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

TL;DR

This paper develops a variational method to accurately compute relativistic corrections to the Bethe logarithm in two-center systems, enabling precise predictions of molecular transition frequencies.

Contribution

It introduces a novel variational approach to evaluate relativistic Bethe logarithm corrections for the ground state of two-center Coulomb problems, improving accuracy for small bond lengths.

Findings

Achieved 3-4 significant digits for relativistic corrections at R=0.2 bohr.

Enabled potential 10^{-10} relative uncertainty in transition frequencies.

Provided a method applicable to molecular ions and exotic atoms.

Abstract

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 to estimate the radiative contribution at m\alpha^7 order in molecular-like three-body systems such as hydrogen molecular ions H_2^+ and HD^+, or antiprotonic helium atoms. While we get 10 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 3-4 significant digits starting from R=0.2 bohr, that will allow to reach 10^{-10} relative uncertainty on transition frequencies.