12.12.1 Introduction

The many-body expansion (MBE) for a system of $N$ monomers is given by

in which $E_I$ represents the energy of monomer $I$, $\mathrm{\Delta }{E}_{IJ}$ = $E_{IJ}$ $-$ $E_I$ $-$ $E_J$ is a two-body correction for dimer $IJ$, and $\mathrm{\Delta }{E}_{IJK}$ = $E_{IJK}$ $-$ $\mathrm{\Delta }{E}_{IJ}$ $-$ $\mathrm{\Delta }{E}_{IK}$ $-$ $\mathrm{\Delta }{E}_{JK}$ $-$ $E_I$ $-$ $E_J$ $-$ $E_k$ is a three-body correction for trimer $IJK$, etc. In a large system and/or a large basis set, truncation of this expression at the two- or three-body level may dramatically reduce the amount of computer time that is required to compute the energy. Convergence of the MBE can be accelerated by embedding the monomer ($E_I$), dimer ($E_{IJ}$), trimer ($E_{IJK}$), $\mathrm{\dots }$ calculations in some representation of the electrostatic potential of the rest of the system. A simple means to do this is via atom-centered point charges that could be obtained when the $E_I$ terms are calculated; this is the so-called electrostatically-embedded many-body expansion (EE-MBE), ²⁸¹ Dahlke E. E., Truhlar D. G.
J. Chem. Theory Comput.
(2007), 3, pp. 46. Link ^{, 1096} Richard R. M., Lao K. U., Herbert J. M.
J. Chem. Phys.
(2014), 141, pp. 014108. Link ^{, 1097} Richard R. M., Lao K. U., Herbert J. M.
Acc. Chem. Res.
(2014), 47, pp. 2828. Link ^{, 726} Lao K. U. et al.
J. Chem. Phys.
(2016), 144, pp. 164105. Link ^{, 528} Herbert J. M.
J. Chem. Phys.
(2019), 151, pp. 170901. Link which we will denote as EE-MBE($n$) when the expansion is truncated at $n$-body terms. MBE($n$) and EE-MBE($n$) are available in Q-Chem, with analytic gradients, up to five-body terms ($n=5$).

It is well known that the interaction energies of non-covalent clusters are usually overestimated—often substantially—owing to basis-set superposition error (BSSE), which disappears only very slowly as the basis sets approach completeness. The widely used Boys-Bernardi counterpoise procedure corrects for this by computing all energies, cluster and individual monomers, using the full cluster basis set. (In clusters with more than two monomers, the obvious generalization of the Boys-Bernardi counterpoise correction is sometimes called the “site–site function counterpoise” correction or SSFC.) Note, however, that basis-set extrapolation is still necessary for high-quality binding energies. In ${({\mathrm{H}}_2\mathrm{O})}_6$, for example, a counterpoise-corrected MP2/aug-cc-pVQZ calculation is still $\approx 1$ kcal/mol from the MP2 basis-set limit. ¹⁰⁹⁴ Richard R. M., Lao K. U., Herbert J. M.
J. Phys. Chem. Lett.
(2013), 4, pp. 2674. Link Fortunately, the MBE allows for use of large basis sets in order to perform basis-set extrapolations in sizable clusters, ¹⁰⁹⁴ Richard R. M., Lao K. U., Herbert J. M.
J. Phys. Chem. Lett.
(2013), 4, pp. 2674. Link ^{, 1095} Richard R. M., Lao K. U., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 224102. Link and one can employ a counterpoise correction that is consistent with an $n$-body expansion in order to obtain an $n$-body approximation to the Boys-Bernardi counterpoise-corrected supersystem energy. Two such corrections have been proposed: the many-body counterpoise correction, MBCP($n$), ¹⁰⁹⁴ Richard R. M., Lao K. U., Herbert J. M.
J. Phys. Chem. Lett.
(2013), 4, pp. 2674. Link ^{, 1095} Richard R. M., Lao K. U., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 224102. Link and the $n$-body Valiron-Mayer function counterpoise correction, VMFC($n$). ⁶²⁷ Kamiya M., Hirata S., Valiev M.
J. Chem. Phys.
(2008), 128, pp. 074103. Link The two approaches are equivalent for $n=2$ but the MBCP($n$) method requires far fewer subsystem calculations starting at $n=3$ and is thus significantly cheaper, while affording very similar results as compared to VMFC($n$). ¹⁰⁹⁴ Richard R. M., Lao K. U., Herbert J. M.
J. Phys. Chem. Lett.
(2013), 4, pp. 2674. Link ^{, 1095} Richard R. M., Lao K. U., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 224102. Link