“semilocal” a functional for which the energy density depends onrin the infinites-
imal neighborhood of the point, and use nonlocal to describe a functional for which
the energy density at the point is determined byrat finite distances from the point.
It is more important to know how the functionals behave than to worry about their
strict adherence to mathematical definitions.
The local density approximation is based on the assumption that at every point
in the molecule theenergy densityhas the value that would be given by a
homogeneous electron gas which had the same electron densityrat that point.
The energy density is the energy (exchange plus correlation) per electron. Note
that the LDA does not assume that the electron density in a molecule is homoge-
neous (uniform); that drastic situation would be true of a “Thomas-Fermi mole-
cule”, which, as we said above, cannot exist [ 22 ] (Section7.2.2). The termlocal
was used to contrast the method with ones in which the functional depends not
just onrbut also on the gradient (first derivative) ofr, the contrast apparently
arising from the assumption that a derivative is a nonlocal property. However,
under the mathematical definition above a gradient is local, and in fact DFT
methods formerly called “nonlocal” are now commonly designated asgradient-
corrected(Section7.2.3.4c). LDA functionals have been largely replaced by a
family representing an extension of the method, local spin density approximation
(LSDA; below) functionals. In fact, in extolling the virtues of a systematic
nonempirical ascent of the DFT Jacob’s ladder, Perdew et al. [ 47 ] slight LDA
and assign to the lowest rung LSDA functionals.
7.2.3.4b The Local Spin Density Approximation (LSDA)
The “spin” here means that electrons of opposite spin are placed in different
Kohn–Sham orbitals, analogously to the Hartree–Fock UHF method (end of
Section 5.2.3.6.5). LSDA functionals are occasionally called LSD functionals.
The elaboration of the LDA method to the LSDA assigns electrons ofaandb
spin to different spatial KS orbitalscKSa andcKSb , from which different electron
density functionsraandrbfollow. LSDA has the advantage that it can handle
systems with one or more unpaired electrons, like radicals, and systems in which
electrons are becoming unpaired, such as molecules far from their equilibrium
geometries; even for ordinary molecules it appears to be more forgiving toward the
use of (necessarily) inexactEXCfunctionals [ 50 ]. For species in which all the
electrons are securely paired, the LSDA is equivalent to the LDA. LSDA geome-
tries, frequencies and electron-distribution properties tend to be reasonably good,
but (as with HF calculations) the dissociation energies, including atomization
energies, are very poor. A popular LSDA functional was the SVWN (Slater
exchange plus Vosko, Wilk, Nusair) [ 51 ]. Atomization energies are often used as
a kind of touchstone for the goodness of a method: for example, they are one of the
criteria for parameterizing and evaluating the high-accuracy energy multistep “ab
462 7 Density Functional Calculations