  Embedding theories became important approaches used for accurate calculations
of both molecules and solids. In these theories, a small chosen subset of
orbitals is treated with an accurate method, called an impurity solver, capable
of describing higher correlation effects. Ideally, such a chosen fragment
should contain multiple orbitals responsible for the chemical and physical
behavior of the compound. Handing a large number of chosen orbitals presents a
very significant challenge for the current generation of solvers used in the
physics and chemistry community. Here, we develop a Green's function coupled
cluster singles doubles and triples (GFCCSDT) solver that can be used for a
quantitative description in both molecules and solids. This solver allows us to
treat orbital spaces that are inaccessible to other accurate solvers. At the
same time, GFCCSDT maintains high accuracy of the resulting self-energy.
Moreover, in conjunction with the GFCCSD solver, it allows us to test the
systematic convergence of computational studies. Developing the CC family of
solvers paves the road to fully systematic Green's function embedding
calculations in solids. In this paper, we focus on the investigation of GFCCSDT
self-energies for a strongly correlated problem of SrMnO$_3$ solid.
Subsequently, we apply this solver to solid MnO showing that an approximate
variant of GFCCSDT is capable of yielding a high accuracy orbital resolved
spectral function.
