The discrete variable–finite basis approach to quantum scattering

Abstract

A discrete variable representation for scattering problems is developed. In this representation the potential matrix is diagonal, with elements being the potential evaluated at the proper quadrature points. The angular momentum operators may be treated exactly up to truncation of the basis set and provide the coupling in the coordinate-labeled discrete variable representation. The definition of the inner product over the internal coordinates as quadratures rather than integrations allows a discrete matrix transformation to be used to diagonalize any potential matrix. This framework allows one to obtain approximate solutions in a particularly simple and efficient manner and is presented in detail for atom–diatom collisions. At large values of the scattering distance the coupled equations may be solved to a high degree of accuracy using the distorted wave approximation in the finite basis representation. At small values of the scattering distance an exactly analogous technique may be used to obtain an approximate solution in the discrete variable representation. The solutions may be matched exactly and the exact scattering boundary conditions applied. Numerical results from atom–rigid rotor collisions are presented.