Essentials of Computational Chemistry

8.5 ADVANTAGES AND DISADVANTAGES OF DFT COMPARED TO MO THEORY 273
contamination exhibited by the HF wave function. This behavior can mitigate the utility of
hybrid functionals in some open-shell systems.
Finally, one clear utility of a wave function is that excited states can be generated as linear
combinations of determinants derived from exciting one or more electrons from occupied to
virtual orbitals (see Section 14.1). Although the Hohenberg–Kohn theorem makes it clear
that the density alone carries sufficient information to determine the excited-state wave
functions, it is only very recently that progress has been made on applying DFT to excited
states (the exception being in symmetric molecules, where the lowest energy state in each
spatial irreducible representation is amenable to a simple SCF treatment as already noted in
Section 8.2.1). Additional discussion on this subject is deferred to Section 14.2.1.
8.5.2 Computational Efficiency
The formal scaling behavior of DFT has already been noted to be in principle no worse
than N 3 ,whereN is the number of basis functions used to represent the KS orbitals. This
is better than HF by a factor of N, and very substantially better than other methods that,
like DFT, also include electron correlation (see Table 7.4). Of course, scaling refers to how
time increases with size, but says nothing about the absolute amount of time for a given
molecule. As a rule, for programs that use approximately the same routines and algorithms
to carry out HF and DFT calculations, the cost of a DFT calculation on a moderately sized
molecule, say 15 heavy atoms, is double that of the HF calculation with the same basis set.
However, it is possible to do very much better than that in programs optimized for DFT.
One area where DFT enjoys a clear advantage over HF is in its ability to use basis functions
that are not necessarily contracted Gaussians. Recall that the motivation for using contracted
GTOs is that arbitrary four-center two-electron integrals can be solved analytically. In most
electronic structure programs where DFT was added as a new feature to an existing HF code,
the representation of the density in the classical electron-repulsion operator is carried out
using the KS orbital basis functions. Thus, the net effect is to create a four-index integral,
and these codes inevitably continue to use contracted GTOs as basis functions. However,
if the density is represented using an auxiliary basis set, or even represented numerically,
other options are readily available for the KS orbital basis set, including Slater-type functions.
STOs enjoy the advantage that fewer of them are required (since, inter alia, they have correct
cusp behavior at the nuclei) and certain advantages associated with symmetry can more
readily be taken, so they speed up calculations considerably. The widely used Amsterdam
Density Functional code (ADF) makes use of STO basis functions covering atomic numbers
1 to 118 (Snijders, Baerends, and Vernooijs 1982; van Lenthe and Baerends 2003; Chong
et al. 2004).
Another interesting possibility is the use of plane waves as basis sets in periodic infinite
systems (e.g., metals, crystalline solids, or liquids represented using periodic boundary
conditions). While it takes an enormous number of plane waves to properly represent the
decidedly aperiodic densities that are possible within the unit cells of interesting chemical
systems, the necessary integrals are particularly simple to solve, and thus this approach sees
considerable use in dynamics and solid-state physics (Dovesi et al. 2000).