On the Convergence of the Hylleraas Variational Method

Abstract

The result of Bartlett, Gibbons and Dunn, that the wave function for He cannot be expressed as a power series in the particle separations $r_1$, $r_2$, $r_{12}$, is discussed in relation to the validity of the Hylleraas variational attack. By a simple analogous example, it is shown that this result does not establish the impossibility of using polynomials in these variables to represent the function as closely as desired. It is further shown that if a formal solution of the wave equation for He exists, then the energy given by the Hylleraas method will converge upon the correct energy and the function will converge in the mean upon the correct function. Even if there exists no formal solution of the wave equation, there is a lower bound to the energy which can be computed with any function, and upon this bound the Hylleraas method will converge. In either case, therefore, the method is justified.