Essentials of Computational Chemistry

274 8 DENSITY FUNCTIONAL THEORY
Even in cases where contracted GTOs are chosen as basis sets, DFT offers the advantage
that convergence with respect to basis-set size tends to be more rapid than for MO techniques
(particularly correlated MO theories). Thus, polarized valence double-ζ basissetsarequite
adequate for a wide variety of calculations, and very good convergence in many properties
can be seen at the level of employing polarized triple-ζ basis sets. Extensive studies of basis
set effects on functional performance and parameterization have been carried out by Jensen
(2002a, 2002b, 2003) and Boese, Martin, and Handy (2003). They found, inter alia, that
for most functionals Pople-type basis sets provide much better accuracy than cc-pVnZ basis
sets of similar size, that adding diffuse functions offers substantial improvement over using
the non-augmented analog basis (a point also made by Lynch, Zhao, and Truhlar (2003),
particularly for the computation of barrier heights or conformational energies in molecules
containing multiple lone pairs of electrons), that satisfactory convergence is generally arrived
at for most properties of interest by the time triple-ζ basis sets are used, and finally that
the optimal values for parameters that are included in various functionals are sensitive to
choice of basis set size. Thus, the optimal percent HF exchange for HCTH/407 was about
28% with double-ζ basis sets, but about 18% with triple-ζ basis sets. Jensen (2002b, 2003)
found that, with reoptimization of the polarization exponents for DFT, the pc-n basis sets
were always able to provide the best accuracy for a given basis set size.
Besides issues associated with basis sets, considerable progress has been made in
developing linear-scaling algorithms for DFT. In this regard, DFT is somewhat simpler
than MO theoretical techniques because all potentials are local (this refers to ‘pure’
DFT – incorporation of HF exchange introduces the non-local exchange operator). Thus,
one promising technique is the ‘divide-and-conquer’ formalism of Yang and co-workers,
where a large system is divided up into a number of smaller regions, within each of which
a KS SCF is carried out representing the other regions in a simplified fashion (Yang and
Lee 1995). The total cost of matrix diagonalization is thereby reduced from N 3 scaling
to M(N/M) 3 scaling where M is the number of sub-regions. Since the number of basis
functions in each sub-region (N/M) tends to be close to some fixed value irrespective of
N, the overall scaling goes as order M, i.e., linear. Of course, all the algorithms developed
to facilitate linear scaling in computing Coulomb interactions in HF and MD calculations
(e.g., fast multipole methods) can be used in DFT calculations as well.
As a final point with regard to efficiency, note that SCF convergence in DFT is sometimes
more problematic than in HF. Because of the similarities between the KS and HF orbitals,
this problem can often be very effectively alleviated by using the HF orbitals as an initial
guess for the KS orbitals. Because the HF orbitals can usually be generated quite quickly, the
extra step can ultimately be time-saving if it sufficiently improves the KS SCF convergence.
8.5.3 Limitations of the KS Formalism
It is important to emphasize that nearly all applications of DFT to molecular systems are
undertaken within the context of the Kohn–Sham SCF approach. The motivation for this
choice is that it permits the kinetic energy to be computed as the expectation value of
the kinetic-energy operator over the KS single determinant, avoiding the tricky issue of