What's the relationship between Density Functional Theory (DFT) and Kohn-Sham equations?

It seems Kohn-Sham equations are approximate methods to solve many body Schrodinger's equation. They directly split a multi-electron Schrodinger equation into many single-electron Schrodinger equations with exchange-correlation energy to eliminate errors.

Why there is a necessity to introduce Density Functional Theory before give Kohn-Sham equations? Or if there is no Density Functional Theory, can the Kohn-Sham equations still be used?

• Is it not similar to relation between Hartree Fock approach and exact many-body time independent Schrödinger equation ? Oct 19 '17 at 15:54

The Hohenberg-Kohn theorem states that the observables of an interacting-electron system are functionals of its ground-state density and that this density minimizes the total energy of the system. The ground-state density is also the connection between the Kohn-Sham system and the interacting-electron system: It is an auxiliary system that is constructed such that its ground-state density is equal to the one of the interacting-electron system.

Besides this there are not so many connections between the Kohn-Sham system and the interacting-electron system. For example, the energy of the highest occupied state in the Kohn-Sham system has a physical meaning as ionization energy, but the other energy eigenvalues don't have such a strict physical meaning. As a consequence the bandgap of the Kohn-Sham system is not equal to the bandgap of the interacting-electron system. Therefore for each property of the Kohn-Sham system one has to be careful how to interpret it in terms of the interacting-electron system.

Fortunately, within the limits of the chosen exchange-correlation functional, an expression for the total energy functional in terms of the density is known and with this also derived quantities like forces, phonon dispersion relations and so on. With these one can already perform many predictions like ground-state molecular and bulk structures, elastic properties, or magnetic configurations. For many other quantities the functionals are not known, especially for excited-state properties.

Note also that under the assumption that the exact exchange-correlation functional is known the Kohn-Sham formalism is not an approximation but an exact procedure to obtain the ground-state density. Unfortunately we only know approximations to this functional. But even if it would be known in an exact way this would not give more meaning to properties of the Kohn-Sham system that don't have a direct connection to the interacting-electron system.

• All observables or only the energy?
– user153036
Oct 18 '18 at 16:48
• @santimirandarp: According to the Hohenberg-Kohn theorem besides a constant potential shift you can exactly reconstruct the many-body Hamiltonian from the ground-state density. With this the many-body wavefunction is exactly defined and with the wavefunction you can calculate all observables. Therefore all observables are at least indirect functionals of the ground-state density. Of course this is only a theoretical statement since it does not provide a practically feasible computation scheme and for most observables no efficient computation schemes are known. Oct 19 '18 at 18:44

DFT tells you how to derive Kohn-Sham equations by variation principle, which is a nonlinear equation and hence you should solve it self-consistently.