Hyperspherical parametrization

We can pass from tree a to b using the suitable Clebsch-Gordan coeficient (eq. 12). The tree (c) illustrates the hyperspherical parametrization that leads to the hyperspherical harmonics Yn- Xm(, W 9) They are related to the harmonics of tree a through the Z coeficient defined in eg. (15). The connection between (b) and (c) requires a Clebsch Gordan coefficient and a phase change related to a (see eq. (14)). 

A historical account of the development of orthogonal coordinates for elementary chemical reactions has been given by one of the protagonists [12], The early hyper-spherical treatment for the helium atom as a tree-body quantum-mechanical problem [13,14], reviewed in Morse and Feshbach s treatise [15], was taken up in two basic papers by Fock [16]. They essentially used the parametrization referred to as asymmetrical in the following. Further important work in atomic physics [17,18] used the same hyperspherical parametrization, as did the investigations by Delves [19] on the breakdown of systems of many particles. 

If a hyperspherical parametrization can be represented by a tree, the coordinates can be shown to form an orthogonal set. This implies that the Laplacian on the hypersphere will contain no cross terms, the corresponding Laplace equations are separable and the hyperspherical harmonics can be constructed in closed form [24]. 

Figure 5 The tree represents the hyperspherical parametrization for the components of the Jacobi vectors describing three particles in space. li,mi eind l2,m2 can be put into correspondence with j,my and l,mi of section 4, and the upper part of the graph to the ordinary spherical harmonics. ...

We consider now the consequences of extending the concept of near separability to the hyperspherical parametrization. The hyperradius can be identified as the near separable variable and it is then possible to expand the wave function as a product of a hyperradial term and h3rperspherical functions [36,37] ... 

In this context, there is a real advantage in using geometrically defined coordinates, one of which (the reaction coordinate) can be viewed as a fair approximation to some steepest-descent curvilinear coordinate. It should be emphasized that, for polyatomic systems, there are always several sets of internal coordinates that uniquely describe the shape and size of the system. For instance, for the simple case of a triatomic system, once again excluding the hyperspherical parametrization, there are at least three sets of coordinates describing unambiguously the system ... 

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

Mass-scaled Jacobi coordinates associated to a generic arrangement X — a for A -I- BC, /I for B + CA and ) for C + AB) cU c denoted by r (diatom vector) and R (atom-molecule vector). They are used in the definition of hyperspherical coordinates which parametrize the nuclear motion of the system, namely the principal axis body frame hyperspherical coordinates [3, 4, 5]. These coordinates are ... 

Gallina et al. [20] introduced the hyperspherical symmetrical parametrization in a particle-physics context, as did Zickendraht later [21, 22], At the same time, F.T. Smith [23] gave the definitions of internal coordinates following Fock s work already mentioned [16], Clapp [24, 25] and others and established, for the symmetrical and asymmetrical parametrization, the basic properties and the notation we follow. Since then, applications have been extensive, especially for bound states. For example, the symmetrical coordinates have often been used in atomic [26], nuclear [27] and molecular [28-31] physics. This paper accounts for modem applications, with particular reference to the field of reaction dynamics, in view of the prominent role played by these coordinates for dealing with rearrangement problems. 

The symmetric parametrization can be achieved by taking as internal reference system the one that diagonalizes the inertia tensor, placing the principal axis in correspondence with that of maximal inertia [6,64], The symmetric hyperspherical coordinates can be calculated from the asymmetric hyperspherical coordinates ... 

G. Grossi, Angular parametrizations in the hyperspherical description of elementary chemical reactions. J. Chem. Phys., 81 3355-3356, 1984. 

Hyperspherical mapping of potential energy surfaces. Alternative parametrization of hyperangles. 

In the application of hyperspherical techniques to solve quantum-mechanical problems, solutions to the A -particle Schrodinger equation are searched for in a 3A -dimensional space, parametrized as a radius and the 3N — I angles of a hypersphere. Recent years have seen considerable progress in the applications of hyperspherical techniques, notably in atomic. 

H by omitting Tp(p), In other words, H is the hamiltonian of a particle of mass // confined to move on a 5-dimensional hypersphere of radius p, subject to the potential V. It depends on p only parametrically, and is given explicitly by... 

As already acknowledged, the h.o. expansion (as well as the hyperspherical expansion of chapter 5) suffers from being a little heavy, especially in the case of unequal masses. Its main advantage is that convergence can eventually be reached, provided one pushes the calculation far enough. Now, there exist many alternative parametrizations of the trial wave function which immediately provide a dramatic accuracy, though they are a little empirical in nature. We shall present below the example of the parametrization of the wave function in terms of Gaussian functions. This method is copiously used in theoretical molecular physics. 

---