Coordinate systems and analytic expansions for three-body atomic wavefunctions. III. Derivative continuity via solutions to Laplace's equation

Terms in a few-particle wavefunction, written as an expansion of homogeneous functions, are derived by a method which resembles standard techniques for solving differential equations in one variable. The addition of solutions to the homogeneous equation, i.e. Laplace's equation, converts a particular solution to a physically acceptable form consistent with the boundary conditions. Some earlier workers had derived expansions which, although satisfying the differential equations, were not consistent with those boundary conditions. A study of these inconsistencies assisted the development of this simpler method for obtaining the terms in the wavefunction in their reduced form. The power of the method is demonstrated by its application to nsms 3 S and npmp 1 S helium wavefunctions up to and including terms of fourth order in the hyper-radius.