The basic idea of ​​this theory is to take the electron density of the ground state ρ (r) as the main variable, and write any other magnitude as a function of it. This theory is based on a variational principle. which requires the total energy to be a single density functional, and that this energy is minimal for the density of the ground state. The best procedure for performing the DFT is Kohn-Sham, they treated the N-body problem using the single-particle Schrõdinger equations called the Kohn-Sham equations. Solving these equations leads normally to the energy E(ρ) and the densityρ (r), of the ground state.The functional E (ρ) contains a non-classical contribution, called the exchange and correlation energy Exc (ρ) and its derivative by relation to ρ (r) which represents the exchange and correlation potential Vxc(ρ). The formalism of DFT is based on the Hohenberg - Kohn theorems. First, Hohenberg and Kohn showed that the total energy of an electron gas in the presence of an external potential, created by the nuclei is a unique functional of electron density ρ (r): E = E[ρ(r) ], Second, they showed that the minimum value of this functional is the exact energy of the ground state and that the density which leads to this energy is the exact density of the ground state. The other properties of the ground state are also functional of this density: E(ρ0) = min E(ρ ). ρ0 : The density of the ground state.

The functional of the total energy of the ground state is written as follows: E[ρ(r)]= F[ρ(r)] + ∫ VXC (r) ρ (r) d3 r. The functional F [ρ (r)] is universal for any system with several electrons. If this functional is known, then it will be relatively easy to use the variational principle. To determine the total energy and electron density of the ground state for a given external potential. Unfortunately, the Hohenberg - Kohn theorem gives no indication of the form of F [ρ (r)].