  Within the self-energy embedding theory (SEET) framework, we study coupled
cluster Green's function (GFCC) method in two different contexts: as a method
to treat either the system or environment present in the embedding
construction. Our study reveals that when GFCC is used to treat the environment
we do not see improvement in total energies in comparison to the coupled
cluster method itself. To rationalize this puzzling result, we analyze the
performance of GFCC as an impurity solver with a series of transition metal
oxides. These studies shed light on strength and weaknesses of such a solver
and demonstrate that such a solver gives very accurate results when the size of
the impurity is small. We investigate if it is possible to achieve a systematic
accuracy of the embedding solution when we increase the size of the impurity
problem. We found that in such a case, the performance of the solver worsens,
both in terms of finding the ground state solution of the impurity problem as
well as the self-energies produced. We concluded that increasing the rank of
GFCC solver is necessary to be able to enlarge impurity problems and achieve a
reliable accuracy. We also have shown that natural orbitals from weakly
correlated perturbative methods are better suited than symmetrized atomic
orbitals (SAO) when the total energy of the system is the target quantity.
