Essentials of Computational Chemistry

250 8 DENSITY FUNCTIONAL THEORY
number of electrons N, i.e.,
N = ρ(r)dr (8.1)
Moreover, because the nuclei are effectively point charges, it should be obvious that their
positions correspond to local maxima in the electron density (and these maxima are also
cusps), so the only issue left to completely specify the Hamiltonian is the assignment of
nuclear atomic numbers. It can be shown that this information too is available from the
density, since for each nucleus A located at an electron density maximum rA
∂ρ(rA)
=−2ZAρ(rA) (8.2)
∂rA
rA=0
where Z is the atomic number of A, rA is the radial distance from A, and ρ is the spherically
averaged density.
Of course, the arguments above do not provide any simpler formalism for finding the
energy. They simply indicate that given a known density, one could form the Hamiltonian
operator, solve the Schrödinger equation, and determine the wave functions and energy
eigenvalues. Nevertheless, they suggest that some simplifications might be possible.
8.1.2 Early Approximations
Energy is separable into kinetic and potential components. If one decides apriori to try to
evaluate the molecular energy using only the electron density as a variable, the simplest
approach is to consider the system to be classical, in which case the potential energy
components are straightforwardly determined. The attraction between the density and the
nuclei is
Vne[ρ(r)] =
nuclei
and the self-repulsion of a classical charge distribution is
k
Vee[ρ(r)] = 1
ρ(r1)ρ(r2)
2 |r1 − r2| dr1dr2
Zk
ρ(r)dr (8.3)
|r − rk|
(8.4)
where r1 and r2 are dummy integration variables running over all space.
The kinetic energy of a continuous charge distribution is less obvious. To proceed, we
first introduce the fictitious substance ‘jellium’. Jellium is a system composed of an infinite
number of electrons moving in an infinite volume of a space that is characterized
by a uniformly distributed positive charge (i.e., the positive charge is not particulate in
nature, as it is when represented by nuclei). This electronic distribution, also called the
uniform electron gas, has a constant non-zero density. Thomas and Fermi, in 1927, used