Analytical solution for a Yukawa-type potential

We use an iterative process to get an analytical solution for a screened Thomas-Fermi (Yukawa) potential. The analytical solution is not exact, but the result agrees with the best variational and numerical integration results. The critical Thomas-Fermi wave vector ${\mathit{q}}_{\mathit{c}}$ above which there is no bound state is obtained within 0.03% accuracy. We also show that the critical effective charge vanishes.