Contribution of ${\alpha }^2$ 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 the interaction of elementary hydrogenlike atoms with target atoms, reported in a previously published paper [S. Mrówczyński, Phys. Rev. D 36, 1520 (1987)], do not include some terms of the same order of magnitude. This 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.