Solving the vibrational Schr?dinger equation using bases pruned to include strongly coupled functions and compatible quadratures

摘要

In this paper, we present new basis pruning schemes and compatible quadrature grids for solving the vibrational Schr?dinger equation. The new 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 Schr?dinger equation without approximating the potential, one must use quadrature to compute potential matrix elements. For a molecule with more than five atoms, the use of iterative methods is imperative, due to the size of the basis and the quadrature grid. 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 new basis and also the quadrature for doing integrals with the basis have enough structure to make efficient matrix-vector products possible. When results obtained with a multimode approximation to the potential are accurate enough, full-dimensional quadrature is not necessary. Using the quadrature methods of this paper, we evaluate the accuracy of calculations made by making multimode approximations.