On direct product based discrete variable representations for angular coordinates and the treatment of singular terms in the kinetic energy operator

Abstract

The use of curvilinear coordinates results in kinetic energy operators with singular terms. These singularities can be in the dynamically relevant region. The present work specifically considers 1/sin 2 θ singularities and wavefunctions showing significant amplitude close to θ = 0. It investigates the use of direct product-type discrete variable representations (DVRs) to describe such situations in quantum dynamics calculations and discusses the regularization of such repulsive singularities. A new scheme, the cot-DVR, is developed which can be used to construct well converging direct product DVRs for Hamiltonians showing a 1/sin 2 θ singularity. The cot-DVR is based on the diagonalization of the cot θ-matrix and adds two additional basis function, sin( θ) and sin(2 θ), to the standard basis of Legendre polynomials. It is particularly well suited for multi-configurational time-dependent Hartree calculations which employ one-dimensional (bottom layer) single-particle functions.