Computing vibrational energy levels of CH4 with a Smolyak collocation method.

In this paper, we demonstrate that it is possible to apply collocation to compute vibrational energy levels of a five-atom molecule using an exact kinetic energy operator (with cross terms and coordinate-dependent coefficients). This is made possible by using (1) a pruned basis of products of univariate functions; (2) a Smolyak grid made from nested sequences of grids for each coordinate; (3) a collocation method that obviates the need to solve a generalized eigenvalue problem; (4) an efficient sequential transformation between the (nondirect product) grid and the (nondirect product) basis representations; and (5) hierarchical univariate functions that make it possible to avoid storing large intermediate vectors. The accuracy of the method is confirmed by computing 500 vibrational energy levels of methane.