Editing Density functional theory (section)

===Origins===
Although density functional theory has its  roots in the [[Thomas–Fermi model]] for the electronic structure of materials, DFT was first put on a firm theoretical footing by [[Walter Kohn]] and [[Pierre Hohenberg]] in the framework of the two '''Hohenberg–Kohn theorems''' (HK).<ref name='Hohenberg1964'>{{cite journal |title=Inhomogeneous electron gas |journal=Physical Review |year=1964 |first1=Pierre |last1=Hohenberg |last2=Walter |first2=Kohn |volume=136 |issue=3B |pages=B864–B871 |doi=10.1103/PhysRev.136.B864 |bibcode = 1964PhRv..136..864H |doi-access=free}}</ref> The original HK theorems held only for non-degenerate ground states in the absence of a [[magnetic field]], although they have since been generalized to encompass these.<ref>{{cite journal |last=Levy |first=Mel |year=1979 |title=Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the ''v''-representability problem |journal=Proceedings of the National Academy of Sciences |volume=76 |issue=12 |pages=6062–6065 |doi=10.1073/pnas.76.12.6062 |pmid=16592733 |bibcode=1979PNAS...76.6062L |pmc=411802|doi-access=free }}</ref><ref name="vignale">{{cite journal |last1=Vignale |first1=G. |first2=Mark |last2=Rasolt |title=Density-functional theory in strong magnetic fields |year = 1987 |journal = Physical Review Letters |volume=59 |issue=20 |pages=2360–2363 |doi=10.1103/PhysRevLett.59.2360 |pmid=10035523 |bibcode=1987PhRvL..59.2360V}}</ref>

The first HK theorem demonstrates that the [[ground-state]] properties of a many-electron system are uniquely determined by an [[electronic density|electron density]] that depends on only three spatial coordinates. It set down the groundwork for reducing the many-body problem of {{mvar|N}} electrons with {{math|3''N''}} spatial coordinates to three spatial coordinates, through the use of [[Functional (mathematics)|functionals]] of the electron density. This theorem has since been extended to the time-dependent domain to develop [[time-dependent density functional theory]] (TDDFT), which can be used to describe excited states.

The second HK theorem defines an energy functional for the system and proves that the ground-state electron density minimizes this energy functional.

In work that later won them the [[Nobel prize in chemistry]], the HK theorem was further developed by [[Walter Kohn]] and [[Lu Jeu Sham]] to produce [[Kohn–Sham equations|Kohn–Sham DFT]] (KS DFT). Within this framework, the intractable [[many-body problem]] of interacting electrons in a static external potential is reduced to a tractable problem of noninteracting electrons moving in an effective [[potential]]. The effective potential includes the external potential and the effects of the [[Coulomb's law|Coulomb interactions]] between the electrons, e.g., the [[exchange interaction|exchange]] and [[electron correlation|correlation]] interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the [[local-density approximation]] (LDA), which is based upon exact exchange energy for a uniform [[Fermi gas|electron gas]], which can be obtained from the [[Thomas–Fermi model]], and from fits to the correlation energy for a uniform electron gas. Non-interacting systems are relatively easy to solve, as the wavefunction can be represented as a [[Slater determinant]] of [[molecular orbitals|orbitals]]. Further, the [[kinetic energy]] functional of such a system is known exactly. The exchange–correlation part of the total energy functional remains unknown and must be approximated.

Another approach, less popular than KS DFT but arguably more closely related to the spirit of the original HK theorems, is [[orbital-free density functional theory]] (OFDFT), in which approximate functionals are also used for the kinetic energy of the noninteracting system.