Accurate basis sets for the calculation of bound and continuum wave functions of the Schrödinger equation

Abstract

The major objective of this paper is to make some simple, but numerically useful generalizations of the classical orthogonal functions. The motivation for doing this is to develop accurate, orthornormal basis sets for the expansion of solutions of the Schr\"odinger equation in multichannel problems. The generalizations are needed for two reasons. First, it is often useful for the set to have a mathematical structure which is unrelated to the classical functions and second, the boundary conditions which need to be satisfied by the solutions to the Schr\"odinger equation are often not easily represented by the classical functions. In contrast to certain other techniques, such as those based on diagonalizing the overlap matrix, the method we propose is capable of generating very large, orthonormal subspaces in a numerically stable fashion. While it is not possible to demonstrate a one-to-one correspondence between this finite basis representation of the Hamiltonian and a representation based on the points and weights of a Gauss quadrature, the so-called discrete variable representation (DVR), as is true for the classical orthogonal functions, it is still possible to transform to a representation which preserves all of the essential features of the DVR. The method is illustrated by applying it to a few simple one- and two-dimensional problems.