8.9.1 Overview

When calculating intermolecular interaction energies, a naïve calculation of the energy difference

usually results in severe overestimation of the interaction energy, even if all three energies in Eq. (8.5) are computed at a good level of theory. This phenomenon, known as basis set superposition error (BSSE), is an artifact of an unbalanced approximation, namely, that the dimer energy $E_{AB}$ is computed in a more flexible basis set as compared to the two monomer energies. Although BSSE disappears in the complete basis-set limit, it does so extremely slowly: in ${({\mathrm{H}}_2\mathrm{O})}_6$, for example, an MP2/aug-cc-pVQZ calculation of the interaction energy is still a bit more than 1 kcal/mol away from the MP2 complete-basis limit. ¹⁰²⁶ Richard R. M., Lao K. U., Herbert J. M.
J. Phys. Chem. Lett.
(2013), 4, pp. 2674. Link Short of computing all energies in very large basis sets and extrapolating to the complete-basis limit, the conventional solution to the BSSE problem is the counterpoise correction, originally proposed by Boys and Bernardi. ¹²⁹ Boys S. F., Bernardi F.
Mol. Phys.
(1970), 19, pp. 553. Link Here, one corrects for BSSE by computing the monomer energies $E_A$ and $E_B$ in the dimer basis set, with the idea being that this results in a more balanced treatment of $\mathrm{\Delta }{E}_{AB}$.

In truth the average of the counterpoise-corrected and uncorrected results is often a better approximation than either of them individually, but in any case one needs the counterpoise-corrected result. This requires basis functions to be placed at arbitrary points in space, not just those defined by the nuclear centers; these are usually termed “floating centers” or “ghost atoms”. Ghost atoms have zero nuclear charge but can support a user-defined basis set. Their positions are specified in the $molecule section alongside all the other atoms (atomic symbol: Gh), and their intended basis functions are specified in one of two ways:

1. 1.

   Via a user-defined $basis section, using BASIS = MIXED.

2. 2.

   Placing “$\mathrm{@}$” next to an atomic symbol in the $molecule section designates it as a ghost atom supporting the same basis functions as the corresponding atom, so that a $basis section is not required.

Examples of either procedure appear below.

The calculation of $\mathrm{\Delta }{E}_{AB}$ in Eq. (8.5) requires three separate electronic structure calculations but this process can be performed automatically using the Q-Chem’s machinery based on absolutely-localized molecular orbitals (ALMOs). This machinery is much more versatile and is described in detail later so we will not discuss the automatic procedure here; see Section 12.4.4 for that.