  The $GW$ approximation is based on the neglect of vertex corrections, which
appear in the exact self-energy and the exact polarizability. Here, we
investigate the importance of vertex corrections in the polarizability only. We
calculate the polarizability with equation-of-motion coupled-cluster theory
with single and double excitations (EOM-CCSD), which rigorously includes a
large class of diagrammatically-defined vertex corrections beyond the random
phase approximation (RPA). As is well-known, the frequency-dependent
polarizability predicted by EOM-CCSD is quite different and generally more
accurate than that predicted by the RPA. We evaluate the effect of these vertex
corrections on a test set of 20 atoms and molecules. When using a Hartree-Fock
reference, ionization potentials predicted by the $GW$ approximation with the
RPA polarizability are typically overestimated with a mean absolute error of
0.3 eV. However, those predicted with a vertex-corrected polarizability are
typically underestimated with an increased mean absolute error of 0.5 eV. This
result suggests that vertex corrections in the self-energy cannot be neglected,
at least for molecules. We also assess the behavior of eigenvalue
self-consistency in vertex-corrected $GW$ calculations, finding a further
worsening of the predicted ionization potentials.
