  The many-body expansion (MBE) of energies of molecular clusters or solids
offers a way to detect and analyze errors of theoretical methods that could go
unnoticed if only the total energy of the system was considered. In this
regard, the interaction between the methane molecule and its enclosing
dodecahedral water cage, CH$_4$(H$_2$O)$_{20}$, is a stringent test for
approximate methods, including density-functional theory (DFT) approximations.
Hybrid and semilocal DFT approximations behave erratically for this system,
with three- and four-body nonadditive terms having neither the correct sign nor
magnitude. Here we analyze to what extent these qualitative errors in different
MBE contributions are conveyed to post-Kohn-Sham random-phase approximation
(RPA). The results reveal a correlation between the quality of the DFT input
states and the RPA results. Moreover, the renormalized singles energy (RSE)
corrections play a crucial role in all orders of MBE. For dimers, RSE corrects
the RPA underbinding for every tested Kohn-Sham model: generalized-gradient
approximation (GGA), meta-GGA, (meta-)GGA hybrids, as well as the optimized
effective potential at the correlated level. Remarkably, the inclusion of
singles in RPA can also correct the wrong signs of three- and four-body
nonadditive energies as well as mitigate the excessive higher-order
contributions to the MBE. The RPA errors are dominated by the contributions of
compact clusters. As a workable method for large systems, we propose to replace
those compact contributions with CCSD(T) energies and to sum up the remaining
many-body contributions up to infinity with supermolecular or periodic RPA. As
a demonstration of this approach, we show that for RPA(PBE0)+RSE it suffices to
apply CCSD(T) to dimers and 30 compact, hydrogen-bonded trimers to get the
methane-water cage interaction energy to within 1.6% of the reference value.
