Vacuum Polarization Contribution to Hydrogen and Positronium Energies

Relative order \alpha(Z\alpha)^3 shift of the energy levels induced by the vacuum polarization is reexamined for a bound system of two particles with masses m and M. Recent results for hydrogen and for positronium are shown to contain an error due to the inadequate procedure of the infrared divergence handling. Numerically, the correction to the ground state energy constitutes 0.647 kHz for hydrogen and 46.7 kHz for positronium.