Hyperspherical formalism

V. Aquilanti, S. Cavalli, G. Grossi, and A. Lagana, A semiclassical approach to the dynamics of chemical reactions within the hyperspherical formalism. J. Mol. Struct., 93 319-323, 1983. 

This matrix was introduced by F. T. Smith [25] for the treatment of non-adiabatic (diabatic) couplings in atomic collisions. It is now familiar also in molecular structure problems, to indicate local breakdowns of the Born-Oppenheimer approximation. Within the hyperspherical formalism, it has been introduced in the three-body Coulomb problem [20] and in chemical reactions [21-24], see also Section 3. Also, from equation (A4)... 

These expressions can be obtained either by direct angular integration of eq. (8.29), or by considering the special case V[,( ) = and using the formula of section 8.1 for the potentials u(, y) = with n = 0,4 or 6. This enables us to switch on one after the other the L = 0, L = 4 and L = 6 multipoles of the potential. This is one more illustration of the many links between the hyperspherical formalism and the harmonic oscillator [43]. 

Let us now consider the overlap between the spherical and the Stark basis. For the latter, the momentum space eigenfimctions, which in configuration space correspond to variable separation in parabolic coordinates, are similarly related to alternative hyperspherical harmonics [2]. The connecting coefficient between spherical and 5 torA basis is formally identical to a usual vector coupling coefficient (from now on n is omitted from the notation) ... 

The hyperspherical method, from a formal viewpoint, is general and thus can be applied to any N-body Coulomb problem. Our analysis of the three body Coulomb problem exploits considerations on the symmetry of the seven-dimensional rotational group. The matrix elements which have to be calculated to set up the secular equation can be very compactly formulated. All intervals can be written in closed form as matrix elements corresponding to coupling, recoupling or transformation coefficients of hyper-angular momenta algebra. 

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... 

Fock[ll] showed that a formal solution for the helium wave function could be obtained in terms of hyperspherical coordinates and that this would take the form of a power series in R = (r2 + r ) 2 and Ini , multiplied by functions of the hyperspherical angles, a and i2. The lowest-order terms of Fock s expansion for the ground state wave function are,... 

The formalism and methodologies for these two cases are otherwise exactly the same. As a result, using the symmetrized hyperspherical coordinates p, 0, 9, inclusion of the effect of the GP on scattering calculations for H, is extremely simple, and is accomplished by imposing a simple boundary condition on the basis set for only one of the six coordinates which describe the nuclear motion of the system. This entails no increase in computational effort. 

Along fhose lines, Cordes and Altick [80] also implemenfed multichannel Cl theory with a basis set of hyperspherical coordinates for the determination of properfies of fhe He (3,3b) resonance. The first Cl multichannel implementation of Fano s formalism was done by Ramaker and Schrader [81] with application to the He (nt ) S autoionizing states. 

Although the emphasis of this paper is on the formalism we used in applying the method of hyper spherical coordinates and local hyperspherical surface functions to the e + H elastic and inelastic scattering problem, we present now a sampling of the results obtained. 

Hyperspherical coordinates were introduced by Delves [52] and the formalism of hyperspherical expansion was further developed by many authors [40,53,54] for three-body or more complicated bound states. The usefulness of this method for baryon spectroscopy was shown by several groups [55]. The basic idea is rather simple the two relative coordinates are merged into a single six-dimensional vector. The three-body problem in ordinary space becomes equivalent to a two-body problem in six dimensions, with a noncentral potential. A generalized partial wave expansion leads to an infinite set of coupled radial equations. In practice, however, a very good convergence is achieved with a few partial waves only. 

In this section we will describe the hyperspherical method in more detail. The description of the formalism is based on extractions taken from the work of Launay [2]. 

---