  A method for precise calculation of the energy corrections due to second
order electric quadrupole interactions, as well as mixed electric
quadrupole-vacuum polarization in the framework of the dynamic hyperfine
structure in heavy muonic atoms is presented. For this, a multipole expansion
of the Uehling potential is performed. The approach is applicable for an
arbitrary nuclear electric charge distribution. By performing these
calculations for muonic Rhenium and Uranium using a deformed Fermi
distribution, it is shown that both corrections contribute on a level
presumably visible in upcoming experiments.
