  The renormalization method of Bogoljubov-Parasiuk-Hepp-Zimmermann (BPHZ) is
used in order to derive the renormalized energy shift due to the gauge
invariant K\"all\'en-Sabry diagram of the two-photon vacuum polarization (VPVP)
as well as the self energy vacuum polarization S(VP)E beyond the Uehling
approximation. It is outlined, that no outer renormalization is required for
the two-photon vacuum polarization and that only the inner renormalization has
to b e accomplished. It is shown that the so-called nongauge invariant spurious
term is absent for a wide class of vacuum polarization (VP) diagrams if one
applies the widely used spherical expansion of bound and free-electron propagat
or. This simplifies significantly calculations in bound state quantum
electrodynamis . As one result of our paper the use of the BPHZ-approach in
bound state QED is established.
