\hat H \Psi = \left{=}\ n(\vec r).
Thus, one can solve the socalled Kohn–Sham equations of this auxiliary noninteracting system,

\left[\frac{\hbar^2}{2m}\nabla^2+V_s(\vec r)\right] \phi_i(\vec r) = \epsilon_i \phi_i(\vec r)
which yields the orbitals \,\!\phi_i that reproduce the density n(\vec r) of the original manybody system

n(\vec r )\ \stackrel{\mathrm{def}}{=}\ n_s(\vec r)= \sum_i^N \left\phi_i(\vec r)\right^2.
The effective singleparticle potential can be written in more detail as

V_s(\vec r) = V(\vec r) + \int \frac{e^2n_s(\vec r\,')}{\vec r\vec r\,'} {\rm d}^3r' + V_{\rm XC}[n_s(\vec r)]
where the second term denotes the socalled Hartree term describing the electronelectron Coulomb repulsion, while the last term \,\!V_{\rm XC} is called the exchangecorrelation potential. Here, \,\!V_{\rm XC} includes all the manyparticle interactions. Since the Hartree term and \,\!V_{\rm XC} depend on n(\vec r ), which depends on the \,\!\phi_i, which in turn depend on \,\!V_s, the problem of solving the Kohn–Sham equation has to be done in a selfconsistent (i.e., iterative) way. Usually one starts with an initial guess for n(\vec r), then calculates the corresponding \,\!V_s and solves the Kohn–Sham equations for the \,\!\phi_i. From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached. A noniterative approximate formulation called Harris functional DFT is an alternative approach to this.
NOTE1: The onetoone correspondence between electron density and singleparticle potential is not so smooth. It contains kinds of nonanalytic structure. E_s[n] contains kinds of singularities, cuts and branches. This may indicate a limitation of our hope for representing exchangecorrelation functional in a simple analytic form.
NOTE2: It is possible to extend the DFT idea to the case of Green function G instead of the density n. It is called as LuttingerWard functional (or kinds of similar functionals), written as E[G]. However,G is determined not as its minimum, but as its extremum. Thus we may have some theoretical and practical difficulties.
NOTE3: There is no onetoone correspondence between onebody density matrix n({\vec r},{\vec r}') and the onebody potential V({\vec r},{\vec r}'). (Remember that all the eigenvalues of n({\vec r},{\vec r}') is unity). In other words, it ends up with a theory similar as the HartreeFock (or hybrid) theory.
Approximations (exchangecorrelation functionals)
The major problem with DFT is that the exact functionals for exchange and correlation are not known except for the free electron gas. However, approximations exist which permit the calculation of certain physical quantities quite accurately. In physics the most widely used approximation is the localdensity approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:

E_{\rm XC}^{\rm LDA}[n]=\int\epsilon_{\rm XC}(n)n (\vec{r}) {\rm d}^3r.
The local spindensity approximation (LSDA) is a straightforward generalization of the LDA to include electron spin:

E_{\rm XC}^{\rm LSDA}[n_\uparrow,n_\downarrow]=\int\epsilon_{\rm XC}(n_\uparrow,n_\downarrow)n (\vec{r}){\rm d}^3r.
Highly accurate formulae for the exchangecorrelation energy density \epsilon_{\rm XC}(n_\uparrow,n_\downarrow) have been constructed from quantum Monte Carlo simulations of jellium.^{[13]}
Generalized gradient approximations^{[14]}^{[15]}^{[16]} (GGA) are still local but also take into account the gradient of the density at the same coordinate:

E_{XC}^{\rm GGA}[n_\uparrow,n_\downarrow]=\int\epsilon_{XC}(n_\uparrow,n_\downarrow,\vec{\nabla}n_\uparrow,\vec{\nabla}n_\downarrow) n (\vec{r}) {\rm d}^3r.
Using the latter (GGA) very good results for molecular geometries and groundstate energies have been achieved.
Potentially more accurate than the GGA functionals are the metaGGA functionals, a natural development after the GGA (generalized gradient approximation). MetaGGA DFT functional in its original form includes the second derivative of the electron density (the Laplacian) whereas GGA includes only the density and its first derivative in the exchangecorrelation potential.
Functionals of this type are, for example, TPSS and the Minnesota Functionals. These functionals include a further term in the expansion, depending on the density, the gradient of the density and the Laplacian (second derivative) of the density.
Difficulties in expressing the exchange part of the energy can be relieved by including a component of the exact exchange energy calculated from Hartree–Fock theory. Functionals of this type are known as hybrid functionals.
Generalizations to include magnetic fields
The DFT formalism described above breaks down, to various degrees, in the presence of a vector potential, i.e. a magnetic field. In such a situation, the onetoone mapping between the groundstate electron density and wavefunction is lost. Generalizations to include the effects of magnetic fields have led to two different theories: current density functional theory (CDFT) and magnetic field density functional theory (BDFT). In both these theories, the functional used for the exchange and correlation must be generalized to include more than just the electron density. In current density functional theory, developed by Vignale and Rasolt,^{[11]} the functionals become dependent on both the electron density and the paramagnetic current density. In magnetic field density functional theory, developed by Salsbury, Grayce and Harris,^{[17]} the functionals depend on the electron density and the magnetic field, and the functional form can depend on the form of the magnetic field. In both of these theories it has been difficult to develop functionals beyond their equivalent to LDA, which are also readily implementable computationally. Recently an extension by Pan and Sahni ^{[18]} extended the HohenbergKohn theorem for non constant magnetic fields using the density and the current density as fundamental variables.
Applications
In general, density functional theory finds increasingly broad application in the chemical and material sciences for the interpretation and prediction of complex system behavior at an atomic scale. Specifically, DFT computational methods are applied for the study of systems exhibiting high sensitivity to synthesis and processing parameters. In such systems, experimental studies are often encumbered by inconsistent results and nonequilibrium conditions. Examples of contemporary DFT applications include studying the effects of dopants on phase transformation behavior in oxides, magnetic behaviour in dilute magnetic semiconductor materials and the study of magnetic and electronic behavior in ferroelectrics and dilute magnetic semiconductors.^{[19]}^{[20]}
In practice, Kohn–Sham theory can be applied in several distinct ways depending on what is being investigated. In solid state calculations, the local density approximations are still commonly used along with plane wave basis sets, as an electron gas approach is more appropriate for electrons delocalised through an infinite solid. In molecular calculations, however, more sophisticated functionals are needed, and a huge variety of exchangecorrelation functionals have been developed for chemical applications. Some of these are inconsistent with the uniform electron gas approximation, however, they must reduce to LDA in the electron gas limit. Among physicists, probably the most widely used functional is the revised Perdew–Burke–Ernzerhof exchange model (a direct generalizedgradient parametrization of the free electron gas with no free parameters); however, this is not sufficiently calorimetrically accurate for gasphase molecular calculations. In the chemistry community, one popular functional is known as BLYP (from the name Becke for the exchange part and Lee, Yang and Parr for the correlation part). Even more widely used is B3LYP which is a hybrid functional in which the exchange energy, in this case from Becke's exchange functional, is combined with the exact energy from Hartree–Fock theory. Along with the component exchange and correlation funсtionals, three parameters define the hybrid functional, specifying how much of the exact exchange is mixed in. The adjustable parameters in hybrid functionals are generally fitted to a 'training set' of molecules. Unfortunately, although the results obtained with these functionals are usually sufficiently accurate for most applications, there is no systematic way of improving them (in contrast to some of the traditional wavefunctionbased methods like configuration interaction or coupled cluster theory). Hence in the current DFT approach it is not possible to estimate the error of the calculations without comparing them to other methods or experiments.
Thomas–Fermi model
The predecessor to density functional theory was the Thomas–Fermi model, developed independently by both Thomas and Fermi in 1927. They used a statistical model to approximate the distribution of electrons in an atom. The mathematical basis postulated that electrons are distributed uniformly in phase space with two electrons in every h^{3} of volume.^{[21]} For each element of coordinate space volume d^{3}r we can fill out a sphere of momentum space up to the Fermi momentum p_f ^{[22]}

\frac43\pi p_f^3(\vec{r}).
Equating the number of electrons in coordinate space to that in phase space gives:

n(\vec{r})=\frac{8\pi}{3h^3}p_f^3(\vec{r}).
Solving for p_{f} and substituting into the classical kinetic energy formula then leads directly to a kinetic energy represented as a functional of the electron density:

t_{TF}[n] = \frac{p^2}{2m_e} \propto \frac{(n^\frac13)^2}{2m_e} \propto n^\frac23(\vec{r})

T_{TF}[n]= C_F \int n(\vec{r}) n^\frac23(\vec{r}) d^3r =C_F\int n^\frac53(\vec{r}) d^3r

where C_F=\frac{3h^2}{10m_e}\left(\frac{3}{8\pi}\right)^\frac23.
As such, they were able to calculate the energy of an atom using this kinetic energy functional combined with the classical expressions for the nuclearelectron and electronelectron interactions (which can both also be represented in terms of the electron density).
Although this was an important first step, the Thomas–Fermi equation's accuracy is limited because the resulting kinetic energy functional is only approximate, and because the method does not attempt to represent the exchange energy of an atom as a conclusion of the Pauli principle. An exchange energy functional was added by Dirac in 1928.
However, the Thomas–Fermi–Dirac theory remained rather inaccurate for most applications. The largest source of error was in the representation of the kinetic energy, followed by the errors in the exchange energy, and due to the complete neglect of electron correlation.
Teller (1962) showed that Thomas–Fermi theory cannot describe molecular bonding. This can be overcome by improving the kinetic energy functional.
The kinetic energy functional can be improved by adding the Weizsäcker (1935) correction:^{[23]}^{[24]}

T_W[n]=\frac{\hbar^2}{8m}\int\frac{\nabla n(\vec{r})^2}{n(\vec{r})}d^3r.
Hohenberg–Kohn theorems
1.If two systems of electrons, one trapped in a potential v_1(\vec r) and the other in v_2(\vec r), have the same groundstate density n(\vec r) then necessarily v_1(\vec r)v_2(\vec r) = const.
Corollary: the ground state density uniquely determines the potential and thus all properties of the system, including the manybody wave function. In particular, the "HK" functional, defined as F[n]=T[n]+U[n] is a universal functional of the density (not depending explicitly on the external potential).
2. For any positive integer N and potential v(\vec r), a density functional F[n] exists such that E_{(v,N)}[n] = F[n]+\int{v(\vec r)n(\vec r)d^3r} obtains its minimal value at the groundstate density of N electrons in the potential v(\vec r). The minimal value of E_{(v,N)}[n] is then the ground state energy of this system.
Pseudopotentials
The many electron wavefunction with angular momentum l_., and pp_. and AE_. denote, respectively, the pseudo wavefunction and the true (allelectron) wavefunction. The index n in the true wavefunctions denotes the valence level. The distance beyond which the true and the pseudo wavefunctions are equal, rl_., is also l_.dependent.
Software supporting DFT
DFT is supported by many Quantum chemistry and solid state physics software packages, often along with other methods.
See also
Lists
References

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Key papers
External links

Walter Kohn, Nobel Laureate Freeview video interview with Walter on his work developing density functional theory by the Vega Science Trust.

Klaus Capelle, A bird'seye view of densityfunctional theory

Walter Kohn, Nobel Lecture

Density functional theory on arxiv.org

FreeScience Library > Density Functional Theory

Density Functional Theory – an introduction

Electron Density Functional Theory – Lecture Notes

Density Functional Theory through Legendre Transformationpdf

Kieron Burke : Book On DFT : " THE ABC OF DFT " http://dft.uci.edu/doc/g1.pdf

Modeling Materials Continuum, Atomistic and Multiscale Techniques, Book