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  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  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 .