Calculation of the relativistic Bethe logarithm in the two-center problem

Abstract

We present a variational approach to evaluate relativistic corrections of order ${\ensuremath{\alpha}}^{2}$ to the Bethe logarithm for the ground electronic state of the Coulomb two-center problem. That allows us to estimate the radiative contribution at $m{\ensuremath{\alpha}}^{7}$ 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 ${10}^{\ensuremath{-}10}$ relative uncertainty on transition frequencies.