The hyperspherical coordinate method

In the present section, we give brief references to the hyperspherical coordinate method as described by Macek [324] and Fano [327] for the description of doubly excited states and the mechanism which drive doubleexcitations. A good introduction to the method is given by Fano [327]. 

The basic idea is a very simple one. Instead of describing doubly-excited states by patching up the independent electron description, we start 

Hyperspherical methods have the merit of providing a dynamical picture of double excitation and double escape for which the central field approximation is inappropriate. Initially, very accurate calculations were not achieved in this way, and so the hyperspherical method was mainly used as a framework to understand the results obtained by other methods. This situation was transformed by the work of Tang and Shimamura [330] who have performed the most detailed calculations to date on systems with two electrons. 

The radial equation in hyperspherical coordinates (we might call it the hyperradial equation) becomes 

In principle, the hyperspherical method corresponds rather well to the strategy advocated by Langmuir (see [308] and section 7.5) namely that one should seek to quantise the motion of more than just one electron in a many-electron system. The coordinates R and a describe the combined motion of an electron pair, and so the quantum numbers which arise in the solution are radial correlation quantum numbers. 

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It should be added that the hyperspherical coordinate method has been extended in further different directions, such as reactive and nonreactive molecular interactions [128-133], three- and four-electron systems [134-137], three-boson systems [94, 138-143], and extension for four-body and even larger systems [144-150]. In this regard, a remark on the so-called Efimov effects [151-153] is due here. 

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. 

There is a range of iterative diagonalization routines to choose between, including classical orthogonal polynomial expansion methods [48], Davidson iteration[58] and Krylov subspace iteration methods. Here the popular Lanezos method[59] will be discussed in the context of finding the eigenstates of the surface Hamiltonian appearing in the hyperspherical coordinate method. 

To date only three reaction-dissociation calculations have been reported by Manz and Romelt /Kaye and Kuppermann and Leforestier. The former two used the hyperspherical coordinates method to study model X-X2 systems,bearing respectively one and two bound states asymptotically. The latter one,whose results are presented on figure 3,will be discussed below it corresponds to a model H-HD system,HD and H2 bearing respectively 7 and 6 bound states. 

The hyperspherical coordinate method described in Section 4 is characterized more by its generality and reliability than by its simplicity or computational efficiency. It is equally applicable to both triatomic and tetratomic reactions, and it has also been used in conjunction with reduced dimensionality approximations to study even larger polyatomic reactions (see Reactive Scattering of Polyatomic Molecules). However,... 

The time-independent variational methods described in Section 5 are equally reliable as the hyperspherical coordinate method, although it is probably fair to say they have not yet been used to study quite such a diverse variety of chemical reactions. Their main advantage lies in their simplicity, and indeed their implementation boils down to performing little more than a standard computational quantum chemistry calculation involving basis sets, matrix elements, and linear algebra.The cost of this simplicity, however, is that the size of the matrices involved in these methods is one full dimension p) larger than the size of the matrices that arise in the hyperspherical coordinate method, and it can rapidly become difficult to fit them into computer memory. 

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]. 

At present the two leading approaches to quantum reactive scattering are hyperspherical coordinate methods, and variational methods based on 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. 

V. Aquilanti and S. Cavalli, Hyperspherical coordinates for molecular dynamics by the methods of trees and the mapping of potential energy surfaces for triatomic systems. J. Chem. Phys. 85, 1362-1375 (1986). 

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