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By applying the hypervirial relations with the Hellman-Feynman theorem to screened Coulomb potentials, the authors derive the energy levels of atoms correct to the sixth order of the perturbation parameter X explicitly without any calculation of perturbed wave functions. The energies of atoms up to the twentieth order of X are also calculated by computer. In the case of the Yukawa potential, the authors demonstrate by explicit calculation that the K-shell energy converges, at least for Z &5, with the value of Xo = 0.85, and that the L-shell energy converges, at least for Z )24, with Xo = 0.70.

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