Using collocation and a hierarchical basis to solve the vibrational Schrödinger equation

We show that it is possible to compute vibrational energy levels of polyatomic molecules with a collocation method and a basis of products of one-dimensional harmonic oscillator functions pruned so that it does not include functions for which the indices of many of the one-dimensional functions are nonzero. Functions with many nonzero indices are coupled only by terms that depend simultaneously on many coordinates, and they are typically small. The collocation equation is derived without invoking differences of interpolation operators, which simplifies implementation of the method. This, however, requires inverting a matrix whose elements are values of the pruned basis functions at the collocation points. The collocation points are the points on a Smolyak grid whose size is equal to the size of the pruned basis set. The Smolyak grid is built from symmetrized Leja points. Because both the basis and the grid are not tensor products, the inverse is not straightforward. It can be done by using so-called hierarchical 1-D basis functions. They are defined so that the matrix whose elements are the 1-D hierarchical basis functions evaluated at points is lower triangular. We test the method by applying it to compute 100 energy levels of CH₂NH with an iterative eigensolver.