Essentials of Computational Chemistry

280 8 DENSITY FUNCTIONAL THEORY
also lead to very poor predictions along coordinates for bond dissociation (Bally and Sastry
1997; Zhang and Wang 1998; Gräfenstein, Kraka, and Cremer 2004), nucleophilic substitution
(Adamo and Barone 1998; Gritsenko et al. 2000), competing cycloaddition pathways
(Jones et al. 2002), and rotation about single bonds in conjugated systems, like the benzylic
bond in styrene (Choi, Kertesz, and Karpfen 1997).
Note that, because electron correlation often stabilizes delocalized electronic structures
over localized ones, HF theory tends to be inaccurate for such systems in the opposite direction
from DFT, and thus, again, hybrid ACM functionals tend to show improved performance
by an offsetting of errors.
A number of different methods have been proposed to introduce a self-interaction correction
into the Kohn–Sham formalism (Perdew and Zunger 1981; Kümmel and Perdew 2003;
Gräfenstein, Kraka, and Cremer 2004). This correction is particularly useful in situations
with odd numbers of electrons distributed over more than one atom, e.g., in transition-state
structures (Patchkovskii and Ziegler 2002). Unfortunately, the correction introduces an additional
level of self-consistency into the KS SCF process because it depends on the KS
orbitals, and it tends to be difficult and time-consuming to converge the relevant equations.
However, future developments in non-local correlation functionals may be able to correct
for self-interaction error in a more efficient manner.
8.6 General Performance Overview of DFT
While the cases noted in the immediately preceding section illustrate certain pathological
failures of current DFT functionals, the general picture for DFT is really quite bright. For
the ‘average’ problem, DFT is the method of choice to achieve a particular level of accuracy
at lowest cost. With the appearance of each new functional, there has tended to be at least
one paper benchmarking the performance of that functional on a variety of standard test sets
(for energies, structures, etc.) and there is now a rather large body of data that is somewhat
scattered and disjoint with respect to individual functional performance. The comparisons
made below are designed to provide as broad a coverage as possible without becoming
unwieldy, and as such are not necessarily exhaustive.
8.6.1 Energetics
Exact DFT is an ab initio theory (even if most modern implementations may be regarded
as having a semiempirical flavor) and like other such theories its quality with respect to
energetic predictions is usually judged based on its performance for atomization energies.
Table 8.1 collects average unsigned and maximum absolute errors in atomization energies
as computed for various functionals, and for some other computational methodologies, over
several different test sets of increasing complexity. The G2/97 and G3/99 test sets (columns
D and E) include substituted hydrocarbons, radicals, inorganic hydrides, unsaturated ring
hydrocarbons, and polyhalogenated organics and inorganics. While effort is made to describe
all levels of theory accurately, it should be noted that in many cases geometries are optimized
(and zero-point vibrational energies computed) using a basis set smaller than that used to
compute the atomization energies. In addition, some results for the larger test sets include