Random-Phase Approximation in Many-Body Noncovalent Systems: Methane in a Dodecahedral Water Cage

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, CH4···(H2O)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), which uses
approximate Kohn–Sham orbitals as its input. 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 the many-body expansion. 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 many-body
expansion. 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.