Ground-State Energies of the He Atom and theLi+Ion in the Faddeev Approach

The ground-state energies of the He atom and the ${\mathrm{Li}}^{+}$ ion have been calculated using the Faddeev formalism. For the off-shell two-body collision amplitude $t$ matrix), Sturmian-function (SF) as well as Coulomb-function representations have been employed. For the He atom, calculations have been performed retaining $1s$, $1s$ and $2s$, and $1s$, $2s$, and $3s$ states in the SF representation for the two-body $t$ matrix, whereas only the $1s$ term has been retained in the mixed-mode (MM) representation. For the ${\mathrm{Li}}^{+}$ ion, computations have been done after retaining terms up to $2s$ in the SF representation and only the $1s$ term in the MM representation. The results obtained by retaining only the $1s$ term are in marked disagreement with the experimental values in both representations. It is noticed that the results for the MM representation are almost as much below the experimental values as those for the SF representation are above. The results for the MM representation are slightly better than those for the SF representation. This is seen more clearly in the case of the ${\mathrm{Li}}^{+}$ ion. The results in the SF representation are found to be in good agreement with the experimental findings after inclusion of the $2s$ term.