A perturbative treatment for the bound states of the Hellmann potential

A new approximation formalism is applied to study the bound states of the Hellmann potential, which represents the superposition of the attractive Coulomb potential −a/r and the Yukawa potential b exp (−δr)/r of arbitrary strength b and screening parameter δ. Although the analytic expressions for the energy eigenvalues En,ℓ yield quite accurate results for a wide range of n, ℓ in the limit of very weak screening, the results become gradually worse as the strength b and the screening coefficient δ increase. This is because that the expansion parameter is not sufficiently small enough to guarantee the convergence of the expansion series for the energy levels.