Essentials of Computational Chemistry

252 8 DENSITY FUNCTIONAL THEORY
for the former. In particular, he suggested that the exchange hole about any position could
be approximated as a sphere of constant potential with a radius depending on the magnitude
of the density at that position. Within this approximation, the exchange energy Ex is
determined as
Ex[ρ(r)] =− 9α
8
1/3
3
π
ρ 4/3 (r)dr (8.7)
Within Slater’s derivation, the value for the constant α is 1, and Eq. (8.7) defines so-called
‘Slater exchange’.
Starting from the uniform electron gas, Bloch and Dirac had derived a similar expression
several years previously, except that in that case α = 2
3 (Bloch, F. 1929 and Dirac,
P. A. M. 1930). The combination of this expression with Eqs. (8.3)–(8.5) defines the
Thomas–Fermi–Dirac model, although it too remains sufficiently inaccurate that it fails
to see any modern use.
Given the differing values of α in Eq. (8.7) as a function of different derivations, many
early workers saw fit to treat it as an empirical value, and computations employing Eq. (8.7)
along these lines are termed Xα calculations (or sometimes Hartree–Fock–Slater calculations
in the older literature). Empirical analysis in a variety of different systems suggests that α = 3
4
provides more accurate results than either α = 1orα = 2
3 . This particular DFT methodology
has largely fallen out of favor in the face of more modern functionals, but still sees occasional
use, particularly within the inorganic community.
8.2 Rigorous Foundation
The work described in the previous section was provocative in its simplicity compared to
wave-function-based approaches. As a result, early DFT models found widespread use in
the solid-state physics community (where the enormous system size required to mimic the
properties of a solid puts a premium on simplicity). However, fairly large errors in molecular
calculations, and the failure of the theories to be rigorously founded (no variational principle
had been established), led to their having little impact on chemistry. This state of affairs was
set to change when Hohenberg and Kohn (1964) proved two theorems critical to establishing
DFT as a legitimate quantum chemical methodology. Each of the two theorems will be
presented here in somewhat abbreviated form.
8.2.1 The Hohenberg–Kohn Existence Theorem
In the language of DFT, electrons interact with one another and with an ‘external potential’.
Thus, in the uniform electron gas, the external potential is the uniformly distributed positive
charge, and in a molecule, the external potential is the attraction to the nuclei given by
the usual expression. As noted previously, to establish a dependence of the energy on the
density, and in the Hohenberg–Kohn theorem it is the ground-state density that is employed,
it is sufficient to show that this density determines the Hamiltonian operator. Also as noted
previously, integration of the density gives the number of electrons, so all that remains to