Scattering-coordinates

Figure 4.3. The initial wavepacket superimposed on an Li+HF potential energy surface. Also shown (in a schematic manner) is the analysis line, marked R o, in the product channel. In an actual calculation the analysis line would be placed at a much larger value of the product scattering coordinate.

Figure 4.4. One-dimensional wavepackets in the reactant scattering coordinate for use in constructing the initial wavepacket. The wavepackets are shown as a function of the initial translational energy, (a) A Gaussian wavepacket (b) a sine wavepacket.

The action of the radial differential operator is accomplished through the use of discrete Fourier transforms as proposed by Kosloff [20,37]. If we take the example of the action of the differential operator in the scattering coordinate, then... 

Following Ref. 75, we define an analysis line at a large fixed value of the dissociation or scattering coordinate R = R o, which should lie in the asymptotic... 

The coupling between the angle y and the dissociation coordinate R is always large if Jacobi coordinates are used. At low energies deep inside the well, this coupling is linear and normal coordinates are usually better suited for interpretation and assignment than are Jacobi coordinates. However, if the molecular dynamics above the dissociation threshold is studied, the normal-mode picture breaks down and scattering coordinates have to be employed. 

W. R. Thorson and J. B. Delos, Phys. Rev., A18, 135 (1978). Theory of Near-Adiabatic Collisions. II. Scattering Coordinate Method. 

For energies below the dissociation threshold we can use various coordinate systems to solve the nuclear Schrodinger equation (2.32). If the displacement from equilibrium is small, normal coordinates are most appropriate (Wilson, Decius, and, Cross 1955 ch.2 Weissbluth 1978 ch.27 Daudel et al. 1983 ch.7 Atkins 1983 ch.ll). However, if the vibrational amplitudes increase so-called local coordinates become more advantageous (Child and Halonen 1984 Child 1985 Halonen 1989). Eventually, the molecular vibration becomes unbound and the molecule dissociates. Under such circumstances, Jacobi or so-called scattering coordinates are the most suitable coordinates they facilitate the definition of the boundary conditions of the continuum wavefunctions at infinite distances which we need to determine scattering or dissociation cross sections (Child 1991 ch.l0). Normal coordinates become less and less appropriate if the vibrational amplitudes increase they are completely impractical for the description of unbound motion in the continuum. 

Tennyson, J. (1986). The calculation of the vibration-rotation energies of triatomic molecules using scattering coordinates, Computer Physics Reports 4, 1-36. 

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