The proof of the Hohenberg-Kohn theorem is not constructive, hence the form of the functional in Equation 2.1 is not known. Kohn and Sham postulated [4] that could be written in the form
where is the Kinetic energy of a system of non-interacting electrons with density . Representing the electron density by a set of single particle orbitals gives
We must find the minimum of with respect to subject to the constraint that the system must contain a fixed number of electrons. Thus by the method of Lagrange multipliers
where is the effective potential given by
and is the exchange correlation potential .
Thus the difficult problem for a system interacting electrons has been mapped onto that of a system of non-interacting electrons moving in an effective potential given by Equation 2.6.