This work presents a new fragment method, the electrostatically embedded many-body expansion of the nonlocal energy (EE-MB-NE), and shows that it, along with the previously proposed electrostatically embedded many-body expansion of the correlation energy (EE-MB-CE), produces accurate results for large systems at the level of CCSD(T) coupled cluster theory. We primarily study water 16-mers, but we also test the EE-MB-CE method on water hexamers. We analyze the distributions of two-body and three-body terms to show why the many-body expansion of the electrostatically embedded correlation energy converges faster than the many-body expansion of the entire electrostatically embedded interaction potential. The average magnitude of the dimer contributions to the pairwise additive (PA) term of the correlation energy (which neglects cooperative effects) is only one-half of that of the average dimer contribution to the PA term of the expansion of the total energy; this explains why the mean unsigned error (MUE) of the EE-PA-CE approximation is only one-half of that of the EE-PA approximation. Similarly, the average magnitude of the trimer contributions to the three-body (3B) term of the EE-3B-CE approximation is only one-fourth of that of the EE-3B approximation, and the MUE of the EE-3B-CE approximation is one-fourth that of the EE-3B approximation. Finally, we test the efficacy of two- and three-body density functional corrections. One such density functional correction method, the new EE-PA-NE method, with the OLYP or the OHLYP density functional (where the OHLYP functional is the OptX exchange functional combined with the LYP correlation functional multiplied by 0.5), has the best performance-to-price ratio of any method whose computational cost scales as the third power of the number of monomers and is competitive in accuracy in the tests presented here with even the electrostatically embedded three-body approximation.
