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.
