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| {{Electronic structure methods}}
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| '''Density functional theory (DFT)''' is a [[quantum mechanics|quantum mechanical]] modelling method used in [[physics]] and [[chemistry]] to investigate the [[electronic structure]] (principally the [[ground state]]) of [[Many-body problem|many-body systems]], in particular atoms, molecules, and the [[condensed phase]]s. With this theory, the properties of a many-electron system can be determined by using [[Functional (mathematics)|functionals]], i.e. functions of another [[Function (mathematics)|function]], which in this case is the spatially dependent [[electronic density|electron density]]. Hence the name density functional theory comes from the use of functionals of the electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, [[computational physics]], and [[computational chemistry]].
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| DFT has been very popular for calculations in [[solid-state physics]] since the 1970s. However, DFT was not considered accurate enough for calculations in [[quantum chemistry]] until the 1990s, when the approximations used in the theory were greatly refined to better model the [[Exchange interaction|exchange]] and [[Electronic correlation|correlation]] interactions. In many cases the results of DFT calculations for solid-state systems agree quite satisfactorily with experimental data. Computational costs are relatively low when compared to traditional methods, such as [[Hartree–Fock method|Hartree–Fock theory]] and [[post-Hartree–Fock|its descendants]] based on the complex many-electron [[wavefunction]].
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| Despite recent improvements, there are still difficulties in using density functional theory to properly describe [[intermolecular force|intermolecular interactions]], especially [[van der Waals force]]s (dispersion); charge transfer excitations; transition states, global potential energy surfaces and some other [[strongly correlated material|strongly correlated]] systems; and in calculations of the [[band gap]] in [[semiconductors]]. Its incomplete treatment of dispersion can adversely affect the accuracy of DFT (at least when used alone and uncorrected) in the treatment of systems which are dominated by dispersion (e.g. interacting [[noble gas]] atoms)<ref>{{cite journal |first=Tanja |last=Van Mourik |coauthors=Gdanitz, Robert J. |year=2002 |title=A critical note on density functional theory studies on rare-gas dimers |journal=Journal of Chemical Physics |volume=116 |issue=22 |pages=9620–9623 |doi=10.1063/1.1476010|bibcode = 2002JChPh.116.9620V }}</ref> or where dispersion competes significantly with other effects (e.g. in [[biomolecule]]s).<ref>{{cite journal |first=Jiří |last=Vondrášek |coauthors=Bendová, Lada; Klusák, Vojtěch; and Hobza, Pavel |year=2005 |title=Unexpectedly strong energy stabilization inside the hydrophobic core of small protein rubredoxin mediated by aromatic residues: correlated ab initio quantum chemical calculations |journal=Journal of the American Chemical Society |pmid=15725017 |volume=127 |issue=8 |pages=2615–2619 |doi=10.1021/ja044607h}}</ref> The development of new DFT methods designed to overcome this problem, by alterations to the functional<ref>{{cite journal |first=Stefan |last=Grimme |year=2006 |title=Semiempirical hybrid density functional with perturbative second-order correlation |journal=Journal of Chemical Physics |pmid=16438568 |volume=124 |issue=3 |page=034108 |doi=10.1063/1.2148954|bibcode = 2006JChPh.124c4108G }}</ref> or by the inclusion of additive terms,<ref>{{cite journal |first=Urs |last=Zimmerli |coauthors=Parrinello, Michele; and Koumoutsakos, Petros |year=2004 |title=Dispersion corrections to density functionals for water aromatic interactions |journal=Journal of Chemical Physics |pmid=15268413 |volume=120 |issue=6 |pages=2693–2699 |doi=10.1063/1.1637034|bibcode = 2004JChPh.120.2693Z }}</ref><ref>{{cite journal |first=Stefan |last=Grimme |year=2004 |title=Accurate description of van der Waals complexes by density functional theory including empirical corrections |journal=Journal of Computational Chemistry |pmid=15224390 |volume=25 |issue=12 |pages=1463–1473 |doi=10.1002/jcc.20078}}</ref><ref>{{cite journal |first=O. Anatole |last=Von Lilienfeld |coauthors=Tavernelli, Ivano; Rothlisberger, Ursula; and Sebastiani, Daniel |year=2004 |title=Optimization of effective atom centered potentials for London dispersion forces in density functional theory |journal=Physical Review Letters |pmid=15524874 |volume=93 |issue=15 |page=153004 |doi=10.1103/PhysRevLett.93.153004 |bibcode=2004PhRvL..93o3004V}}</ref><ref>{{cite journal |first=Alexandre |last=Tkatchenko |coauthors=Scheffler, Matthias|year=2009 |title=Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data |journal=Physical Review Letters |volume=102 |pages=073005 |doi=10.1103/PhysRevLett.102.073005 |issue=7 |pmid=19257665|bibcode = 2009PhRvL.102g3005T }}</ref> is a current research topic.
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| ==Overview of method==
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| Although density functional theory has its conceptual roots in the [[Thomas–Fermi model]], DFT was put on a firm theoretical footing by the two '''Hohenberg–[[Walter Kohn|Kohn]] theorems''' (H–K).<ref name='Hohenberg1964'>{{cite journal|title=Inhomogeneous electron gas|journal=Physical Review|year=1964|first=Pierre|last=Hohenberg |coauthors=Walter Kohn|volume=136|issue=3B|pages=B864–B871| doi=10.1103/PhysRev.136.B864|bibcode = 1964PhRv..136..864H }}</ref> The original H–K 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 |publisher=United States National Academy of Sciences |volume=76 |issue=12 |pages=6062–6065 |doi=10.1073/pnas.76.12.6062|bibcode = 1979PNAS...76.6062L }}</ref><ref name="vignale">{{cite journal |last=Vignale |first=G. |coauthors=Mark Rasolt |title=Density-functional theory in strong magnetic fields |year = 1987 |journal = Physical Review Letters |volume = 59 |issue=20 |pages=2360–2363|publisher=[[American Physical Society]] |doi=10.1103/PhysRevLett.59.2360 |pmid=10035523 |bibcode=1987PhRvL..59.2360V}}</ref>
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| The first H–K theorem demonstrates that the [[Stationary state#ground state|ground state]] properties of a many-electron system are uniquely determined by an [[electronic density|electron density]] that depends on only 3 spatial coordinates. It lays the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to 3 spatial coordinates, through the use of [[Functional (mathematics)|functionals]] of the electron density. This theorem can be extended to the time-dependent domain to develop [[time-dependent density functional theory]] (TDDFT), which can be used to describe excited states.
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| The second H–K theorem defines an energy functional for the system and proves that the correct ground state electron density minimizes this energy functional.
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| Within the framework of [[Kohn–Sham equations|Kohn–Sham DFT]] (KS DFT), the intractable [[many-body problem]] of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting 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 [[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.
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| Another approach, less popular than KS DFT but arguably more closely related to the spirit of the original H-K theorems, is [[orbital-free density functional theory]] (OFDFT), in which approximate functionals are also used for the kinetic energy of the non-interacting system.
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| '''Note:''' Recently, another foundation to construct the DFT without the Hohenberg–Kohn theorems is getting popular, that is, as a Legendre transformation from external potential to electron density. See, e.g., [http://arxiv.org/abs/physics/9806013 Density Functional Theory – an introduction],
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| [http://dx.doi.org/10.1103/RevModPhys.78.865 Rev. Mod. Phys. 78, 865–951 (2006)], and references therein. A book, [http://www.ifw-dresden.de/institutes/itf/members-groups/helmut-eschrig/dft.pdf 'The Fundamentals of Density Functional Theory'] written by H. Eschrig, contains detailed mathematical discussions on the DFT; there is a difficulty for ''N''-particle system with infinite volume; however, we have no mathematical problems in finite periodic system (torus).
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| ==Derivation and formalism==
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| As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the [[Born–Oppenheimer approximation]]), generating a static external potential ''V'' in which the electrons are moving. A [[Stationary state|stationary electronic state]] is then described by a wavefunction <math>\Psi(\vec r_1,\dots,\vec r_N)</math> satisfying the many-electron time-independent [[Schrödinger equation]]
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| :<math> \hat H \Psi = \left[{\hat T}+{\hat V}+{\hat U}\right]\Psi = \left[\sum_i^N \left(-\frac{\hbar^2}{2m_i}\nabla_i^2\right) + \sum_i^N V(\vec r_i) + \sum_{i<j}^N U(\vec r_i, \vec r_j)\right] \Psi = E \Psi </math>
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| where, for the <math> \ N </math>-electron system, <math> \hat H </math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian]], <math> \ E </math> is the total energy, <math> \hat T </math> is the kinetic energy, <math> \hat V </math> is the potential energy from the external field due to positively charged nuclei, and <math> \hat U </math> is the electron-electron interaction energy. The operators <math> \hat T </math> and <math> \hat U </math> are called universal operators as they are the same for any <math> \ N </math>-electron system, while <math> \hat V </math> is
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| system dependent. This complicated many-particle equation is not separable into simpler single-particle equations because of the interaction term <math> \hat U </math>.
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| There are many sophisticated methods for solving the many-body Schrödinger equation based on the expansion of the wavefunction in [[Slater determinant]]s. While the simplest one is the [[Hartree–Fock]] method, more sophisticated approaches are usually categorized as [[post-Hartree–Fock]] methods. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger, more complex systems.
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| Here DFT provides an appealing alternative, being much more versatile as it provides a way to systematically map the many-body problem, with <math> \hat U </math>, onto a single-body problem without <math> \hat U </math>. In DFT the key variable is the particle density <math>n(\vec r),</math> which for a [[Normalisable wave function|normalized]] <math>\,\!\Psi</math> is given by
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| :<math>n(\vec r) = N \int{\rm d}^3r_2 \int{\rm d}^3r_3 \cdots \int{\rm d}^3r_N \Psi^*(\vec r,\vec r_2,\dots,\vec r_N) \Psi(\vec r,\vec r_2,\dots,\vec r_N).</math>
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| This relation can be reversed, i.e. for a given ground-state density <math>n_0(\vec r)</math> it is possible, in principle, to calculate the corresponding ground-state wavefunction <math>\Psi_0(\vec r_1,\dots,\vec r_N)</math>. In other words, <math>\,\!\Psi</math> is a unique [[functional (mathematics)|functional]] of <math>\,\!n_0</math>,<ref name='Hohenberg1964' />
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| :<math>\,\!\Psi_0 = \Psi[n_0]</math>
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| and consequently the ground-state [[Expectation value (quantum mechanics)|expectation value]] of an observable <math>\,\hat O</math> is also a functional of <math>\,\!n_0</math>
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| :<math> O[n_0] = \left\langle \Psi[n_0] \left| \hat O \right| \Psi[n_0] \right\rangle.</math>
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| In particular, the ground-state energy is a functional of <math>\,\!n_0</math>
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| :<math>E_0 = E[n_0] = \left\langle \Psi[n_0] \left| \hat T + \hat V + \hat U \right| \Psi[n_0] \right\rangle</math>
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| where the contribution of the external potential <math>\left\langle \Psi[n_0] \left|\hat V \right| \Psi[n_0] \right\rangle</math> can be written explicitly in terms of the ground-state density <math>\,\!n_0</math>
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| :<math>V[n_0] = \int V(\vec r) n_0(\vec r){\rm d}^3r. </math>
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| More generally, the contribution of the external potential <math>\left\langle \Psi \left|\hat V \right| \Psi \right\rangle</math> can be written explicitly in terms of the density <math>\,\!n</math>,
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| :<math>V[n] = \int V(\vec r) n(\vec r){\rm d}^3r. </math>
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| The functionals <math>\,\!T[n]</math> and <math>\,\!U[n]</math> are called universal functionals, while <math>\,\!V[n]</math> is called a non-universal functional, as it depends on the system under study. Having specified a system, i.e., having specified <math>\hat V</math>, one then has to minimize the functional
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| :<math> E[n] = T[n]+ U[n] + \int V(\vec r) n(\vec r){\rm d}^3r </math>
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| with respect to <math>n(\vec r)</math>, assuming one has got reliable expressions for <math>\,\!T[n]</math> and <math>\,\!U[n]</math>. A successful minimization of the energy functional will yield the ground-state density <math>\,\!n_0</math> and thus all other ground-state observables.
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| The variational problems of minimizing the energy functional <math>\,\!E[n]</math> can be solved by applying the Lagrangian method of undetermined multipliers.<ref name='Kohn1965'>{{cite journal|title=Self-consistent equations including exchange and correlation effects|journal=Physical Review|year=1965|first=W.| last=Kohn|coauthors=Sham, L. J.|volume=140|issue=4A |pages=A1133–A1138|doi=10.1103/PhysRev.140.A1133|bibcode = 1965PhRv..140.1133K }}</ref> First, one considers an energy functional that doesn't explicitly have an electron-electron interaction energy term,
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| :<math>E_s[n] = \left\langle \Psi_s[n] \left| \hat T + \hat V_s \right| \Psi_s[n] \right\rangle</math>
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| where <math>\hat T</math> denotes the kinetic energy operator and <math>\hat V_s</math> is an external effective potential in which the particles are moving, so that <math>n_s(\vec r)\ \stackrel{\mathrm{def}}{=}\ n(\vec r)</math>.
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| Thus, one can solve the so-called Kohn–Sham equations of this auxiliary non-interacting system,
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| :<math>\left[-\frac{\hbar^2}{2m}\nabla^2+V_s(\vec r)\right] \phi_i(\vec r) = \epsilon_i \phi_i(\vec r) </math>
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| which yields the [[molecular orbital|orbital]]s <math>\,\!\phi_i</math> that reproduce the density <math>n(\vec r)</math> of the original many-body system
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| :<math>n(\vec r )\ \stackrel{\mathrm{def}}{=}\ n_s(\vec r)= \sum_i^N \left|\phi_i(\vec r)\right|^2. </math>
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| The effective single-particle potential can be written in more detail as | |
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| :<math>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)]</math>
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| where the second term denotes the so-called Hartree term describing the electron-electron Coulomb repulsion, while the last term <math>\,\!V_{\rm XC}</math> is called the exchange-correlation potential. Here, <math>\,\!V_{\rm XC}</math> includes all the many-particle interactions. Since the Hartree term and <math>\,\!V_{\rm XC}</math> depend on <math>n(\vec r )</math>, which
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| depends on the <math>\,\!\phi_i</math>, which in turn depend on <math>\,\!V_s</math>, the problem of solving the Kohn–Sham equation has to be done in a self-consistent (i.e., [[iteration|iterative]]) way. Usually one starts with an initial guess for <math>n(\vec r)</math>, then calculates the corresponding <math>\,\!V_s</math> and solves the Kohn-Sham equations for the <math>\,\!\phi_i</math>. From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached. A non-iterative approximate formulation called [[Harris functional]] DFT is an alternative approach to this.
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| '''NOTE:''' The one-to-one correspondence between electron density and single-particle potential is not so smooth. It contains kinds of non-analytic structure. <math>E_s[n]</math> contains kinds of singularities. This may indicate a limitation of our hope for representing exchange-correlation functional in a simple form.
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| ==Approximations (exchange-correlation functionals)==
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| 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 [[local-density approximation]] (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:
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| :<math>E_{\rm XC}^{\rm LDA}[n]=\int\epsilon_{\rm XC}(n)n (\vec{r}) {\rm d}^3r.</math>
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| The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electron [[spin (physics)|spin]]:
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| :<math>E_{\rm XC}^{\rm LSDA}[n_\uparrow,n_\downarrow]=\int\epsilon_{\rm XC}(n_\uparrow,n_\downarrow)n (\vec{r}){\rm d}^3r.</math>
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| Highly accurate formulae for the exchange-correlation energy density
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| <math>\epsilon_{\rm XC}(n_\uparrow,n_\downarrow)</math> have been constructed
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| from [[quantum Monte Carlo]] simulations of [[jellium]].<ref>{{cite journal|title = Prescriptions for the design and selection of density functional approximations: More constraint satisfaction with fewer fits|author = John P. Perdew, Adrienn Ruzsinszky, Jianmin Tao, Viktor N. Staroverov, Gustavo Scuseria and Gábor I. Csonka|journal = Journal of Chemical Physics|volume = 123|page = 062201|year = 2005|doi = 10.1063/1.1904565|pmid = 16122287|issue = 6|bibcode = 2005JChPh.123f2201P }}</ref>
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| Generalized gradient approximations (GGA) are still local but also take into account the [[gradient]] of the density at the same coordinate:
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| :<math>E_{XC}^{\rm GGA}[n_\uparrow,n_\downarrow]=\int\epsilon_{XC}(n_\uparrow,n_\downarrow,\vec{\nabla}n_\uparrow,\vec{\nabla}n_\downarrow)
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| n (\vec{r}) {\rm d}^3r.</math>
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| Using the latter (GGA) very good results for molecular geometries and ground-state energies have been achieved.
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| Potentially more accurate than the GGA functionals are the meta-GGA functionals, a natural development after the GGA (generalized gradient approximation). Meta-GGA 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 exchange-correlation potential.
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| 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.
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| 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 functional]]s.
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| ==Generalizations to include magnetic fields==
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| 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 one-to-one mapping between the ground-state 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,<ref name="vignale" /> 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,<ref name='GrayceHarris94'>{{cite journal|title=Magnetic-field density-functional theory|journal=Physical Review A|year=1994|first=Christopher|last=Grayce |coauthors=Robert Harris|volume=50|issue=4|pages=3089–3095| doi=10.1103/PhysRevA.50.3089 |bibcode = 1994PhRvA..50.3089G|pmid=9911249 }}</ref> 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.
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| ==Applications==
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| [[File:c60 isosurface.png|thumb|right|160px|[[fullerene|C<sub>60</sub>]] with [[isosurface]] of ground-state electron density as calculated with DFT.]]
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| 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 exchange-correlation 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 generalized-gradient parametrization of the free electron gas with no free parameters); however, this is not sufficiently calorimetrically accurate for gas-phase 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 [[wavefunction]]-based 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.
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| ==Thomas–Fermi model==
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| The predecessor to density functional theory was the '''[[Thomas–Fermi model]]''', developed independently by both [[Llewellyn Thomas|Thomas]] and [[Enrico Fermi|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 <math>h^{3}</math> of volume.<ref name='ParrYang1989p47'>{{harv|Parr|Yang|1989|p=47}}</ref> For each element of coordinate space volume <math>d^{3}r</math> we can fill out a sphere of momentum space up to the Fermi momentum <math>p_f</math> <ref>{{cite book|last=March|first= N. H. |title=Electron Density Theory of Atoms and Molecules|publisher=Academic Press|year= 1992|isbn =0-12-470525-1|page=24}}</ref>
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| :<math>(4/3)\pi p_f^3(\vec{r}).\ </math> | |
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| Equating the number of electrons in coordinate space to that in phase space gives:
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| :<math>n(\vec{r})=\frac{8\pi}{3h^3}p_f^3(\vec{r}).\ </math>
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| Solving for <math>p_{f}</math> and substituting into the [[Classical mechanics|classical]] [[kinetic energy]] formula then leads directly to a kinetic energy represented as a [[functional (mathematics)|functional]] of the electron density:
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| :<math>t_{TF}[n] = \frac{p^2}{2m_e} \propto \frac{(n^{1/3})^2}{2m_e} \propto n^{2/3}(\vec{r})\ </math>
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| :<math>T_{TF}[n]= C_F \int n(\vec{r}) n^{2/3}(\vec{r}) d^3r =C_F\int n^{5/3}(\vec{r}) d^3r\ </math>
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| :where <math>C_F=\frac{3h^2}{10m_e}\left(\frac{3}{8\pi}\right)^{2/3}.\ </math>
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| As such, they were able to calculate the [[energy]] of an atom using this kinetic energy functional combined with the classical expressions for the nuclear-electron and electron-electron interactions (which can both also be represented in terms of the electron density).
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| 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 [[Paul Dirac|Dirac]] in 1928.
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| 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]].
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| [[Edward Teller|Teller]] (1962) showed that Thomas–Fermi theory cannot describe molecular bonding. This can be overcome by improving the kinetic energy functional.
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| The kinetic energy functional can be improved by adding the [[Carl Friedrich von Weizsäcker|Weizsäcker]] (1935) correction:<ref name='Weizsäcker1935'>{{cite journal|title=Zur Theorie der Kernmassen|journal=Zeitschrift für Physik|year=1935|first=C. F. v.|last=Weizsäcker|volume=96|issue=7–8|pages=431–58|doi= 10.1007/BF01337700|bibcode = 1935ZPhy...96..431W }}</ref><ref name='ParrYang1989p127'>{{harv|Parr|Yang|1989|p=127}}</ref>
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| :<math>T_W[n]=\frac{1}{8}\frac{\hbar^2}{m}\int\frac{|\nabla n(\vec{r})|^2}{n(\vec{r})}d^3r.\ </math>
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| ==Hohenberg–Kohn theorems==
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| 1.If two systems of electrons, one trapped in a potential <math> v_1(\vec r)</math> and the other in <math> v_2(\vec r)</math>, have the same ground-state density <math> n(\vec r)</math> then necessarily <math> v_1(\vec r)-v_2(\vec r) = const</math>.
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| Corollary: the ground state density uniquely determines the potential and thus all properties of the system, including the many-body wave function. In particular, the "HK" functional, defined as <math> F[n]=T[n]+U[n] </math> is a universal functional of the density (not depending explicitly on the external potential).
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| 2. For any positive integer <math> N </math> and potential <math> v(\vec r)</math> it exists a density functional <math>F[n]</math> such that <math> E_{(v,N)}[n] = F[n]+\int{v(\vec r)n(\vec r)d^3r} </math> obtains its minimal value at the ground-state density of <math> N </math> electrons in the potential <math> v(\vec r)</math>. The minimal value of <math> E_{(v,N)}[n] </math> is then the ground state energy of this system.
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| ==Pseudo-potentials==
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| The many electron [[Schrödinger equation]] can be very much simplified if electrons are divided in two groups: [[valence electrons]] and inner core [[electrons]]. The electrons in the inner shells are strongly bound and do not play a significant role in the chemical binding of [[atoms]]; they also partially [[screening effect|screen]] the nucleus, thus forming with the [[nucleus (atomic structure)|nucleus]] an almost inert core. Binding properties are almost completely due to the valence electrons, especially in metals and semiconductors.
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| This separation suggests that inner electrons can be ignored in a large number of cases, thereby reducing the atom to an ionic core that interacts with the valence electrons. The use of an effective interaction, a [[pseudopotential]], that approximates the potential felt by the valence electrons, was first proposed by Fermi in 1934 and Hellmann in 1935. In spite of the simplification pseudo-potentials introduce in calculations, they remained forgotten until the late 50’s.
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| '''''Ab initio'' Pseudo-potentials'''
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| A crucial step toward more realistic pseudo-potentials was given by Topp and Hopfield and more recently Cronin, who suggested that the pseudo-potential should be adjusted such that they describe the valence charge density accurately. Based on that idea, modern pseudo-potentials are obtained inverting the free atom Schrödinger equation for a given reference electronic configuration and forcing the pseudo wave-functions to coincide with the true valence wave functions beyond a certain distance <math>rl_.</math>. The pseudo wave-functions are also forced to have the same norm as the true valence wave-functions and can be written as
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| :<math>R_{\rm l}^{\rm pp}(r)=R_{\rm nl}^{\rm AE}(r).</math>
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| :<math>\int_{0}^{rl}dr|R_{\rm l}^{\rm PP}(r)|^2r^2=\int_{0}^{rl}dr|R_{\rm nl}^{\rm AE}(r)|^2r^2.</math>
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| where <math>R_{\rm l}(r).</math> is the radial part of the [[wavefunction]] with [[angular momentum]] <math>l_.</math>, and <math>pp_.</math> and <math>AE_.</math> denote, respectively, the pseudo wave-function and the true (all-electron) wave-function. The index n in the true wave-functions denotes the [[valence (chemistry)|valence]] level. The distance beyond which the true and the pseudo wave-functions are equal, <math>rl_.</math>, is also <math>l_.</math>-dependent.
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| ==Software supporting DFT==
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| DFT is supported by many [[List of quantum chemistry and solid-state physics software|Quantum chemistry and solid state physics software]], often along with other methods.
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| ==See also==
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| * [[Harris functional]]
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| * [[Basis set (chemistry)]]
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| * [[Gas in a box]]
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| * [[Helium atom]]
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| * [[Kohn–Sham equations]]
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| * [[Local density approximation]]
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| * [[Molecule]]
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| * [[Molecular design software]]
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| * [[Molecular modelling]]
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| * [[Quantum chemistry]]
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| * [[List of quantum chemistry and solid state physics software]]
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| * [[List of software for molecular mechanics modeling]]
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| * [[Thomas–Fermi model]]
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| * [[Time-dependent density functional theory]]
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| * [[Dynamical Mean Field Theory]]
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| ==References==
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| {{reflist|2}}
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| ==Bibliography==
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| * {{cite book |first1=R. G.|last1=Parr|first2=W.|last2=Yang| title = Density-Functional Theory of Atoms and Molecules | publisher = Oxford University Press | year = 1989 |location=New York| url = http://books.google.com/?id=mGOpScSIwU4C&printsec=frontcover&dq=Density-Functional+Theory+of+Atoms+and+Molecules&cd=1#v=onepage&q | isbn = 0-19-504279-4 | ref=harv}}{{cite book| isbn = 0-19-509276-7 (paperback)}}
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| ==Key papers==
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| * {{Cite journal|first=L. H. |last=Thomas|title= The calculation of atomic fields|doi=10.1017/S0305004100011683|journal= Proc. Camb. Phil. Soc|volume=23|year=1927|pages=542–548|issue=5|bibcode = 1927PCPS...23..542T }}
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| * {{cite journal|doi=10.1103/PhysRev.136.B864|title=Inhomogeneous Electron Gas|year=1964|last1=Hohenberg|first1=P.|last2=Kohn|first2=W.|journal=Physical Review|volume=136|issue=3B|pages=B864|bibcode = 1964PhRv..136..864H }}
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| * {{Cite journal|doi=10.1103/PhysRev.140.A1133|title=Self-Consistent Equations Including Exchange and Correlation Effects|year=1965|last1=Kohn|first1=W.|last2=Sham|first2=L. J.|journal=Physical Review|volume=140|issue=4A|pages=A1133|bibcode = 1965PhRv..140.1133K }}
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| * {{cite journal|doi=10.1063/1.464913|title=Density-functional thermochemistry. III. The role of exact exchange|year=1993|last1=Becke|first1=Axel D.|journal=The Journal of Chemical Physics|volume=98|issue=7|page=5648|bibcode = 1993JChPh..98.5648B }}
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| * {{Cite journal|doi=10.1103/PhysRevB.37.785|title=Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density|year=1988|last1=Lee|first1=Chengteh|last2=Yang|first2=Weitao|last3=Parr|first3=Robert G.|journal=Physical Review B|volume=37|issue=2|page=785|bibcode = 1988PhRvB..37..785L }}
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| * {{cite journal|doi=10.1063/1.1904586 |arxiv=cond-mat/0410362|title=Time-dependent density functional theory: Past, present, and future|year=2005|last1=Burke|first1=Kieron|last2=Werschnik|first2=Jan|last3=Gross|first3=E. K. U.|journal=The Journal of Chemical Physics|volume=123|issue=6|page=062206|bibcode = 2005JChPh.123f2206B }}
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| ==External links==
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| * [http://www.vega.org.uk/video/programme/23 Walter Kohn, Nobel Laureate] Freeview video interview with Walter on his work developing density functional theory by the Vega Science Trust.
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| * Klaus Capelle, [http://arxiv.org/abs/cond-mat/0211443 A bird's-eye view of density-functional theory]
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| * Walter Kohn, [http://nobelprize.org/chemistry/laureates/1998/kohn-lecture.pdf Nobel Lecture]
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| *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=447765 Density functional theory on arxiv.org]
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| *[http://freescience.info/books.php?id=30 FreeScience Library -> Density Functional Theory]
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| *[http://arxiv.org/abs/physics/9806013 Density Functional Theory – an introduction]
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| *[http://www.fh.huji.ac.il/~roib/LectureNotes/DFT/DFT_Course_Roi_Baer.pdf Electron Density Functional Theory – Lecture Notes ]
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| *[http://ptp.ipap.jp/link?PTP/92/833/ Density Functional Theory through Legendre Transformation][http://ann.phys.sci.osaka-u.ac.jp/~kotani/pap/924-09.pdf pdf]
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| * Kieron Burke : Book On DFT : " THE ABC OF DFT " http://dft.uci.edu/doc/g1.pdf
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| *[http://www.modelingmaterials.org/the-books Modeling Materials Continuum, Atomistic and Multiscale Techniques, Book]
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| [[Category:Density functional theory| ]]
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| [[Category:Electronic structure methods]]
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| [[Category:Physics theorems]]
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