The application of hypervirial theorems to the perturbation theory for screened Coulomb potentials

By applying the hypervrial relation with the Hellmann-Feynman theorem to screened Coulomb potentia!s,we derived the energy levels of atoms correct to the sixty order of the perturbation paremeter A explicitly without any calculation of perturbed wave functions. In the case of the Yukawa potential,we demonstrated explicit calculation that the K-shell energy converges,at least for Z > 5,with the value of Ao = 0.85,and that the L-shell converges,at least for Z > 24, with A0 = 0.70.