  We return to the problem of evaluation of the light-by-light contribution to
the energy levels of the hydrogen atom. We find an additional contribution
directly related to the Delbr\"uck scattering amplitude. The new correction is
larger than the previously included light-by-light terms at order~$\alpha^2
(Z\alpha)^6\ln(Z\alpha)\, m_e$. We consider the effective potential in position
space using an effective field theory approach and evaluate light-by-light
corrections to the energy levels of states with non-zero orbital momentum as
well as to the weighted difference of $s$ states. We also determine the large
distance asymptotic behaviour of the effective potential induced by the
light-by-light scattering in muonic atoms.
