  Green's function methods within many-body perturbation theory provide a
general framework for treating electronic correlations in excited states. Here
we investigate the cumulant form of the one-electron Green's function based on
the coupled-cluster equation of motion approach in an extension of our previous
study. The approach yields a non-perturbative expression for the cumulant in
terms of the solution to a set of coupled first order, non-linear differential
equations. The method thereby adds non-linear corrections to traditional
cumulant methods linear in the self energy. The approach is applied to the
core-hole Green's function and illustrated for a number of small molecular
systems. For these systems we find that the non-linear contributions lead to
significant improvements both for quasiparticle properties such as core-level
binding energies, as well as the satellites corresponding to inelastic losses
observed in photoemission spectra.
