Some remarks on Jacobi coordinates and scattering applications

1 Some remarks on Jacobi coordinates and scattering applications 

It should be clear from Fig. 2b (Part I) that either set of mass-scaled Jacobi coordinates alone provides a complete description of the available collinear coordinate space. However, it should be equally clear that while Ra and Va are better suited to describing translational and vibrational motions in the reactant channel, Rc and Tc are more appropriate for a corresponding description of the products. It therefore seems natural to retain both sets of coordinates at once, using each set for convenience as required. Moreover, formulations of quantum reactive scattering based on this idea are quite easy to construct. Indeed a comprehensive account of such a formulation, for the 

The three variational principles in common use in scattering theory are due to Kohn [9], Schwinger [11] and Newton [12]. Two of these variational principles, those due to Kohn and Newton, have been successfully developed and applied to reactive scattering problems in recent years there is the S-matrix Kohn method of Zhang, Chu, and Miller, the related log derivative Kohn method of Manolopoulos, D Mello, and Wyatt and the L - Amplitude Density Generalized Newton Variational Principle (L -AD GNVP) method of Schwenke, Kouri, and Truhlar. 

The Kohn variational principle is perhaps the simplest of the three scattering variational principles mentioned above [9]. In particular, it requires that one calculate matrix elements only over the total Hamiltonian H of the system, and not over the Green s function Gq E) of some reference Hamiltonian Hq. While matrix elements of H between energy-independent basis functions are also energy-independent, all matrix elements of G E) have to be re-evaluated at each new scattering energy E. The Kohn variational principle is therefore somewhat easier to apply than the Schwinger and Newton 

In their original paper Miller and Jansen op de Haar only applied their method to simple elastic and inelastic scattering tests [16]. The first practical demonstration of its efficiency came, when Zhang and Miller used the method to calculate J=0 reaction probabilities, over quite a wide energy range, for the already heavily studied H-I-H2 reaction [102]. Their results agreed well with the earlier calculations, and appeared to be quite easy to converge. 

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