Understanding the many-body expansion for large systems. I. Precision considerations

Abstract

Electronic structure methods based on low-order "n-body" expansions are an increasingly popular means to defeat the highly nonlinear scaling of ab initio quantum chemistry calculations, taking advantage of the inherently distributable nature of the numerous subsystem calculations. Here, we examine how the finite precision of these subsystem calculations manifests in applications to large systems, in this case, a sequence of water clusters ranging in size up to (H₂O)₄₇. Using two different computer implementations of the n-body expansion, one fully integrated into a quantum chemistry program and the other written as a separate driver routine for the same program, we examine the reproducibility of total binding energies as a function of cluster size. The combinatorial nature of the n-body expansion amplifies subtle differences between the two implementations, especially for n ⩾ 4, leading to total energies that differ by as much as several kcal/mol between two implementations of what is ostensibly the same method. This behavior can be understood based on a propagation-of-errors analysis applied to a closed-form expression for the n-body expansion, which is derived here for the first time. Discrepancies between the two implementations arise primarily from the Coulomb self-energy correction that is required when electrostatic embedding charges are implemented by means of an external driver program. For reliable results in large systems, our analysis suggests that script- or driver-based implementations should read binary output files from an electronic structure program, in full double precision, or better yet be fully integrated in a way that avoids the need to compute the aforementioned self-energy. Moreover, four-body and higher-order expansions may be too sensitive to numerical thresholds to be of practical use in large systems.

Understanding the many-body expansion for large systems. III. Critical role of four-body terms, counterpoise corrections, and cutoffs

“…This is perhaps counterintuitive, but the increasingly large number of subsystem calculations (each with error in the last digits) that are required as n increases engenders loss-of-precision issues that necessitate the use of far tighter convergence thresholds and drop tolerances than would ordinarily be required in a single electronic structure calculation. 27,28 In our experience, mainly with water clusters, these issues do not manifest in a significant way until the number of fragments reaches N ≈ 30. Perhaps because supersystem calculations on large systems are required in order to notice this problem, it has largely been overlooked in most previous work on the MBE.…”

“…Four-body and higher-order terms are typically neglected, despite having been shown to be important in predicting relative conformational energies of proteins. 14 These terms are also definitely not negligible in water clusters, [26][27][28] where many-body polarization effects are significant.…”

“…It has been argued [29][30][31] that embedding the n-body subsystem quantum chemistry calculations in an environment of classical point charges, which serve to mimic the remaining fragments, will accelerate convergence of the MBE by replicating some portion of the many-body polarization effects that are neglected when higher-order terms in the MBE are omitted. Papers I and II 27,28 strongly contest this idea, however, and suggest that much of the "conventional wisdom" regarding the MBE is either incorrect or at best does not generalize beyond the rather small systems (say, N 10 fragments) that have generally been used to benchmark truncated MBEs. Notably, small water clusters of this size were used as benchmarks in Refs.…”

Accelerating wavefunction in density-functional-theory embedding by truncating the active basis set

“…3-body and higher-order effects explicitly (which is time consuming and in polar systems may not work at all 44 ) but by modelling their effect on monomer and dimer energetics with increasingly sophisticated embedding methods. 45,46 In our previous work, 29 it was shown that the CCSD(T)-in-HF embedded many-body expansion for a model water trimer could accurately reproduce CCSD(T) calculations on the whole system.…”

An Adaptive Multiscale Approach for Electronic Structure Methods

“…This is also the underlying idea of various decomposition and fragmentation approaches, where the full global electronic structure problem is decomposed into appropriate local subproblems, while the local results are linearly combined to generate a consistent energy expression for the global system [31]. For example, there exist the sum of interactions between fragments computed ab initio procedure (SIBFA) [32], the fragmentation reconstruction method (FRM) [3], the fragment molecular orbital method (FMO) [49,51,58], additive model approaches [16,19] and many-body expansions [1,12,17,18,30,53,66,67,72]. Note that the linear scaling QM methods and fragmentation based QM methods all take advantage of a data locality principle which involves the so-called nearsightedness of electronic matter [50,65].…”