Essentials of Computational Chemistry

272 8 DENSITY FUNCTIONAL THEORY
Another key example is in the area of dynamics, where transition probabilities depend
on matrix elements between different wave functions. Because densities do not have phases
as wave functions do, multistate resonance effects, interference effects, etc., are not readily
evaluated within a DFT formalism.
Because of the mechanical details of the KS formalism, it is easy to become confused
about whether there is a KS ‘wave function’. Early work in the field tended to resist any
attempts to interpret the KS orbitals, viewing them as pure mathematical constructs useful
only in construction of the density. In practice, however, the shapes of KS orbitals tend to be
remarkably similar to canonical HF MOs, and they can be quite useful in qualitative analysis
of chemical properties. If we think of the procedure by which they are generated, there are
indeed a number of reasons to prefer KS orbitals to HF orbitals. For instance, all KS orbitals,
occupied and virtual, are subject to the same external potential. HF orbitals, on the other
hand, experience varying potentials, and, in particular, HF virtual orbitals experience the
potential that would be felt by an extra electron being added to the molecule. As a result,
HF virtual orbitals tend to be too high in energy and anomalously diffuse compared to KS
virtual orbitals. (In exact DFT, it can also be shown that the eigenvalue of the highest KS
MO is the exact first ionization potential, i.e., there is a direct analogy to Koopmans’ theorem
for this orbital – in practice, however, approximate functionals are quite bad at predicting
IPs in this fashion without applying some sort of correction scheme, e.g., an empirical linear
scaling of the eigenvalues).
In point of fact, there is a DFT wave function; it is just not clear how useful it should
be considered to be. Recall that the Slater determinant formed from the KS orbitals is the
exact wave function for the fictional non-interacting system having the same density as the
real system. This KS Slater determinant has certain interesting properties by comparison
to its HF analogs. In open-shell systems, KS determinants usually show extremely low
levels of spin contamination, even for cases where HF determinants are pathologically bad
(Baker, Scheiner, and Andzelm 1993). For instance, the spin contamination in planar triplet
phenylnitrenium cation (PhNH + ) is very high at the UHF/cc-pVDZ level (see Section 6.3.3)
as judged by an expectation value for S 2 of 2.50. At the BLYP/cc-pVDZ level, on the
other hand the expectation value for S 2 over the KS determinant is 2.01, very close to the
proper eigenvalue of 2.0. The high spin contamination at the UHF level leads to the planar
structure being erroneously determined to be a minimum, while at the BLYP level it is
correctly identified as a TS structure for rotation about the C–N bond (Cramer, Dulles, and
Falvey 1994).
While it is by no means guaranteed that the expectation value for S 2 over the KS
determinant has any bearing at all on its expectation value over the exact wave function
corresponding to the KS density (see Gräfenstein and Cremer 2001), it is an empirical
fact that DFT is generally much more robust in dealing with open-shell systems where
HF methods show high spin contamination (recall that high HF spin contamination makes
post-HF methods of questionable utility, so DFT can be a happy last resort). Note, incidentally,
that expectation values of S 2 are sensitive to the amount of HF exchange in the
functional. A ‘pure’ functional nearly always shows very small spin contamination, and
each added percent of HF exchange tends to titrate in a corresponding percentage of the spin