TUM Quantum Chemistry Laboratory
Prof. N. Rösch

Counterpoise Correction

The basis set superposition error (BSSE) is a consequence of basis set truncation, i.e. of the unavoidable fact that finite basis sets have to be chosen.

If a system becomes a part of a larger system, e.g. as ligand in a metal complex, it usually enjoys an improved description even for the same basis set. For the fragment is able to utilize, at least in part, the basis functions of its interaction partners. This is not the case when the subsytem (ligand) is treated alone, e.g. when a binding or interaction energy is calculated. Therefore, the energy of the whole system is computed lower in comparison to the separated subsystems which do not benefit from the basis functions of their interaction partners.

In the counterpoise correction the basis set for the subsystems contain also the basis functions of the whole molecule, see figure.
 

Formally

In the uncorrected calculation of a dimer AB, the dimer basis set is the union of the two monomer basis sets. The uncorrected interaction energy is

VAB(G) = EAB(G,AB) - EA(A) - EB(B)

where G denotes the coordinates that specify the geometry of the dimer and EAB(G,AB) the total energy of the dimer AB calculated with the full basis set AB of the dimer at that geometry. Similarly, EA(A) and EB(B) denote the total energies of the monomers A and B, each calculated with the appriate monomer basis sets A and B, respectively. This is the "straightforward" procedure for calculating an interaction energy.

The counterpoise corrected interaction energy is

VABcc(G) = EAB(G,AB) - EA(G,AB) - EB(G,AB)

where EA(G,AB) and EB(G,AB) denote the total energies of monomers A and B, respectively, computed with the dimer basis set AB at geometry G, i.e. in the calculation of monomer A the basis set of the "other" monomer B is present at the same location as in dimer A, but the nuclei of B are not. In this way, the basis set for each monomer is extended by the functions of the other monomer.

The counterpoise correction provides only an estimate of the BSSE since the monomer basis set is enhanced not only by empty orbitals, but also by orbitals occupied by electrons of the other monomer molecule.
 

Dissociation energy

For the calculation of the dissociation energy one has to take the zero point energy (ZPE) into account. Imagine now that A and B are molecular species, not just atoms as above, such that they have internal vibrations. Then, the dissociation energy D0 (without counterpoise correction) and D0cc (with counterpoise correction) can be calculated as follows:

D0 = -VAB(G) - ZPEAB(G) + ZPEA + ZPEB
D0cc = -VABcc(G) - ZPEAB(G) + ZPEA + ZPEB

The zero point energy ZPEAB of the dimer can be calculated by a regular frequency calculation with the dimer basis set, using the dimer geometry G . To calculate the ZPE of the monomer a frequency calculation has to be carried out for each monomer using