To use quantum mechanics to study the motion of nuclei in a molecule or a reacting system one must solve a Schrödinger equation. It is a high-dimensional equation because each nucleus moves in 3-D space. If there are N nuclei, the dimensionality of the equation is 3N. Problems of molecular quantum dynamics are therefore well-suited to sparse grid methods. To discretize the Schrödinger equation, one represents its solutions, called wavefunctions, as a linear combination of basis functions. The best univariate basis functions are not the commonly used piece-wise linear “hat” functions, but polynomials. The appropriate polynomials depend on the choice of the coordinates used to specify the positions of the nuclei. The choice of the basis functions is motivated by the physics of the problem. The polynomial univariate functions are combined using the sparse grid recipe to make a “sparse basis”. The underlying structure of the sparse basis is critical for efficiently evaluating the matrix-vector products required to use a Krylov method to solve the linear algebra problem obtained by discretizing the Schrödinger equation. The sparse basis used in most of our calculations is a pruned tensor product basis. For a 12-D problem, it is orders of magnitude smaller than the tensor product basis from which it is extracted. If one uses a Galerkin approach, it is not enough to have a sparse basis because matrix elements required to build the discretized problem are themselves high-dimensional integrals. A sparse grid is also required. One option is to compute the matrix elements with a sparse grid (Smolyak) quadrature. In this talk, I concentrate instead on collocation methods. They require computing matrix-vector products with, B, the matrix whose elements are (multi-dimensional) basis functions evaluated at points (in a high-dimensional space) and with B^{-1}. It is well known that, if the univariate basis functions are “hat” functions, these matrix-vector products can be evaluated efficiently by using what is known as the unidirectional principle. The matrix-vector products can also be efficiently evaluated with any (e.g. non-“hat”) univariate basis functions. Collocation methods obviate all the integrals required when using a Galerkin approach. In this talk, I present a collocation method with more points than basis functions, that systematically approaches Galerkin accuracy. This means using rectangular matrices. Excellent results are obtained for molecules with as many as six atoms without any need to optimize points.