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The Bethe logarithm is calculated for the lowest rotational-vibrational states of H 2 + and HD + hydrogen molecular ions in a wide range of vibrational (v = 0-4) and total orbital momentum (L = 0-4) quantum numbers. Numerical results with eight to nine significant digits are obtained for all the states within this range. This allows us to reduce an error in the leading-order radiative contribution, which results eventually in the relative uncertainty of rovibrational frequency intervals at a level lower than 10 −11. This high precision is important for the rovibrational spectroscopy experiments of hydrogen molecular ions aiming to determine the electron-to-proton mass ratio.

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