Aiming for benchmark accuracy with the many-body expansion.

Conspectus The past 15 years have witnessed an explosion of activity in the field of fragment-based quantum chemistry, whereby ab initio electronic structure calculations are performed on very large systems by decomposing them into a large number of relatively small subsystem calculations and then reassembling the subsystem data in order to approximate supersystem properties. Most of these methods are based, at some level, on the so-called many-body (or "n-body") expansion, which ultimately requires calculations on monomers, dimers, ..., n-mers of fragments. To the extent that a low-order n-body expansion can reproduce supersystem properties, such methods replace an intractable supersystem calculation with a large number of easily distributable subsystem calculations. This holds great promise for performing, for example, "gold standard" CCSD(T) calculations on large molecules, clusters, and condensed-phase systems. The literature is awash in a litany of fragment-based methods, each with their own working equations and terminology, which presents a formidable language barrier to the uninitiated reader. We have sought to unify these methods under a common formalism, by means of a generalized many-body expansion that provides a universal energy formula encompassing not only traditional n-body cluster expansions but also methods designed for macromolecules, in which the supersystem is decomposed into overlapping fragments. This formalism allows various fragment-based methods to be systematically classified, primarily according to how the fragments are constructed and how higher-order n-body interactions are approximated. This classification furthermore suggests systematic ways to improve the accuracy. Whereas n-body approaches have been thoroughly tested at low levels of theory in small noncovalent clusters, we have begun to explore the efficacy of these methods for large systems, with the goal of reproducing benchmark-quality calculations, ideally meaning complete-basis CCSD(T). For high accuracy, it is necessary to deal with basis-set superposition error, and this necessitates the use of many-body counterpoise corrections and electrostatic embedding methods that are stable in large basis sets. Tests on small noncovalent clusters suggest that total energies of complete-basis CCSD(T) quality can indeed be obtained, with dramatic reductions in aggregate computing time. On the other hand, naive applications of low-order n-body expansions may benefit from significant error cancellation, wherein basis-set superposition error partially offsets the effects of higher-order n-body terms, affording fortuitously good results in some cases. Basis sets that afford reasonable results in small clusters behave erratically in larger systems and when high-order n-body expansions are employed. For large systems, and (H2O)N≳30 is large enough, the combinatorial nature of the many-body expansion presents the possibility of serious loss-of-precision problems that are not widely appreciated. Tight thresholds are required in the subsystem calculations in order to stave off size-dependent errors, and high-order expansions may be inherently numerically ill-posed. Moreover, commonplace script- or driver-based implementations of the n-body expansion may be especially susceptible to loss-of-precision problems in large systems. These results suggest that the many-body expansion is not yet ready to be treated as a "black-box" quantum chemistry method.