functionalhas, while the exchange-correlation potentialvXC(r), the functional deriva-
tive ofEXC[r(r)], is a function of the variabler, i.e. ofx,y,z. Clearly,vXC(r) depends
onr(r) and, liker(r), varies from point to point in the molecule. The functional is a
recipe for transformingrinto the exchange-correlation energyEXC. Actually, as
hinted in connection with Eq.7.13, this energy ideally also compensates for the
classical self-repulsion in the charge cloud ofr, and for the deviation of the kinetic
energy of the noninteracting KS electrons from that of real electrons. Thus a good
functional handles not only exchange and correlation errors, but also self-repulsion
and kinetic energy errors. The functional is normally tackled as an exchange term and
a correlation term; for example in the B3LYP functional (below) B3 denotes the
Becke 88 3-parameter exchange functional and LYP the Lee, Yang, Parr correlation
functional, and in the TPSS functional (below), both functionals enshrine the names
Tao, Perdew, Staroverov, Scuseria, and some programs require TPSS be denoted
TPSSTPSS. Devising good functionalsEXC[r(r)] is the main problem in density
functional theory, for all the theoretical difficulties of Kohn–Sham DFT have been
swept into the functional.
Below we look briefly at functionals based, in order of increasing sophistication
(although not quite invariably smoothly increasing excellence), on these methods:
(a) the local density approximation (LDA), (b) the local spin density approximation
(LSDA), (c) the generalized gradient approximation (GGA), (d) meta-GGA
(MGGA), (e) hybrid GGA or adiabatic connection methods (ACM methods), (f)
hybrid meta-GGA (hybrid MGGA) methods, and (g) “fully nonlocal” theory. This
hierarchy of theory has been likened to the biblical ladder reaching up to heaven
[ 40 ], and this DFT Jacob’s ladder [ 41 ] will, one hopes, culminate in what has been
appropriately called the divine functional [ 42 ]. Jensen has listed some of the proper-
ties that the divine functional must on theoretical grounds possess [ 43 ]. Some
valuable reviews of DFT are summarized here:
- Sousa et al. 2007 [ 44 ]; 14 pp. A concise historical introduction to the various
methods and extensive comparisons of many functionals for various purposes;
see especially Table 3; highlights the predominance of B3LYP. - Zhao and Truhlar 2007 [ 45 ]; 11 pp. An extensive comparison of the very popular
B3LYP functional with some new functionals; focuses on overcoming problems
of transition metals, barrier heights, and weak interactions. A class of func-
tionals, M06, “with better across-the-board average performance than B3LYP”
is presented.^2 A restrained choice of data is clearly presented. Clear recommen-
dations for various kinds of calculations. - Riley et al. 2007 [ 46 ]; 27 pp. The efficacy of DFT is examined “for small
molecules containing elements commonly found in proteins, DNA, and RNA.”
The results are very clearly presented with figures. Very extensive comparison:
37 DFT methods (functional/basis set pairs) are compared with ab initio HF
and MP2. The Pople 6-31G* basis (sometimes used with one or two sets
(^2) M06:“M zero six”, or colloquially “M oh six”. A descendant of M05, Minnesota ‘05 (2005):
Y. Zhao, N.E. Schultz, D.E. Truhlar, J. Chem. Phys., 2005, 123 , 161103.
460 7 Density Functional Calculations