The electron density corresponding to the ground state can be calculated using a variational principle, we know that every
. In this way Kohn and Sham added a tremendous addendum in 1965 , within their variational scheme an auxiliary system of non-interacting electrons is introduced. It is assumed that the ground state density of this hypothetical non-interacting electrons system is the same as that of the original interacting system.
In addition, the Kohn-Sham approach to the interacting electrons problem is to rewrite the Hohenberg-Kohn expression for the energy functional corresponding to the ground state as follows:
where is the kinetic energy of the non-interacting electron system. is the energy due to the external potential (external potential means that it is due to the atomic nuclei), and it has the following form:
is the Hartree energy, which can be expressed as a functional of the electron density of the system as follows:
and finally, is the so called exchange correlation energy term which includes "all remaining" contributions.
We have to note that in practice the correct term is unknown, for this reason the determination of such functional requires some approximation, and it is this approximation which determines the accuracy of the method.