The n is the vibrational coordinate, i.e., the...

A large part of the computational work has been influenced by the introduction of curvilinear coordinates, designed to take advantage of the topography of potential surfaces. These coordinates allow for a smooth change from reactant to product conformations and in effect transform the rearrangement problem into the much simpler one of inelastic collisions. The various treatments have employed reaction-path (or natural collision) coordinates less restricted reaction coordinates atom-transfer coordinates, somewhat analogous to those used for electron-transfer and, for planar and spatial motion, bifurcation coordinates. 

A study of reactive H + H2 collisions has also been carried out by Wu and Levine (1971) using the close-coupling method and natural collision coordinates. They employed a Porter-Karplus (1964) surface and investigated the collision energy range between 9 and 35 kcal/mole. For the interaction they used... 

The natural collision coordinates for spatial motion introduced by Marcus were made the starting point for a development (Wyatt, 1972) of reactive collision equations for AB + C -> A + BC. The treatment may be regarded as an extension of previous work (Curtis and Adler, 1952) for inelastic collisions. The kinetic energy operator was simplified by introducing two approximations appropriate for linear intermediates, and the interaction was chosen of form... 

A. Intrinsic Reaction Paths and Natural Collision Coordinates Variations on the Same Theme Potentials for Distinguished Coordinate Paths IRP Potentials in Internal (Valence) Coordinates Reaction Surfaces V. Dynamics on Path-Based Surfaces... 

A. Intrinsic Reaction Paths and Natural Collision Coordinates... 

Figure 10. Reaction path on a contour diagram of the PES of Fig. 1. The reaction coordinate s is the distance along the path. Points R off the path are related to s in natural collision coordinates, as indicated [see Eq. (4.2) for the corresponding formula in MWC coordinates].

Figure 11. In natural collision coordinates the relation between points off the path and the reaction coordinate s may be ambiguous due to curvature of the path.

In developing an ab initio PES for a chemical reaction in terms of a single reaction path potential, we first located the stationary points on the PES, then followed the reaction path by the steepest descent or distinguished coordinate methods, then chose coordinates, natural collision coordinates or internals, and finally expressed the PES as a series expansion about the... 

Natural collision coordinates are clearly hopeless. They are based firmly on one path (which one to choose ) and will probably be undefined, due to path curvature, at large distances from the defining path, where other important paths may reside. Moreover, a Taylor expansion in Cartesian (or MWC) displacements from one path will not give us a PES that is properly symmetric with respect to the CNPI group or even symmetric under rotation and inversion of the molecule. 

Natural collision coordinates were introduced so that we would know on which point on the path to base the Taylor expansion. Why Because we expect that the point on the path closest to R will probably give the most accurate truncated Taylor expansion, while points farther away would result in less accuracy. But we do not need to base the expansion solely on the closest point. We could use some or all of the points if we weight their contributions to favor the closest points. For example, if we have ab initio calculations at points on the reaction path, we can put [203]... 

So Eq. (7.2) is a new type of reaction path potential, with some significant advantages. Natural collision coordinates are not involved, so the PES is well-defined independent of the curvature of the path. For distinguished coordinate paths, the PES of Eq. (4.14) depends critically on that coordinate this is sometimes unreasonable since that may be just one of a number of coordinates that change along the path. Equation... 

In the next section (Sec. 2), we will develop the theory of the BCRLM. We discuss the solution of the coupled-channel equations in both natural collision coordinates " and hyperspherical coordinates. " Both coordinate systems are widely used to treat collinear reactive scattering processes. We will discuss the projection " of the hyperspherical equations on coordinate surfaces appropriate for applying scattering boundary conditions and review the definition of integral and differential scattering cross sections in this model. 

Early treatments of col linear reaction dynamics addressed the coordinate problems associated with different asymptotic arrangement channels by using natural collision coordinates.The generic... 

Figure 1. Col linear configuration space, subdivided into regions (1-IV) in which different coordinate systems are used. Regions I and II are for reactants, and III and IV are for products. M is a matching line between reactants and products, and TC is tlie origin of the polar natural collision coordinates used in Regions II and III.

Whether the numerical problem is solved in natural collision coordinates or in hyperspherical coordinates, we still must express boundary conditions in the appropriate asymptotic Jacobi coordinates. 

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