  We discuss diagrammatic modifications to the coupled cluster doubles (CCD)
equations, wherein different groups of terms out of rings, ladders,
crossed-rings and mosaics can be removed to form approximations to the coupled
cluster method, of interest due to their similarity with various types of
random phase approximations. The finite uniform electron gas is benchmarked for
14- and 54-electron systems at the complete basis set limit over a wide density
range and performance of different flavours of CCD are determined. These
results confirm that rings generally overcorrelate and ladders generally
undercorrelate; mosaics-only CCD yields a result surprisingly close to CCD. We
use a recently developed numerical analysis [J. J. Shepherd and A. Gr\"uneis,
Phys. Rev. Lett. 110, 226401 (2013)] to study the behaviours of these methods
in the thermodynamic limit. We determine that the mosaics, on forming the
Brueckner Hamltonian, open a gap in the effective one-particle eigenvalues at
the Fermi energy. Numerical evidence is presented which shows that methods
based on this renormalisation have convergent energies in the thermodynamic
limit including mosaic-only CCD, which is just a renormalised MP2. All other
methods including only a single channel, namely ladder-only CCD, ring-only CCD
and crossed-ring-only CCD, appear to yield divergent energies; incorporation of
mosaic terms prevents this from happening.
