Computational Chemistry

ofþfunctions) is competitive with or better than the much bigger Dunning aug-

cc-pVDZ and cc-pVTZ sets. An all-round best functional was not found but

B1B95 and B98 were among the best.

4. Perdew et al. 2005 [ 47 ]; 9 pp. Nicely prescriptive exposition of “personal
   preferences and metaphysical principles” for designing and choosing func-
   tionals. Exhorts developers to adopt a nonempirical methodology of climbing
   the DFT Jacob’s ladder by proceeding to the next higher rung by building on
   what works at each tested level, and striving to obey the known theoretical
   constraints. Holds that with these provisos DFT is not semiempirical, but rather
   a “middle way” between semiempirical and ab initio. Favors functionals without
   empirical parameters. Defends the LSDA as a still useful method and as a
   limiting case to which more sophisticated functionals should devolve in the
   uniform electron gas limit. Summarizes some known exact constraints on the
   ideal functional. They recommend functionals with “few fitted parameters” like
   PBE or TPSS.


5. Mattsson 2002 [ 42 ]; a very brief (two pages) sketch of the development of DFT.


6. Kurth et al. 1999 [ 48 ]; 21 pp. Delves well into the mathematical background
   behind functionals and discusses solids and metal surfaces in addition to atoms
   and molecules. Examines functionals constructed semiempirically as well as
   purely by considering known theoretical constraints.
   We now consider the rungs of this Jacob’s ladder.

7.2.3.4a The Local Density Approximation (LDA)

The simplest approximation toEXC[r(r)], the bottom rung of the DFT Jacob’s
ladder, results from thelocal density approximation, LDA. In mathematics a
local propertyof a function at a point on the surface (line, or two-dimensional
surface, or hypersurface) that is defined by the function is a property that depends
on the behavior of the function only in the immediate vicinity of the point [ 49 ].
“Immediate vicinity” can be taken to mean the region within an infinitesimal
distance beyond the point. Consider the derivative at some point Pion the line
defined by plottingy¼f(x) againstx. This property, the derivative or gradient, is
the limit

lim

Dx! 0

Dy

Dx

¼

dy

dx

and depends on the behavior of the curve at just an infinitesimal distance away from
Pi, i.e. in the immediate vicinity of Pi. The derivative is a local property and may
exist at Pibut not at someotherpoint, where the curve may have, say, a cusp. The
opposite of a local property is aproperty in the large[ 49 ]. Kurth et al. [ 48 ] define
“locality” somewhat differently: they take a local functional to be one for which the
energy density (below) at a point is determined byrat the point, designate by

7.2 The Basic Principles of Density Functional Theory 461