Jacob’s ladder.” From this it is evident that “fully nonlocal” DFT theory does not
promise a single, sharply defined functional, although the divine functional [ 42 ]
must be fully nonlocal. What does fully nonlocal mean? Kurth et al. uselocal,
semilocal, andnonlocalto refer to properties that are determined at a point, at an
infinitesimal distance beyond the point, and at a finite distance beyond the point,
respectively [ 48 ]. These are not the strict mathematical definitions of the terms [ 49 ],
but they are working intuitive concepts: to determine the gradient at a point one must
move an infinitesimal distance beyond. Exact electron exchange energy is an example
of a nonlocal property ofr, because it arises from “Pauli repulsion” between electrons
a finite distance apart; a fully nonlocal functional would presumably take into account
all such nonlocal phenomena. Attention is paid to the local nature or otherwise of
various functionals in the review by Zhao and Truhlar [ 45 ]. Nonlocal functionals have
been under development for some years [ 59 ], but fully nonlocal ones, with all relevant
properties treated nonlocally, are apparently not yet available for practical molecular
calculations. However, some recent (2006 and later) functionals, for example
B2PLYP [ 60 ], which uses hybrid-GGA with MP2-like (Section 5.4.2) promotion of
electrons into virtual orbitals for treating electron correlation, rival coupled-cluster ab
initio calculations (Section 5.4.3) for certain purposes [ 61 ].
Having just examined the various DFT levels, we can return to the question
posed at the end of Section7.1: is DFT semiempirical, or is it a kind of ab initio
method? We can also ask: does it matter? Addressing the first question: a semiem-
pirical method is one that is parameterized against experiment (but in chemistry we
wisely do not demand that fundamental constants like the velocity of light and
Planck’s constant be calculated from first principles!). Itispossible to develop
functionals that have not been parameterized against experiment, and the review
[ 47 ] in which Perdew et al. “present the case for the nonempirical construction” of
such functionals argues convincingly for the classification of DFT as an ab initio
techniquewhen it follows these strictures. The adjectives “ab initio” and “wave-
function” need not be synonymous: the wavefunction and the operators which act
on it are central concepts in traditional ab initio theory, while in density functional
theory the corresponding central concepts are the electron density function and the
functionals which act on that. The wavefunction is not in principle indispensable in
DFT. Rather, the Kohn–Sham wavefunction (Eq.7.19) is a clever subterfuge for
coming to grips with the functional problem by reducing to a small fraction of the
total energy the energy (Eq.7.21) that an imperfect functional generates, and for
permitting the electron density function to be iteratively refined (Eq.7.22). Regarding
the second question: quite apart from the esthetic value that some see in a purely
nonempirical calculation (recall von Neumann’s jaundiced view of empirical equa-
tions: Section 6.3.7), it may well be true that empirical approaches can reliably
interpolate, but not extrapolate, and that they are, outside their parameterized
domains, susceptible to “catastrophic failure” [ 62 ]. We close this discussion of the
“philosophy” of DFT with a look at a provocative stance by Nooijen, namely that
DFT resembles molecular mechanics in that there “exists an exact force field foreach
electronic state with a given number of electrons” and that “the existence of many
different exact functionals...also suggests that the physical content of DFT is easily
466 7 Density Functional Calculations