Hyperspherical coordinates approach

C.D. Lin, Hyperspherical coordinate approach to atomic and other Coulombic three-body systems, Phys. Rep. 257 (1995) 1. 

A. Ohsaki, H. Nakamura, Hyperspherical coordinate approach to atom-diatom chemical reactions in the sudden and adiabatic approximations, Phys. Rep. 187 (1990) 1. 

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. 

While the detailed calculations gave energies and widths in reasonable agreement with the observed experimental results, they did not provide a simple physical picture of why the He spectrum was the way it was. To address this issue Macek described the He atom using hyperspherical coordinates. In this approach, the two electrons, each described by three coordinates, are replaced by an equivalent single particle in six dimensions.8,9 In Fig. 23.3 we show the He atom in which rt and r2 are the vectors from He2+ to each of the two electrons. The hyperradius Rh is defined by... 

A procedure in this time-independent approach that corresponds to following the time evolution of the collision system would be to study the annihilation function x P p) defined in terms of the hyperspherical coordinates by... 

The model potential displayed in Figure 8.2 had originally been used by Kulander and Light (1980) to study, within the time-independent R-matrix formalism, the photodissociation of linear symmetric molecules like C02. It will become apparent below that in this and similar cases the time-dependent approach, which we shall pursue in this chapter, has some advantages over the time-independent picture. The motion of the ABA molecule can be treated either in terms of the hyperspherical coordinates defined in (7.33) or directly in terms of the bond distances Ri and i 2 The Hamiltonian for the linear molecule expressed in bond distances... 

Time-independent approaches to quantum dynamics can be wxriational where the wavefunction for all coordinates is expanded in some basis set and the parameters optimized. The best knowm variational implementation is perhaps the S-matrix version of Kohn s variational prineiple which was introduced by Miller and Jansen op de Haar in 1987[1]. Another time-independent approach is the so called hyperspherical coordinate method. The name is unfortunate as hyperspher-ical coordinates may also be used in other contexts, for instance in time-dependent wavepacket calculations [2]. 

The hyperspherical coordinate method is the subject of the present article. In this method, one coordinate, the hj-perradius, is treated by a propagator method. This leaves one coordinate less to treat by a bexsis set expansion than in the nri-ational approaches. Thus the eorresponding matrices are one dimension smaller. Here I will focxis on two aspects of the theory, viz. the application of boundary conditions and how the matrices can be diagonalized. A short derivation of a Hamiltonian operator for umbrella type motions is also inchided. I will end with some illustrations of calculations that we have performed and finally there will be some concluding remarks. 

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 ... 

---