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 Bako [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-zeta basis set and a four-body triple-zeta Hartree-Fock computation.