Finite element solution of the S-limit Schrodinger equation of helium

The Schroedinger equation, for all but the simplest systems, is an elliptic partial differential equation. Almost every method of solution is based on the expansion of the unknown solution in terms of a set of known global functions. Only a few calculations, using strictly numerical techniques based on the finite difference method, have been reported. Shorter cycle times and the increasing memory of computer hardware along with ease of programing provide the major impetus in the investigation of strictly numerical techniques in quantum mechanics. Recently, the Finite Element Method CFEM), which has been used extensively in engineering fields, has been applied to equations of quantum chemistry. However, only three of these calculations involved solution of a partial differential equation (PDE). This paper reports the application of the FEM to a 2-dimensional problem, that of the S-wave limit of the He atom. This problem has also been treated by the numerical finite difference method . The purpose of this study is not to establish that the FEM is an efficient method for solving quantum mechanical problems, but merely to explore the procedure to learn what is involved in its application. This problem has been chosen because its simplicity allows examination of the details of the FEM.