Essentials of Computational Chemistry

278 8 DENSITY FUNCTIONAL THEORY
accuracy of the ACM methods can fail to extend to systems where they are methodologically
less stable owing to their Hartree–Fock component.
8.5.4 Systematic Improvability
In molecular orbital theory, there is a clear and well defined path to the exact solution of the
Schrödinger equation. All we need do is express our wave function as a linear combination
of all possible configurations (full CI) and choose a basis set that is infinite in size, and we
have arrived. While such a goal is essentially never practicable, at least the path to it can be
followed unambiguously until computational resources fail.
With density functional theory, the situation is much less clear when it comes to evaluating
how to do a ‘better’ calculation. One thing that seems fairly clear is that, as a general rule,
results from MGGA functionals tend to improve on those from GGA functionals, which in
turn drastically improve on those from LSDA. Somewhat less clear is the status of hybrid
functionals. The best ones are competitive in quality with the best MGGA functionals (and
B3LYP seems to continue to be the ‘magic’ functional in project after project) subject to
the caveat that in certain situations the presence of HF exchange may cause problems that
are associated with the single-determinant KS formalism to become manifest more quickly.
As for basis-set effects, just as with MO theory one can examine convergence with respect
to basis-set size, but there is no guarantee that this may not lead to increased errors since
errors associated with basis-set incompleteness may offset errors associated with approximate
functionals.
All that being said, experience dictates that, across a surprisingly wide variety of systems,
DFT tends to be remarkably robust. Thus, unless a problem falls into one of a few classes
of well characterized problems for DFT, there is good reason to be optimistic about any
particular calculation.
Finally, it seems clear that routes to further improve DFT must be associated with better
defining hole functions in arbitrary systems. In particular, the current generation of functionals
has reached a point where finding efficient algorithms for correction of the classical
self-interaction error are likely to have the largest qualitative (and quantitative) impact.
8.5.5 Worst-case Scenarios
Certain failures of modern DFT should be anticipated, and others are readily explained
after some thought about the forms of current functionals. One clear problem with modern
functionals is that they make the energy a function entirely of the local density and possibly
the density gradient. As such, they are incapable of properly describing London dispersion
forces, which, as noted in Section 2.2.4, derive entirely from electron correlation at ‘long
range’. Adding HF exchange to the DFT functional cannot entirely alleviate this problem,
since the HF level of theory, while non-local, does not account in any way for opposite-spin
electron correlation.
So, even though noble-gas dimers like He2, Ne2, etc., exhibit potential energy minima
at van der Waals contact, DFT predicts the potential energy curve for these diatomics to