The percentage of HF exchange energy to use is a main distinguishing characteristic
of the various hybrid functionals. The first popular, successful hybrid method was
B3LYP. This is the B3PW91 functional first proposed by Becke [ 57 ], modified by
Stephens et al. [ 58 ]. The B3LYP functional has a total of eight purely empirical
parameters. B3LYP has been wildly popular: Sousa et al. [ 44 ] show in their 2007
paper that from 2002 to 2006 in each year it has accounted for ca. 80% of the names
of the functionals in journal articles and abstracts, and Zhao and Truhlar single it
out for special comparison with their new functionals [ 45 ]. This popularity is
despite the fact that evidently, for almost any particular application, one can find
a better functional. The durability of B3LYP and the advisability of its continued
use are discussed later; for now we note that near the end of their extensive
comparison, Sousa et al. [ 44 ] say that “B3LYP still remains a valid and particularly
efficient alternative for the ‘average’ quantum chemistry problem”.
Some hybrid methods base the HF percentage not on experimental parameteri-
zation (“parameter-free” hybrid methods), but on theoretical arguments; this does
not automatically give them superior performance. GGA functionals tend to under-
estimate barriers and HF methods tend to overestimate them, but a happy adjust-
ment of HF exchange for barriers tends to reduce the accuracy for other properties.
7.2.3.4f Hybrid Meta-GGA (HMGGA) Functionals
These are analogous to the hybrid GGA functionals of Section7.2.3.4eabove, but
with Hartree–Fock exchange added on to meta-GGA (Section7.2.3.4d), rather than
GGA, functionals (Section7.2.3.4c). Hybrid MGGA (HMGGA) uses the first
derivative ofrand its second derivative, or the kinetic energy density (Sec-
tion7.2.3.4d), and Hartree–Fock exchange. They are the highest-level functionals
in routine use. Most are, as of mid-2009, fairly recent: in Table 2 of ref. [ 44 ] (2007),
of the 52 “most common” functionals listed and referenced, 14 are HMGGA and of
these one is vintage 1996 and the others 2003–2005; this paper depicts HMGGA on
the fourth rung of the ladder, rather than the sixth implied here, because it effec-
tively collapses on to rung one LDA and LSDA, and places on rung four both
HGGA and HMGGA. The strongpoint of HMGGA seems to be “an improvement
over the previous formalisms in...barrier heights and atomization energies” [ 44 ].
7.2.3.4g Fully Nonlocal Theory
This is the seventh and highest rung in our ordering, the fifth on the “collapsed”
ladder of Sousa et al. [ 44 ], a step above HMGGA functionals. Perdew et al. say [ 47 ]
that “a fully nonlocal functional of the density...can be satisfied on the fourth rung
by hyper-GGAs that use full exact exchange” that “Exact exchange can only be
combined with a fully nonlocal correlation, constructed on the fourth or fifth rungs
of the ladder” and that “there is also continuing interest...in the weighted density
approximation, a nonempirical and fully nonlocal functional that does not fit on
7.2 The Basic Principles of Density Functional Theory 465