Hohenberg and Kohn proved that the total energy of a system including that of the many body effects of electrons (exchange and correlation) in the presence of static external potential (for example, the atomic nuclei) is a unique functional of the charge density. The minimum value of the total energy functional is the ground state energy of the system. The electronic charge density which yields this minimum is then the exact single particle ground state energy.
It was then shown by Kohn and Sham that it is possible to replace the many electron problem by an exactly equivalent set of self consistent one electron equations. The total energy functional can be written as a sum of several terms:
for a fixed set of atomic nuclei at . The first two terms are the classical Coulomb interaction between the electrons and ions and between electrons and other electrons respectively, both of which are simply functions of the electronic charge density . This equation is analogous to the Hartree method, but the term contains the effects of exchange and correlation and also the single particle kinetic energy:
where is the kinetic energy of a system of non-interacting electrons with density and is the energy of exchange and correlation of an interacting system. There is no simple expression for the exchange and correlation - this will be considered in Section 3.4
According to the Hohenberg-Kohn theorem, the total energy function given by equation (3.8) is stationary with respect to variations in the ground state charge density, that is, it is subject to the condition
where is the functional derivative of the exchange-correlation energy with respect to the electronic charge density. There is also the requirement that a variation in the charge density leaves the particle number
unchanged. This can be ensured by the condition
Applying the condition of constant particle number, (3.12) to Equation (3.10) gives the result
is a constant, , which is the Lagrange multiplier associated with the requirement of constant particle number. Comparing this to the corresponding equation for a system with an effective potential but without electron-electron interaction results in
It can be seen that the mathematical representations are similar provided that
The effect of this is to allow an indirect variation in through variation in the Kohn-Sham single particle orbitals, , where the kinetic energy operator can be expressed in terms of the single particle states as
It then follows that the solution can be found by solving the Schrödinger equation for noninteracting particles moving under the influence of an effective potential
which gives the charge density
The solution of the system of equations (3.17) leads to the energy and electronic charge density of the ground state and all quantities which can be derived from them. The minimum of the Kohn-Sham energy functional, (3.8), leads to the ground state charge density of the electronic system which ions at the fixed positions . It is only this minimum which has any physical meaning, therefore the path by which this minimum is found is unimportant. This point will be returned to later when minimisation techniques will be discussed.