Coulomb three-body bound-state problem: Variational calculations of nonrelativistic energies

It is known that variational methods are the most powerful tool for studying the Coulomb three-body bound-state problem. However, they often suffer from loss of stability when the number of basis functions increases. This problem can be cured by applying the multiprecision package designed by D. H. Bailey. We consider variational basis functions of the type $\exp \left(-{\alpha }_n{r}_1-{\beta }_n{r}_2-{\gamma }_n{r}_{12}\right)$ with complex exponents. The method yields the best available energies for the ground states of the helium atom and the positive hydrogen molecular ion as well as many other known atomic and molecular systems.