Pluvinage Method for Systems of Three Charged Particles

A method due to Pluvinage for finding ground-state energies of two-electron systems has been studied. His choice of a trial function, ${\Psi }_k$, is of interest since $H{\Psi }_k$, where $H$ is the Hamiltonian, is free of singularities. However, his calculations had to be done numerically. It has been found possible to utilize the method in such a way that, with little loss in accuracy, the various calculations can be performed analytically. With a trial function which contains no variational parameters, the method just fails to give a bound state for ${\mathrm{H}}^{-}$. (The results for $Z$ greater than unity are quite accurate.) The introduction of a scaling factor gives a bound state. Thus, only one variational parameter is required to prove the existence of a bound state, as compared with the two parameters required for a Hylleraas-type trial function. However, the calculations are much more tedious. The method has been generalized to be applicable to an arbitrary system of three charged particles which interact only through their Coulomb fields. It has been applied to the proton, proton, ${\mu }^{-}$ system, but it does not give very useful results there.