Bethe logarithm and Lamb shift for the hydrogen molecular ion

A variational method is used to evaluate Bethe logarithm for the ground electronic state of the hydrogen molecular ion. The trial function employed in the calculations analytically represents the singular behavior of the exact solution of the variational problem for high virtual photon energies. This ensures that the accuracy of six significant figures can be obtained with basis sets not exceeding 70 functions. For the equilibrium internuclear separation of 2.0 bohr, the calculated value of the Bethe logarithm, equal to 2.31936, lies outside the bounds 2.35 and 2.56, representing the best previous estimation of this quantity. The accurate values of the Bethe logarithm for internuclear separations ranging from 0.6 to 6.0 bohr are tabulated and used to calculate the radiative corrections to the rovibrational energy levels of H₂⁺, HD⁺, and D₂⁺. The influence of these corrections on the transition and dissociation energies is also examined.