  In recent years, Green's function methods have garnered considerable interest
due to their ability to target both charged and neutral excitations. Among
them, the well-established $GW$ approximation provides accurate ionization
potentials and electron affinities and can be extended to neutral excitations
using the Bethe-Salpeter equation (BSE) formalism. Here, we investigate the
connections between various Green's function methods and evaluate their
performance for charged and neutral excitations. Comparisons with other
widely-known second-order wave function methods are also reported.
Additionally, we calculate the singlet-triplet gap of cycl[3,3,3]azine, a model
molecular emitter for thermally activated delayed fluorescence, which has the
particularity of having an inverted gap thanks to a substantial contribution
from the double excitations. We demonstrate that, within the $GW$
approximation, a second-order BSE kernel with dynamical correction is required
to predict this distinctive characteristic.
