Essentials of Computational Chemistry

258 8 DENSITY FUNCTIONAL THEORY
empirical parameters appear, which necessarily introduce some kinetic energy correction if
they are based on experiment.
In discussing the nature of various functionals, it is convenient to adopt some of the
notation commonly used in the field. For instance, the functional dependence of Exc on the
electron density is expressed as an interaction between the electron density and an ‘energy
density’ εxc that is dependent on the electron density, viz.
Exc[ρ(r)] = ρ(r)εxc[ρ(r)]dr (8.22)
The energy density εxc is always treated as a sum of individual exchange and correlation
contributions. Note that there is some potential for nomenclature confusion here because two
different kinds of densities are involved: the electron density is a per unit volume density,
while the energy density is a per particle density. In any case, within this formalism, it is
clear from inspection of Eq. (8.7) that the Slater exchange energy density, for example, is
εx[ρ(r)] =− 9α
8
1/3 3
ρ
π
1/3 (r) (8.23)
Another convention expresses the electron density in terms of an effective radius such that
exactly one electron would be contained within the sphere defined by that radius were it to
have the same density throughout as its center, i.e.,
3
rS(r) =
4πρ(r)
1/3
(8.24)
Lastly, we have ignored the issue of spin up to this point. Spin can be dealt with easily
enough in DFT – one simply needs to use individual functionals of the α and β densities
– but there is again a notational convention that sees widespread use. The spin densities
at any position are typically expressed in terms of ζ , the normalized spin polarization
ζ(r) = ρα (r) − ρ β (r)
ρ(r)
(8.25)
so that the α spin density is simply one-half the product of the total ρ and (ζ + 1), andthe
β spin density is the difference between that value and the total ρ.
8.4.1 Local Density Approximation
The term local density approximation (LDA) was originally used to indicate any density
functional theory where the value of εxc at some position r could be computed exclusively
from the value of ρ at that position, i.e., the ‘local’ value of ρ. In principle, then, the only
requirement on ρ is that it be single-valued at every position, and it can otherwise be wildly
ill-behaved (recall that there are cusps in the density at the nucleus, so some ill-behavior