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Understanding Many-Body Basis Set Superposition Error: Beyond Boys and Bernardi Ryan Matthew Richard, Brandon W. Bakr, and C. David Sherrill J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01232 • Publication Date (Web): 26 Mar 2018 Downloaded from http://pubs.acs.org on March 27, 2018
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Journal of Chemical Theory and Computation
Understanding Many-Body Basis Set Superposition Error: Beyond Boys and Bernardi Ryan M. Richard, Brandon W. Bakr, and C. David Sherrill∗ Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, and School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, USA E-mail:
[email protected] Abstract Fragment based methods promise accurate energetics at a cost that scales linearly with the number of fragments. This promise is founded on the premise that the manybody expansion (or another similar energy decomposition) need only consider spatially local many-body interactions. Experience and chemical intuition suggest that typically at most four-body interactions are required for high accuracy. Bettens and coworkers [J. Chem. Theory Comput. 9, 3699—3707 (2014)] published a detailed study showing that for moderately sized water clusters, basis set superposition error (BSSE) undermines this premise. Ultimately, they were able to overcome BSSE by performing all computations in the supersystem basis set, but such a solution destroys the reduced computational scaling of fragment based methods. Their findings led them to suggest that there is “trouble with the many-body expansion”. Since then, a subsequent follow-up study from Bettens and co-workers [J. Chem. Theory. Comput. 11, 5132–5143 (2015)] as well as a related study by Mayer and Bak´o [J. Chem. Theory Comput. 13, 1883—1886 (2017)] have proposed new frameworks for understanding BSSE in the many-body expansion. Although the two frameworks ultimately propose
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the same working set of equations to the BSSE problem, their interpretations are quite different, even disagreeing on whether or not the solution is an approximation. In this work we propose a more general BSSE framework. We then show that, somewhat paradoxically, the two interpretations are compatible and amount to two different “normalization” conditions. Finally, we consider applications of these BSSE frameworks to small water clusters, where we focus on replicating high-accuracy coupled cluster benchmarks. Ultimately, we show for water clusters, using the present framework, one can obtain results that are within ± 0.5 kcal mol−1 of the coupled cluster complete basis set limit without considering anymore than a correlated three-body computation in a quadruple-zeta basis set and a four-body triple-zeta Hartree-Fock computation.
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Introduction
Canonical electronic structure methods scale as the system size, N , to a power p, such that typically p > 3. The last decade or so has seen an influx of methods known as “fragment based methods” whose primary purpose is to circumvent such unfavorable scaling. 1–5 On some level, these methods all exploit the locality of electronic matter 6,7 to divide a molecular system of interest, called the “supersystem,” into a set of smaller molecular systems, called “fragments” (or equivalently monomers). The properties of the fragments are then used to estimate the properties of the supersystem. Hence for a supersystem fragmented into M monomers of size n such that n < N , fragment methods reduce the original O(N p ) problem to a series of problems, each with smaller size O(np ). Furthermore these M fragment computations are largely independent of each other making fragment methods “pleasantly parallel.” Unfortunately quantum mechanics is an inherently non-local theory and as just described fragment based methods are thus inevitably approximations. Depending on the desired accuracy, one wants a way to systematically improve these approximations. Physically speaking, the fragment description of the supersystem lacks “many-body” interactions among the 2
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fragments. There are at least two ways to build in such interactions: perturbatively, via methods such as symmetry-adapted perturbation theory (SAPT), 8 or via a hierarchy of molecular computations derived from the original system. Presently we are interested in the latter. Methods that take this approach tend to rely on the many-body expansion (MBE), vide infra, and assume that the MBE is rapidly convergent. Surveys focusing on existing fragment based methods, such as those in Refs., 1–3 have shown that the vast majority of the time, applications of these methods consider interactions among at most three fragments. Often this truncation point is justified by citing chemical intuition. Interestingly, some systematic studies 9–13 have numerically shown that quick convergence is often not the case, but rather the MBE tends to slowly oscillate towards convergence. Of the aforementioned studies one by Ouyang, Cvitkovic, and Bettens 13 (henceforth abbreviated OB1) conclusively showed that this slow convergence was not caused by a failure of chemical intuition, but rather failing to account for basis set superposition error (BSSE; the colloquial “borrowing of your neighbors’ basis functions for your own selfish, variational purposes”). Other studies have hinted at similar conclusions, 11,14,15 but never as systematically or in as much detail. For clusters comprised of more than two monomers, OB1 brought to light the importance of BSSE in MBE based methods. Unfortunately, OB1 was only able to remove BSSE by approaching the complete basis set (CBS) limit or by computing the MBE in the supersystem basis set. The former is impractical for large systems and the latter is particularly unappealing as it undermines the main cost savings of the MBE. In a follow-up manuscript Ouyang and Bettens 16 (henceforth abbreviated OB2) reexamined much of the OB1 study in the context of a new “many-ghost many-body expansion” (MGMBE) hoping to avoid the need for supersystem basis sets. Unfortunately the MGMBE still requires supersystem computations to compute a BSSE-free solution. OB2 argued that at the cost of reintroducing some BSSE, the MGMBE can be truncated and supersystem computations avoided. In response to OB2, Mayer and Bak´o 17 (henceforth abbreviated MB) formulated a framework
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in which OB2’s truncated MGMBE is actually BSSE-free. Although it seems paradoxical for both OB2 and MB to be simultaneously correct, the present study conjectures that this is indeed the case. To this end we present a framework for discussing BSSE within the context of the MBE that asserts that because only relative energies matter, the fact that OB2’s and MB’s frameworks differ by an energy shift is immaterial and simply amounts to two different normalization conventions of a more general framework. The development of this more general framework is the primary focus of this study. Ultimately, like MB, we conclude that for truncated expansions MB’s framework offers a theoretically simpler and computationally easier route to the BSSE-free answer.
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Theory
2.1
Many-Body Expansion (MBE)
Our BSSE framework is tightly coupled to the MBE and this section briefly reviews the theory underlying the MBE. Consider two fragments, I and J, with energies EI and EJ respectively. The interaction between them, ∆EIJ can be obtained via:
∆EIJ = EIJ − EI − EJ ,
(1)
where EIJ is the energy of the system formed from the union of fragments I and J, a “dimer.” Eq. (1) defines the supermolecular approach to computing the two-body interaction between fragments I and J. For a system comprised of M fragments, consideration of all such twobody interactions leads to an approximation for the supersystem energy, E:
E≈
M X
M C2
EI +
I=1
where
M C2
X
∆EIJ ,
I
