Essentials of Computational Chemistry

330 9 CHARGE DISTRIBUTION AND SPECTROSCOPIC PROPERTIES
Of these four levels, the computation of the MP2 spin-density matrix is considerably
more time-consuming than the other three. It is thus of interest to examine the accuracy of
DFT methods, which by construction include electron correlation directly into their easily
computed spin-density matrices. For the same geometries, the mean unsigned errors for
the BVWN, BLYP, B3P86, and B3LYP levels of theory were 32.6, 32.6, 29.7, and 28.9 G.
Somewhat surprisingly, these errors increased in every case when geometries were optimized
at the corresponding DFT level, to 60.5, 54.4, 30.9, and 34.3 G. For this particular data
set, several of the radicals seem prone to the DFT overdelocalization problem noted in
Section 8.5.6. Guerra (2000) has shown similarly poor performance of the B3LYP functional
in the context of vinylacyl radicals, where the functional strongly overestimates the stability
of π delocalized radicals relative to σ alternatives, in contravention of experimental data.
In cases where overdelocalization is not a problem, however, DFT methods have proven
to be quite robust for computing h.f.s. constants. For instance, Adamo, Cossi, and Barone
(1999) have reported results for h.f.s. constants in the methyl radical using PW, B3LYP, and
PBE1PBE that are competitive with correlated MO methods (Chipman 1983; Cramer 1991;
Barone et al. 1993). Moreover, if a given system suffers from heavy spin contamination at
the UHF level of theory, DFT may be the only reasonable recourse.
In general, then, DFT methods provide the best combination of accuracy and efficiency so
long as overdelocalization effects do not poison their performance. The MP2 level of theory
also provides a reasonably efficient way of carrying out h.f.s. calculations at a correlated
level of theory. More highly correlated levels of MO theory are generally more accurate,
but can be prohibitively expensive in large systems.
As a final note, although we have focused here on the computation of isotropic h.f.s. values,
it is also straightforward to compute anisotropic hyperfine couplings, although these cannot
be observed experimentally unless the system can be prevented from random tumbling (e.g.,
by freezing in a matrix or single crystal). Similarly, it is possible to calculate the electronic
g value. These subjects are beyond the scope of the text, however, and interested readers
are referred to relevant titles in the bibliography.
9.2 Ionization Potentials and Electron Affinities
As the general utility of semiempirical, HF, and DFT methods for the computation of IPs
and EAs has already been discussed in some detail in Sections 5.6.1, 6.4.1, and 8.6.1, this
section is restricted to a very brief recapitulation of the most important points relative to
these properties.
Koopmans’ theorem suggests that the ionization energies for any orbital (usually ‘IP’
refers specifically to the ionization potential associated with the HOMO) will be equal to the
negative of the eigenvalue of that orbital in HF theory. This provides a particularly simple
method for estimating IPs, and because of canceling errors in basis-set incompleteness and
failure to adequately account for electron correlation, the approach works reasonably well
for the occupied orbitals in the highest energy range in ab initio HF wave functions (with
semiempirical methods, performance is spottier). However, as one ionizes from orbitals