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The Density Functional Theory departs from the principle that the electrons density determines all properties of a model. Hence the electron density determines all properties of a gas of electrons and atomic nuclei.

The Hohenberg-Kohn theorems apply to such a system. These theorems are the basis of the density functional theory.

Theorem one states that the external potential Vext(r) and the total energy is a unique functional of the electron density ρ(r) r is the spacial position vector. The energy functional is :

ρ(r) is the electron density on position r. F(ρ(r)) is an unknown potential that depends only on the electron density.

Theorem two states that the ground state electron density that minimizes E(ρ(r)) is the exact ground state density. The ground state electron density and hence all properties of the model can be obtain with the variation theorem.

In density functional theory the electron density can be calculated with the Kohn-Sham equations. These equations map a gas of strongly interacting particles on a gas of non-interacting particles that move in a effective on particle potential. The Kohn Sham equations are derived from the Hohenberg-Kohn theorems.

The Kohn-Sham equations are

Veff(r) is the effective potential and Φ(i) are the Kohn-Sham oprbitals. The electron density can be calculated from the Kohn-Sham orbitals.

The effective potential is

The first term Vext(r) is the external potential due to the atomic nuclei.

The second term is the potential an electron experiences from the other electrons.

The third term is the exchange correlation potential.

The exchange correlation potential cannot be determined analytical. A lot of research has been done and is been done to find an appropriate expression for the exchange correlation potential. A popular exchange-correlation potential is the B3LYP potential.