A finite basis representation Lanczos calculation of the bend energy levels of methane

We present a method for computing bend energy levels of a five-atom molecule. We use polyspherical coordinates in terms of which both the kinetic energy operator and its associated volume element are simple. The basis functions we use are new. They are parity-adapted combinations of products of spherical harmonics and an associated Legendre function. We show that in the parity-adpated basis it is possible to evaluate matrix-vector products efficiently, despite the fact that the parity-adapted functions are not products of functions of a single variable. The method is applied to compute bend levels of methane. Within each parity block we use the symmetry-adapted Lanczos algorithm to compute levels labeled by irreps of a four-member group that contains the parity operator and the permutation of two of the four hydrogen atoms. In a future publication we shall use the bend wave functions as contracted basis functions to compute numerically exact vibrational levels of methane.