Essentials of Computational Chemistry

8.5 ADVANTAGES AND DISADVANTAGES OF DFT COMPARED TO MO THEORY 275
determining the kinetic energy as a functional of the density. However, as has already been
discussed in the context of MO theory, some chemical systems are not well described by a
single Slater determinant. The application of DFT to such systems is both technically and
conceptually problematic.
To illustrate this point, let us return to the cases of p-benzyne and N-protonated 2,5pyridyne
already discussed at length in Section 7.6.2. When restricted DFT is applied to
the closed-shell singlet states of these molecules, the predicted splittings between the singlet
and triplet states at the BPW91/cc-pVDZ level are 3.1 and 3.7 kcal mol −1 , respectively.
Comparing to the last line of Table 7.2, we see that these predictions are in error by about
8 kcal mol −1 and are qualitatively incorrect about which state is the ground state. A careful
analysis indicates that there is no problem with the triplet state, but that the singlet state
is predicted to be insufficiently stable as a consequence of enforcing a single-determinantal
description as part of the KS formalism (this also results in rather poor predicted geometries
for the singlets).
In cases like this, showing high degrees of non-dynamical correlation, there are two
primary approaches to correcting for inadequacies in the KS treatment. In the first approach,
the remedy is fairly simple: an unrestricted KS formalism is applied and the wave function
for the singlet is allowed to break spin symmetry. That is, even though the singlet is closedshell,
the α and β orbitals are permitted to be spatially different. When this unrestricted
formalism is applied to p-benzyne and N-protonated 2,5-pyridyne, the S–T splittings are
predicted to be −3.6 and−3.9 kcal mol −1 , respectively, in dramatically improved agreement
with experiment/best estimates (singlet geometries are also improved).
Similar results have been obtained in transition-metal compounds containing two metal
atoms that are antiferromagnetically coupled. An adequate description of the singlet state
sometimes requires a broken-symmetry SCF, and inspection of the KS orbitals afterwards
typically indicates the highest energy α electron(s) to be well localized on one metal atom
while the corresponding highest energy β electron(s) can be found on the other metal atom.
The transition from a stable restricted DFT solution to a broken-symmetry one takes place
as the distance between the metal atoms increases and covalent-like bonding gives way to
more distant antiferromagnetic interactions (Lovell et al. 1996; Cramer, Smith, and Tolman
1996; Adamo et al. 1999).
What is to be made of these broken-symmetry singlet KS wave functions? One interpretation
is to invoke the variational principle and assert that, insofar as they lower the energy, they
must provide better densities and one is fully justified in using them. While pragmatic, this
view is somewhat unsatisfying in a number of respects. One troubling issue is that the expectation
value of the total spin operator for the KS determinant is often significantly in excess
of the expected exact value. Thus, in the case of the singlet arynes discussed above, 〈S 2 〉
values of zero are expected, but computed values are on the order of 0.2 for broken-symmetry
solutions. If these were HF wave functions, we would take such a value as being indicative
of a fair degree of spin contamination. However, it is by no means obvious that 〈S 2 〉 for the
non-interacting KS wave function is in any way indicative of what 〈S 2 〉 may be for the interacting
wave function corresponding to the final KS density. It may not be spin contaminated
at all. On the other hand, DFT energies where broken-symmetry KS wave functions have 〈S 2 〉