Understanding the Many-Body Basis Set Superposition Error: Beyond Boys and Bernardi

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 many-body expansion (or another similar energy
decomposition) needs to 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 co-workers
[J. Chem. Theory Comput. 2014 9, 36993707] 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. 2015, 11, 5132–5143] as well as a related study by Mayer and Bakó
[J. Chem. Theory Comput. 2017, 13, 18831886] have proposed new frameworks for understanding
BSSE in the many-body expansion. Although the two frameworks ultimately
propose 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, that 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-ζ
basis set and a four-body triple-ζ Hartree–Fock computation.