8.9 Ghost Atoms and Basis Set Superposition Error

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.⁸²⁰ 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.⁹⁵ 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 12.4.3 for that.