Essentials of Computational Chemistry

6.4 GENERAL PERFORMANCE OVERVIEW OF AB INITIO HF THEORY 195
example, Maksic and Vianello 2002). In practice, the cancellation can be remarkably good;
Koopmans’ theorem IPs are often within 0.3 eV or so of experiment provided basis sets of
polarized valence-double-ζ quality or better are used in the HF calculation. However, this
favorable cancellation begins to break down if IPs are computed for orbitals other then the
HOMO. As more tightly held electrons are ionized, particularly core electrons, the relaxation
effects are much larger than the correlation effects, and Koopmans’ approximation should
not be used.
Koopmans’ theorem can be formally applied to electron affinities (EAs) as well, i.e., the
EA can be taken to be the negative of the orbital energy of the lowest unoccupied (virtual)
orbital. Here, however, relaxation effects and correlation effects both favor the radical anion,
so rather than canceling, the errors are additive, and Koopmans’ theorem estimates will
almost always underestimate the EA. It is thus generally a better idea to compute EAs from
a SCF approach whenever possible.
A key point meriting discussion is the use of HF theory to model systems where two
or more molecules are in contact, held together by non-bonded interactions. Such interactions
in actual physical systems include electrostatic interactions between permanent and
induced charge distributions, dispersion, and hydrogen bonding (the latter includes both of
the prior two in addition to some possible degree of covalent interaction). It is important
to note that HF theory is formally incapable of modeling dispersion, because this
phenomenon is entirely a consequence of electron correlation, for which HF theory fails to
account. Nevertheless, bimolecular interaction energies are often reasonably well predicted
by HF theory, particularly with basis sets like 6-31G(d) and others of similar size. As
might be expected based on preceding discussion, this again reflects a cancellation of
errors.
Clearly, failure to account for dispersion would be expected to strongly reduce intermolecular
interactions, so the remaining errors must be in the direction of overbinding. In
this instance, there are two chief contributors to overbinding. The first is that, as noted in
Section 6.4.3, HF charge distributions tend to be overpolarized, which gives rise to electrostatic
interactions that are somewhat too large. The second effect is more technical in nature,
and is referred to as ‘basis set superposition error’ (BSSE). If we consider a bimolecular
interaction, the HF interaction energy can be trivially defined as
Ebind = E a∪b
HF (AžB) − E a HF (A) − Eb HF (B) (6.12)
where a and b are the basis functions associated with molecules A and B, respectively.
Note that if a and b are not both infinite basis sets, then there are more basis functions
employed in the calculation of the complex than in either of the monomers. The greater
flexibility of the basis set for the complex can provide an artifactual lowering of the energy
when one of the monomers ‘borrows’ basis functions of the other to improve its own wave
function.
One method proposed to correct for this phenomenon is the so-called counterpoise (CP)
correction. Although some variations exist, one popular approach defines the CP corrected