8.8 Ghost Atoms and Basis Set Superposition Error

When calculating intermolecular interaction energies, a na $\"{\i }$ ve calculation of the energy difference

$\begin{equation} \label{eq:DeltaE_ AB} \Delta E_{AB} = E_{AB} - E_ A - E_ B \end{equation}$

(8.4)

usually results in severe overestimation of the interaction energy, even if all three energies in Eq. eq:DeltaE_AB 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 $\rm (H_2O)_6$ , 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 et al.(2013a)Richard, Lao, and Herbert] 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 and Bernardi(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 $\Delta E_{AB}$ .

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 “ $@$ ” 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 $\Delta E_{AB}$ in Eq. eq:DeltaE_AB 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 13.4.3 for that.

Example 8.201 A calculation on a water monomer in the presence of the full dimer basis set. The energy will be slightly lower than that without the ghost atom functions due to the greater flexibility of the basis set.

$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

Example 8.202 A calculation on ammonia in the presence of the basis set of ammonia borane.

$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