Hyperspherical coordinates advantages

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... 

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... 

The main advantage that hyperspherical coordinates have over natural collision coordinates is that the kinetic energy operator and volume element are far simpler in hyperspherical coordinates. The collinear reactive scattering Hamiltonian can be written in terms of p and 9, for example, as... 

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. 

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 ... 

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