Positronium energy levels at order $m \alpha^7$: vacuum polarization corrections in the two-photon-annihilation channel

We have calculated all contributions to the energy levels of parapositronium at order $m \alpha^7$ coming from vacuum polarization corrections to processes involving virtual annihilation to two photons. This work is motivated by ongoing efforts to improve the experimental determination of the positronium ground-state hyperfine splitting.