We present a variational approach to evaluate relativistic corrections of order ${\alpha }^2$ to the Bethe logarithm for the ground electronic state of the Coulomb two-center problem. That allows us to estimate the radiative contribution at $m{\alpha }^7$ order in molecular-like three-body systems such as hydrogen molecular ions ${\text{H}}_2$${}^{+}$ and ${\text{HD}}^{+}$ or antiprotonic helium atoms. While we get ten significant digits for the nonrelativistic Bethe logarithm, calculation of the relativistic corrections is much more involved, especially for small values of bond length $R$. We were able to achieve a level of three to four significant digits starting from $R=0.2$ bohr, which will allow us to reach ${10}^{-10}$ relative uncertainty on transition frequencies.

• Received 16 April 2013

DOI:https://doi.org/10.1103/PhysRevA.87.062506