Density Functional Theory (DFT) is based on the notion of the single
particle electron density as a fundamental variable. This is a
consequence of the Hohenberg - Kohn theorem [11] which
states that the ground state electron density 
minimises
the energy functional
where 
is a universal functional. Furthermore, the minimum
value of E is 
, the ground state electronic energy. This is an
exact result and in principle, means that the ground state energy and
electron density may be found using a variational minimisation over
the electron density 
, a process with scales linearly with
system size.
Unfortunately, the proof of the Hohenberg - Kohn theorem is not
constructive, hence the form of the functional in
Equation 
is not known. Kohn and Sham postulated [12]
that this functional could be written
Here the first term 
is the kinetic energy of a system
of non-interacting electrons with density and the second is the
electron - electron Hartree interaction. The final term,
, is the exchange-correlation energy. By writing the
electron density in terms of a set of single particle wavefunctions 
such that
the kinetic energy term may be written
Minimising 
with respect to , subject to the
constraint that the number of electrons must be constant, leads us to a
set of equations
where 
is an `effective potential',
and 
is the exchange-correlation potential 
. Equation can be seen to be a
set of Schrödinger-like equations for the single particle Kohn-Sham
orbitals
Thus, the problem of a system of interacting electrons has been
mapped onto a system of non-interacting electrons moving in an
effective potential given by Equation . However, the expression of
the density in terms of a set of single particle orbitals has
increased the complexity of the problem. The minimisation must now be
performed over NM degrees of freedom where M is the number of
basis functions used to represent the .
