What is counterpoise correction?

What is counterpoise correction exactly ? Can you explain when it is needed and why ?

Counterpoise correction is a method to limit an error that results when studying an intermolecular reaction using an incomplete basis set.

Typically a basis set is not converged, and a calculation could always be improved with more basis functions. This is especially true for long range interactions, ie one often needs to add diffuse functions to the set to account for the small about of electron density well away from the atomic centers.

When studying bond breaking/forming between two separate reactants, a basis set superposition error (BSSE) can occur. The basis set localized on one reactant can act as diffuse functions for the electrons from the other reactant, and vise versa. The error is largest at some problem-specific intermediate range.

One way to correct this is to use a larger and larger basis set. If one uses a sufficiently accurate description of the atomic orbitals far away from the atomic centers (more and more diffuse functions in a traditional calculation), then it wont matter if additional basis functions (from another reactants atomic orbital description) occupy that same long range region. The added basis functions from the other reactant are unnecessary and wont improve the quality of the calculation.

It is not always possible to use a larger basis set, because it's often too computationally expensive to increase the basis set. Alternatively, one can calculate a counterpoise correction, which approximates the bias to the quality of the calculation that results in the intermediate range. To get the corrected energy involves three steps:

1. Calculate the energy with both reactants, including all electrons and nuclei. This results in the energy of the complex of reactant 1 and reactant 2: $W_{12}$
2. Repeat the calculation for each reactant, on their own, using the same geometry they are in, in the complex. This results in values $W_1$ and $W_2$
3. Repeat the calculation for each reactant, but with a modified basis set: in addition to the individual reactant's basis set, the other reactant's basis is also used. For example, for reactant 1, these added basis functions are localized where reactant 2 is localized in the complex. These calculations do not include the nuclei or the electrons of the other reactant. This results in energies $W_1^{*}$ and $W_2^{*}$

The counterpoise correction is calculated from the energies of the individual reactants: $\Delta W_{c} = (W_1^*-W_1)+(W_2^*-W_2)$. This represents the lowering of the energy due to the addition of other reactant's basis set. Since the energy is lower or the same (with variational methods) with added basis functions, this value should be negative.

The corrected interaction energy is then $$\Delta W_{int} = W_{12}-W_1-W_2-\Delta W_c = W_{12} - W_1^*-W_2^*$$

Why does this matter? This correction will depend on the geometries of the reactants. When they are very far from one another, it will be very small: they don't influence one another. When they are very close, this effect will be small, for the same reasoning. It's the intermediate distances that have the largest BSSE. These are the distances at or approaching the transition state, which serves as the bottleneck for the reaction. If you are not accounting for the artificial improvement near the transition state, then you will get an incorrect approximation of the activation energy, the energy difference between this transition state and the separated-reactant limit.

Some questions to think about are: How can this be done for intramolecular reactions? Is this even important in those cases? Professor David Sherrill addresses these questions and other complicated cases in a freely available, self-published document.

A counterpoise correction is an a posteriori correction that may be applied to correct for the basis set superposition error (BSSE). More specifically, it uses mixed basis sets with "Ghost orbitals". For more information see,

Boys, S.F. and Bernardi, F., "The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors", Mol. Phys., 19 (1970), 553.