ABSTRACT
This chapter deals with some simple finite-difference methods for calculating energy eigenvalues of the one-particle Schrödinger equation. The particular methods which the author uses are mainly his/her own, although they necessarily have much in common with any other finite-difference methods. The eigenvalues obtained are in error by a leading term of order h2, requiring a Richardson extrapolation process to convert them to very accurate energies. Several authors have discussed how to use finite-difference methods for several-electron atoms, using large computers; it will be interesting to see whether their procedures can be simplified sufficiently to work on microcomputers. In the case of atoms or ions in which one or two electrons occupy a valence shell outside a filled shell core it is quite common to use an effective single-particle potential to describe the motion of the outer electrons. Hajj has recently described an approach to the two-dimensional Schrodinger equation which uses finite-difference methods and banded matrix methods.
