When calculating intermolecular interaction energies, a naïve calculation of the energy difference
$$\mathrm{\Delta}{E}_{AB}={E}_{AB}-{E}_{A}-{E}_{B}$$ | (8.4) |
usually results in severe overestimation of the interaction energy, even if all three energies in Eq. (8.4) 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.^{Richard:2013a} 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.^{Boys:1970} 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:
Via a user-defined $basis section, using BASIS = MIXED.
Placing “$\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. (8.4) 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.
$molecule 0 1 O 1.68668 -0.00318 0.000000 H 1.09686 0.01288 -0.741096 H 1.09686 0.01288 0.741096 Gh -1.45451 0.01190 0.000000 Gh -2.02544 -0.04298 -0.754494 Gh -2.02544 -0.04298 0.754494 $end $rem METHOD mp2 BASIS mixed $end $basis O 1 6-31G* **** H 2 6-31G* **** H 3 6-31G* **** O 4 6-31G* **** H 5 6-31G* **** H 6 6-31G* **** $end
$molecule 0 1 N 0.0000 0.0000 0.7288 H 0.9507 0.0001 1.0947 H -0.4752 -0.8234 1.0947 H -0.4755 0.8233 1.0947 @B 0.0000 0.0000 -0.9379 @H 0.5859 1.0146 -1.2474 @H 0.5857 -1.0147 -1.2474 @H -1.1716 0.0001 -1.2474 $end $rem METHOD B3LYP BASIS 6-31G(d,p) PURECART 1112 $end