Hyperspherical coordinate system

Figure 7, Schematic representation of the 1-TS (solid) and 2-TS (dashed) (where TS = transition state) reaction paths in the reaction Ha + HbHc Ha He + Hb- The H3 potential energy surface is represented using the hyperspherical coordinate system of Kuppermann [54], in which the equilateral-triangle geometry of the Cl is in the center (x), and the linear transition states ( ) are on the perimeter of the circle the hyperradius p = 3.9 a.u. The angle is the internal angular coordinate that describes motion around the CL...

An alternative would be the hyperspherical coordinate system, introduced into the study of the ground-state helium by Gronwall [86], developed for nuclear reactions by Delves [87, 88], adopted in molecular reactive collisions by Smith [89], and initiated applications to two-electron excited QBSs by Macek [90]. The hyperradius p and one of the hyperangles, the radial hyperangle a, are defined by... 

The results of HSCC calculations have proved much more rapid convergence with the number of coupled channels than the conventional close-coupling equations in terms of the independent-particle coordinates or the Jacobi coordinates based on them. This is considered to be because of the particle-particle correlations considerably taken into account already in the choice of the hyperspherical coordinate system. The results suggest an approximate adiabaticity with respect to the hyperradius p, even when the mass ratios might appear to violate the conditions for the adiabaticity, for example, for Ps- with three equal masses. Then, it makes sense to study an adiabatic approximation with p adopted as the adiabatic parameter. 

Finally we should mention the hyperspherical coordinate system,30 31... 

The basic idea of the hyperspherical approach is the introduction of the p variable, which plays the role of a radius of a hypersphere. In hyperspherical coordinates systems the hyperradius is a critical quantity and is found to be ... 

Consider then an adiabatic well in the hyperspherical coordinate system. Classically, the motion of the periodic orbit at the well would be an oscillation from a point on the inner equipotential curve in the reactant channel to a point on the same equipotential curve in the product channel. This is qualitatively the motion of what are termed "resonant periodic orbits" (RPO s). For example the RPO s of the IHI system are given in Fig. 5. Thus, finding adiabatic wells in the radial coordinate system corresponds to finding RPO s and quantizing their action. Note that in Fig. 5 we have also plotted all the periodic orbit dividing surfaces (PODS) of the system, except for the symmetric stretch. By definition, a PODS is a periodic orbit that starts and ends on different equi-potentials. Thus the symmetric stretch PODS would be an adiabatic well for an adiabatic surface in reaction path coordinates. However, the PODS in the entrance and exit channels shown in Fig. 5 may be considered as adiabatic barrieres in either the radial or reaction path coordinate systems. Here, the barrier in radial coordinates, has quantally a tunneling path between the entrance and exit channels. 

It is more economical to use hyperspherical coordinate systems " for HLH systems. For collinear configurations, these coordinates are also plane polar coordinates, but the turning center is located at the origin. These coordinates have had a wide application to collinear re-actions,especially those of the HLH variety. The hyperspherical radius p is independent of the arrangement channel index... 

The value of p is independent of the choice of i whereas that of is not. The hyperspherical coordinate system % is defined as the set of coordinates formed by p, Wi, the spaced-fixed polar angles of Rj (or R ) and two polar angles which... 

Since H3 has D31, symmetry, it is advantageous to use an orthogonal coordinate system in which the symmetry operations are easily carried out The hyperspherical coordinate system recently jnesented by Pack4 permits the symmetry decomposition of the problem into the A j, A 2, and E iti< ucible representations to be carried out easily. After wave... 

In the hyperspherical coordinate systems, the scattering coordinate - the hyperradius - is defined to describe simultaneously all the fragmentation channels possible in a collision system. There is no need for troublesome match-... 

More importantly, in contrast to the situation described above for natural collision coordinates, comparatively simple Hamiltonians and volume integrals are also obtained for a variety of triatomic and tetratomic hyperspherical coordinate systems in three-dimensional space. 

We have expressed P in tenns of Jacobi coordinates as this is the coordmate system in which the vibrations and translations are separable. The separation does not occur in hyperspherical coordinates except at p = oq, so it is necessary to interrelate coordinate systems to complete the calculations. There are several approaches for doing this. One way is to project the hyperspherical solution onto Jacobi s before perfonning the asymptotic analysis, i.e. 

Aquilanti V and Cavalli S 1997 The quantum-mechanical Hamiltonian for tetraatomic systems in symmetric hyperspherical coordinates J. Chem. See. Faraday Trans. 93 801... 

Kuppermann A 1996 Reactive scattering with row-orthonormal hyperspherical coordinates. I. Transformation properties and Hamiltonian for triatomic systems J. Phys. Chem. 100 2621... 

In the presence of a phase factor, the momentum operator (P), which is expressed in hyperspherical coordinates, should be replaced [53,54] by (P — h. /r ) where VB creates the vector potential in order to define the effective Hamiltonian (see Appendix C). It is important to note that the angle entering the vector potential is shictly only identical to the hyperangle <]> for an A3 system. 

The location of the crossing seam (or seam) for an X3 system is established from the requirement that /-ab = rec = r c, where j-ab, rec, and fAc are the interatomic distances. Since the goal are the the geometric properties produced by this seam, hyperspherical coordinates (p,0,atomic masses are equal, say iiiB = me, the seam is defined [5] by... 

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