Seminar in Numerical Analysis: Tucker Carrington (Queen's University Kingston)

Sparse grid quadrature and interpolation methods for solving the Schroedinger equation

To compute the vibrational spectrum of a molecule without neglecting coupling and anharmonicity one must calculate eigenvalues and eigenvectors of a large matrix representing the Hamiltonian in an appropriate basis. Iterative algorithms (e.g. Lanczos, Davidson, Filter Diagonalisation) enable one to compute eigenvalues and eigenvectors. It is easy to efficiently implement iterative algorithms when a direct product basis is used. However, for a molecule with more than four atoms, a direct product basis set is large and it is better to reduce the number of basis functions required to obtain converged eigenvalues by pruning. This is done without jeopardizing the efficiency of the matrix-vector products required by all iterative algorithms. In this talk, I shall present new basis-size reduction ideas that are compatible with efficient matrix-vector products. The basis is designed to include the product basis functions coupled by the largest terms in the potential and important for computing low-lying vibrational levels. To solve the vibrational Schrödinger equation without approximating the potential, one must use quadrature to compute potential matrix elements. When using iterative methods in conjunction with quadrature, it is important to evaluate matrix-vector products by doing sums sequentially. This is only possible if both the basis and the grid have structure. Although it is designed to include only functions coupled by the largest terms in the potential, the basis we use and also the (Smolyak-type) quadrature for doing integrals with the basis have enough structure to make efficient matrix-vector products possible. Using the quadrature methods of this paper, we evaluate the accuracy of calculations made by making multimode approximations.

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