Essentials of Computational Chemistry

6.4 GENERAL PERFORMANCE OVERVIEW OF AB INITIO HF THEORY 197
For minimum-energy structures, HF geometries are usually very good when using basis
sets of relatively modest size. For basis sets of polarized valence-double-ζ quality, errors
in bond lengths between heavy atoms average about 0.03 ˚A, and between heavy atoms
and H about 0.015 ˚A. Bond angles are predicted to an average accuracy of about 1.5 ◦ ,
and dihedral angles are also generally well predicted, although available experimental data
in the gas phase are scarce. Even with the 3-21G (∗) basis set, this accuracy is not much
degraded.
To the extent that HF theory is in error, it tends to overemphasize occupation of bonding
orbitals (see Chapter 7). Thus, errors tend to be in the direction of predicting bonds to be
too short, and this effect becomes more pronounced as one proceeds to saturated basis sets;
Feller and Peterson (1998) observed predicted geometries at the HF level to degrade in
quality with increasing basis-set size in the series aug-cc-pVnZ usingn = D, T, Q. A good
example is the case of the monocyclic singlet diradical 1,3-didehydrobenzene (Figure 6.11).
RHF theory erroneously predicts this molecule to be bicyclic with a formal single bond
between the radical positions.
There are some additional pathological cases that must be borne in mind in evaluating the
quality of predicted HF geometries for minima. As already noted, polarization functions are
absolutely required for geometric accuracy in systems characterized by hypervalent bonding;
failure to include polarization functions on heteroatoms with single lone pairs can also cause
them to be insufficiently pyramidalized. Furthermore, in systems crowding many pairs of
non-bonding electrons into small regions of space (e.g., the four oxygen lone pairs in a
peroxide) electron correlation effects on geometries, ignored by HF theory, can begin to be
large, so some caution is warranted here as well. Finally, dative bonds (i.e., those where both
electrons in the bonding pair formally come from only one of the atoms) are often poorly
described at the HF level. For instance, at the HF/6-31G(d) level, the B–C and B–N distances
in the complexes H3BžCO and H3BNH3 are predicted to be too long by about 0.1 ˚A.
Geometries of TS structures are not readily available from experiment, but a fairly
substantial body of theoretical work permits comparisons to be made with very high-level
H
2.1 Å 1.5 Å
H
H
Expt.
H
Figure 6.11 Structures of 1,3-didehydrobenzene (m-benzyne) from experiment and RHF calculations.
Because of its tendency to overemphasize bonding interactions, RHF optimization results in a bicyclic
structure. While the RHF error in bond length is very large, it should be noted that the ‘bond-stretching’
coordinate is known to be very flat (for very detailed analyses on the sensitivity of this system to
different theoretical levels, see Kraka et al. 2001 and Winkler and Sander 2001)
H
H
H
RHF
H