The R-Matrix Method for Electron-Molecule Scattering: Theory and Computation
The R-matrix method has proven itself to be a useful tool for the study of the collisions of electrons with diatomic molecules. In principle the technique can be extended to treat polyatomic species but no applications have been reported to date. The idea behind the method is to divide configuration space into regions or boxes, solve the Schrodinger equation in each box separately and to then match the solutions on the surface bounding adjacent regions. The motivation for this division of space is the recognition that the dominant physics governing the behavior of the incident particle may be different in each spatial region. This allows us to develop numerical methods which are best suited for each box. In the electron-molecule scattering problem when the incident electron is “near” the target electrons and nuclei it is subject to strong, non-central electrostatic forces. As a consequence we must take proper account of electron correlation and anti-symmetry in this internal region. The problem becomes one of calculating the electronic wave-function of a compound (N+1) particle system and is therefore quite similar to the molecular structure problems treated by quantum chemists. When the incident electron is “far” away from the target it is subject to weak electrostatic forces of multi-polar form and rotational and vibrational coupling terms which may now be of comparable strength to the electrostatic terms. Since the electron is far away and thus distinguishable from the target electrons it is possible to ignore anti-symmetry of target and incident particle. The neglect of anti-symmetry leads to a very important simplification; there are no longer any non-local potentials in the problem. The Schrodinger equation can be reduced to the solution of a set of coupled differential equations. The number of channels which must be included in the external region is of course infinite, but in practice can be truncated to a manageable set of equations. If rotational and vibrational terms are smaller than the electrostatic forces in the external region it is possible to reduce the complexity of the problem even further. In any case the solution of even a large set of coupled differential equations on a modern day computer is a manageable task.
KeywordsIncident Electron External Region Schrodinger Equation Hamiltonian Matrix Couple Differential Equation
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