Application of finite element and boundary integral methods in molecular collision theory. I. Introduction and model calculations

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Abstract
The adaptive multigrid technique in the finite element method of the solution of partial differential equations is examined in the context of model problems in atom–atom and collinear atom–diatom collisions. For the problem leading to scattering along an L-shaped region, it yields accurate results for regions of energy far from threshold for excitation of a new channel without inclusion of virtual states. Close to threshold, the cusp-like structure of the transition probability (vs energy) and the time delay associated with the onset of a resonance are recovered only by inclusion of the new (closed) channel in the finite element solution. For atom–diatom collinear collisions, use of an orthogonal coordinate system facilitates discretization and adds no extra labor in the finite element method, compared to the usual mass-weighted system. In such collisions, where the threshold behavior of the transition probability resembles the presence of a bound-state resonance below the threshold, the no-virtual-state solution is shifted in energy relative to the converged one-virtual-state solution. Under certain circumstances, transition probabilities from the former solution are found to deviate substantially from those determined from the latter in energy regions well below threshold. The multigrid method of solution of the finite element equation is found to be accurate provided a sufficiently fine grid is employed for the coarsest level of the multigrid adaptive solution process. However, for the problems studied, when the solution obtained from the coarsest grid does not yield a sufficiently accurate interpolated initial approximation for the relaxation method of solution of the finer grid levels, this method was found to converge to erroneous solutions. In future work, techniques examined herein will be applied to molecular scattering problems involving collisional dissociation.