Contribution of α²-terms to the total interaction cross sections of relativistic elementary atoms with atoms of matter

It is shown that the corrections of $\alpha^2$ order to the total cross sections for interaction of elementary hydrogen-like atoms with target atoms, reported in the previously published paper [S.Mrowczynski, Phys.Rev. D36, 1520 (1987)], do not include some terms of the same order of magnitude. That results in a significant contribution of these corrections in particular cases. The full $\alpha^2$-corrections have been derived and it is shown that they are really small and could be omitted for most practical applications.