Multilevel adaptive technique for quantum reactive scattering

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Abstract
Discretization (with localized basis functions or grid points) of the coupled integral equations for molecular collisions leads to a very large system of linear algebraic equations. New methods, which are well adapted to vector supercomputers and parallel architectures, are developed for solving this large system. The multilevel adaptive technique (MLAT) is combined with recursive and iterative techniques. First, a multichannel solution is obtained on a low level grid. The basis is then adapted to this solution and the coarse solution is projected or interpolated onto the adapted basis. The scattering amplitudes (K-matrix elements) on the high level are then developed through use of either the recursion method (for single amplitudes, or a small batch of them) or the iterative technique (for all transitions from a specified initial state). In both of these methods, the original large system of algebraic equations is projected into a much smaller subspace (an orthonormalized Krylov space) spanned by a few basis vectors. Applications to very large systems are possible because it is not necessary to store or invert a large matrix. Computational results on a model chemical reaction are presented.