Understanding the many-body expansion for large systems. II. Accuracy considerations

To complement our study of the role of finite precision in electronic structure calculations based on a truncated many-body expansion (MBE, or "n-body expansion"), we examine the accuracy of such methods in the present work. Accuracy may be defined either with respect to a supersystem calculation computed at the same level of theory as the n-body calculations, or alternatively with respect to high-quality benchmarks. Both metrics are considered here. In applications to a sequence of water clusters, (H₂O)_N=6−55 described at the B3LYP/cc-pVDZ level, we obtain mean absolute errors (MAEs) per H₂O monomer of ∼1.0 kcal/mol for two-body expansions, where the benchmark is a B3LYP/cc-pVDZ calculation on the entire cluster. Three- and four-body expansions exhibit MAEs of 0.5 and 0.1 kcal/mol/monomer, respectively, without resort to charge embedding. A generalized many-body expansion truncated at two-body terms [GMBE(2)], using 3–4 H₂O molecules per fragment, outperforms all of these methods and affords a MAE of ∼0.02 kcal/mol/monomer, also without charge embedding. GMBE(2) requires significantly fewer (although somewhat larger) subsystem calculations as compared to MBE(4), reducing problems associated with floating-point roundoff errors. When compared to high-quality benchmarks, we find that error cancellation often plays a critical role in the success of MBE(n) calculations, even at the four-body level, as basis-set superposition error can compensate for higher-order polarization interactions. A many-body counterpoise correction is introduced for the GMBE, and its two-body truncation [GMBCP(2)] is found to afford good results without error cancellation. Together with a method such as ωB97X-V/aug-cc-pVTZ that can describe both covalent and non-covalent interactions, the GMBE(2)+GMBCP(2) approach provides an accurate, stable, and tractable approach for large systems.