Essentials of Computational Chemistry

8.1 THEORETICAL MOTIVATION 251
fermion statistical mechanics to derive the kinetic energy for this system as (Thomas 1927;
Fermi 1927)
Tueg[ρ(r)] = 3
10 (3π 2 ) 2/3
ρ 5/3 (r)dr (8.5)
Note that the various T and V terms defined in Eqs. (8.3)–(8.5) are functions of the density,
while the density itself is a function of three-dimensional spatial coordinates. A function
whose argument is also a function is called a ‘functional’, and thus the T and V terms are
‘density functionals’. The Thomas–Fermi equations, together with an assumed variational
principle, represented the first effort to define a density functional theory (DFT); the energy
is computed with no reference to a wave function. However, while these equations are of
significant historical interest, the underlying assumptions are sufficiently inaccurate that they
find no use in modern chemistry (in Thomas–Fermi DFT, all molecules are unstable relative
to dissociation into their constituent atoms...)
One large approximation is the use of Eq. (8.4) for the interelectronic repulsion, since it
ignores the energetic effects associated with correlation and exchange. It is useful to introduce
the concept of a ‘hole function’, which is defined so that it corrects for the energetic errors
introduced by assuming classical behavior. In particular, we write
electrons
i