The Utility of Constraining Basis Function Indices When Using the Lanczos Algorithm to Calculate Vibrational Energy Levels

Direct product basis sets are frequently used to calculate vibrational energy levels of small polyatomic molecules. They have the important advantage of simplicity. However, they have the important disadvantage that a very large number of direct product functions is necessary to obtain converged energy levels. By using an iterative, rather than an explicit, method to calculate eigenvalues of the Hamiltonian matrix, it is possible to calculate energy levels despite the huge size of the direct product basis. Nonetheless, it is natural to attempt to reduce the size of the direct product basis by excluding functions that do not contribute to the wave functions associated with the energy levels of interest. In this paper we present a variational basis representation (VBR) example and a discrete variable representation (DVR) example demonstrating that it is possible to use the Lanczos method and exclude direct product basis functions by restricting basis function indices while maintaining the favorable n f +1 scaling relation for the cost of direct product basis matrix-vector products.