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The $GW$ approximation is based on the neglect of vertex corrections, whichappear in the exact self-energy and the exact polarizability. Here, weinvestigate the importance of vertex corrections in the polarizability only. Wecalculate the polarizability with equation-of-motion coupled-cluster theorywith single and double excitations (EOM-CCSD), which rigorously includes alarge class of diagrammatically-defined vertex corrections beyond the randomphase approximation (RPA). As is well-known, the frequency-dependentpolarizability predicted by EOM-CCSD is quite different and generally moreaccurate than that predicted by the RPA. We evaluate the effect of these vertexcorrections on a test set of 20 atoms and molecules. When using a Hartree-Fockreference, ionization potentials predicted by the $GW$ approximation with theRPA polarizability are typically overestimated with a mean absolute error of0.3 eV. However, those predicted with a vertex-corrected polarizability aretypically underestimated with an increased mean absolute error of 0.5 eV. Thisresult suggests that vertex corrections in the self-energy cannot be neglected,at least for molecules. We also assess the behavior of eigenvalueself-consistency in vertex-corrected $GW$ calculations, finding a furtherworsening of the predicted ionization potentials.

Vertex corrections to the polarizability do not improve the GW approximation for the ionization potential of molecules