Methods for calculating vibrational energy levels

This article reviews new methods for computing vibrational energy levels of small polyatomic molecules. The principal impediment to the calculation of energy levels is the size of the required basis set. If one uses a product basis the Hamiltonian matrix for a four-atom molecule is too large to store in core memory. We discuss iterative methods that enable one to use a product basis to compute energy levels (and spectra) without storing a Hamiltonian matrix. Despite the advantages of iterative methods it is not possible, using product basis functions, to calculate vibrational spectra of molecules with more than four atoms. A very recent method combining contracted basis functions and the Lanczos algorithm with which vibrational energy levels of methane have been computed is described. New ideas, based on exploiting preconditioning, for reducing the number of matrix-vector products required to converge energy levels of interest are also summarized.