  Most currently used approximations for the one-particle Green's function G in
the framework of many-body perturbation theory, such as Hedin's GW
approximation or the cumulant GW+C approach, are based on a linear response
approximation for the screened interaction W. The extent to which such a
hypothesis is valid and ways to go beyond have been explored only very little.
Here we show how to derive a cumulant Green's function beyond linear-response
from the equation of motion of the Green's function in a functional derivative
formulation. The results can be written in a compact form, which opens the
possibility to calculate the corrections in a first principles framework using
time-dependent density functional theory. In order to illustrate the potential
importance of the corrections, numerical results are presented for a model
system with a core level and two valence orbitals.
