Bethe logarithm for the hydrogen molecular ionH2+

The mean excitation energy (the Bethe logarithm) is calculated for the lowest rotational (up to $L=4$) and vibrational (up to $v=4$) states of the hydrogen molecular ion ${{\mathrm{H}}_2}^{+}$. The calculations are based on a method of the direct integration over photon momenta. The estimated accuracy of obtained values is about ${10}^{-6}$. The Araki-Sacher $Q$ term is computed as well, which allows us to evaluate the leading order ${\alpha }^3$ radiative correction to the rovibrational energies of the hydrogen molecular ion.