Order ${\mathrm{\alpha }}^4$ ln${\mathrm{\alpha }}^{\mathrm{-}1}$ ${\mathit{f}}_{\mathrm{RYD}}$ corrections to the n=1 and n=2 energy levels of positronium

The order ${\mathrm{\alpha }}^4$ ln${\mathrm{\alpha }}^{\mathrm{-}1}$ ${\mathit{f}}_{\mathrm{RYD}}$ corrections to the n=1 and n=2 energy levels of positronium are calculated using the conventional Bethe-Salpeter formalism. The Barbieri-Remiddi potential provides a suitable zeroth-order kernel about which perturbation theory is developed. A simple algorithmic method which amounts to an expansion of the interaction kernel in powers of the momentum divided by the mass is shown to suffice to extract the ln${\mathrm{\alpha }}^{\mathrm{-}1}$ terms from most of the interaction terms.