32nd  Top numerical analysis topics 
Density functional theory (DFT) is a quantum mechanical theory used in physics and chemistry to investigate the electronic structure (principally the ground state) of manybody systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a manyelectron system can be determined by using functionals, i.e. functions of another function, which in this case is the spatially dependent 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 condensedmatter physics, computational physics, and computational chemistry.
DFT has been very popular for calculations in solid state physics since the 1970s. In many cases the results of DFT calculations for solidstate systems agreed quite satisfactorily with experimental data. Also, the computational costs were relatively low when compared to traditional ways which were based on the complicated manyelectron wavefunction, such as HartreeFock theory and its descendants. 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 and correlation interactions. DFT is now a leading method for electronic structure calculations in chemistry and solidstate physics.
Despite the improvements in DFT, there are still difficulties in using density functional theory to properly describe intermolecular interactions, especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces and some other strongly correlated systems; and in calculations of the band gap in semiconductors. Its poor treatment of dispersion renders DFT unsuitable (at least when used alone) for the treatment of systems which are dominated by dispersion (e.g., interacting noble gas atoms) or where dispersion competes significantly with other effects (e.g. in biomolecules). The development of new DFT methods designed to overcome this problem, by alterations to the functional or by the inclusion of additive terms, is a current research topic.
Although density functional theory has its conceptual roots in the ThomasFermi model, DFT was put on a firm theoretical footing by the two HohenbergKohn theorems (HK).^{[1]} The original HK theorems held only for nondegenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these.^{[2]}^{[3]}
The first HK theorem demonstrates that the ground state properties of a manyelectron system are uniquely determined by an electron density that depends on only 3 spatial coordinates. It lays the groundwork for reducing the manybody problem of N electrons with 3N spatial coordinates to 3 spatial coordinates, through the use of functionals of the electron density. This theorem can be extended to the timedependent domain to develop timedependent 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 correct ground state electron density minimizes this energy functional.
Within the framework of KohnSham DFT, the intractable manybody 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 interactions between the electrons, e.g., the exchange and correlation interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the localdensity approximation (LDA), which is based upon exact exchange energy for a uniform electron gas, which can be obtained from the ThomasFermi model, and from fits to the correlation energy for a uniform electron gas. Noninteracting systems are relatively easy to solve as the wavefunction can be represented as a Slater determinant of orbitals. Further, the kinetic energy functional of such a system is known exactly. The exchangecorrelation part of the totalenergy functional remains unknown and must be approximated.
Another approach, less popular than KohnSham DFT (KSDFT) but arguably more closely related to the spirit of the original HK theorems, is orbitalfree density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the noninteracting system.
As usual in manybody electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the BornOppenheimer approximation), generating a static external potential V in which the electrons are moving. A stationary electronic state is then described by a wavefunction satisfying the manyelectron Schrödinger equation
where is the electronic molecular Hamiltonian, is the number of electrons, is the electron kinetic energy, is the electron potential energy from the external field, and is the electronelectron interaction energy for the electron system. The operators and are socalled universal operators as they are the same for any system, while is system dependent, i.e. nonuniversal. The difference between having separable singleparticle problems and the much more complicated manyparticle problem arises from the interaction term .
There are many sophisticated methods for solving the manybody Schrödinger equation based on the expansion of the wavefunction in Slater determinants. While the simplest one is the HartreeFock method, more sophisticated approaches are usually categorized as postHartreeFock 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.
Here DFT provides an appealing alternative, being much more versatile as it provides a way to systematically map the manybody problem, with , onto a singlebody problem without . In DFT the key variable is the particle density , which for a normalized is given by
This relation can be reversed, i.e. for a given groundstate density it is possible, in principle, to calculate the corresponding groundstate wavefunction . In other words, is a unique functional of ,^{[1]}
and consequently the groundstate expectation value of an observable is also a functional of
In particular, the groundstate energy is a functional of
where the contribution of the external potential can be written explicitly in terms of the groundstate density
More generally, the contribution of the external potential can be written explicitly in terms of the density ,
The functionals and are called universal functionals, while is called a nonuniversal functional, as it depends on the system under study. Having specified a system, i.e., having specified , one then has to minimize the functional
with respect to , assuming one has got reliable expressions for and . A successful minimization of the energy functional will yield the groundstate density and thus all other groundstate observables.
The variational problems of minimizing the energy functional can be solved by applying the Lagrangian method of undetermined multipliers.^{[4]} First, one considers an energy functional that doesn't explicitly have an electronelectron interaction energy term,
where denotes the noninteracting kinetic energy and is an external effective potential in which the particles are moving. Obviously, if is chosen to be
Thus, one can solve the socalled KohnSham equations of this auxiliary noninteracting system,
which yields the orbitals that reproduce the density of the original manybody system
The effective singleparticle potential can be written in more detail as
where the second term denotes the socalled Hartree term describing the electronelectron Coulomb repulsion, while the last term is called the exchangecorrelation potential. Here, includes all the manyparticle interactions. Since the Hartree term and depend on , which depends on the , which in turn depend on , the problem of solving the KohnSham equation has to be done in a selfconsistent (i.e., iterative) way. Usually one starts with an initial guess for , then calculates the corresponding and solves the KohnSham equations for the . From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached.
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:
The local spindensity approximation (LSDA) is a straightforward generalization of the LDA to include electron spin:
Highly accurate formulae for the exchangecorrelation energy density have been constructed from quantum Monte Carlo simulations of a free electron model.^{[5]}
Generalized gradient approximations (GGA) are still local but also take into account the gradient of the density at the same coordinate:
Using the latter (GGA) very good results for molecular geometries and groundstate energies have been achieved.
Potentially more accurate than the GGA functionals are metaGGA functions. 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 HartreeFock theory. Functionals of this type are known as hybrid functionals.
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 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,^{[3]} 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, 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.
In practice, KohnSham 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 PerdewBurkeErnzerhof 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 HartreeFock 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.
For molecular applications, in particular for hybrid functionals, KohnSham DFT methods are usually implemented just like HartreeFock itself.
The predecessor to density functional theory was the Thomas–Fermi model, developed by 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.^{[6]} 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} ^{[7]}
Equating the number of electrons in coordinate space to that in phase space gives:
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:
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:^{[8]}^{[9]}
1.For Ninteracting electrons,E[n] is only functional of the electron density.
2.E[n_{GS}] = E_{GS}
E_{GS} is the real ground state energy,and n_{GS} is the real ground state electron density.
DFT is supported by many Quantum chemistry and solid state physics codes, often along with other methods.
Electronic structure methods 
Tight binding 
Nearly free electron model 
Hartree–Fock 
Modern valence bond 
Generalized valence bond 
Møller–Plesset perturbation theory 
Configuration interaction 
Coupled cluster 
Multiconfigurational selfconsistent field 
Density functional theory 
Quantum chemistry composite methods 
Quantum Monte Carlo 
k·p perturbation theory 
Muffintin approximation 
LCAO method 
Density functional theory (DFT) is a quantum mechanical theory used in physics and chemistry to investigate the electronic structure (principally the ground state) of manybody systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a manyelectron system can be determined by using functionals, i.e. functions of another function, which in this case is the spatially dependent 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 condensedmatter physics, computational physics, and computational chemistry.
DFT has been very popular for calculations in solid state physics since the 1970s. In many cases the results of DFT calculations for solidstate systems agreed quite satisfactorily with experimental data. Also, the computational costs were relatively low when compared to traditional ways which were based on the complicated manyelectron wavefunction, such as HartreeFock theory and its descendants. 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 and correlation interactions. DFT is now a leading method for electronic structure calculations in chemistry and solidstate physics.
Despite recent improvements, there are still difficulties in using density functional theory to properly describe intermolecular interactions, especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces and some other 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) or where dispersion competes significantly with other effects (e.g. in biomolecules). The development of new DFT methods designed to overcome this problem, by alterations to the functional or by the inclusion of additive terms, is a current research topic.
Although density functional theory has its conceptual roots in the ThomasFermi model, DFT was put on a firm theoretical footing by the two HohenbergKohn theorems (HK).^{[1]} The original HK theorems held only for nondegenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these.^{[2]}^{[3]}
The first HK theorem demonstrates that the ground state properties of a manyelectron system are uniquely determined by an electron density that depends on only 3 spatial coordinates. It lays the groundwork for reducing the manybody problem of N electrons with 3N spatial coordinates to 3 spatial coordinates, through the use of functionals of the electron density. This theorem can be extended to the timedependent domain to develop timedependent 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 correct ground state electron density minimizes this energy functional.
Within the framework of KohnSham DFT (KS DFT), the intractable manybody 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 interactions between the electrons, e.g., the exchange and correlation interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the localdensity approximation (LDA), which is based upon exact exchange energy for a uniform electron gas, which can be obtained from the ThomasFermi model, and from fits to the correlation energy for a uniform electron gas. Noninteracting systems are relatively easy to solve as the wavefunction can be represented as a Slater determinant of orbitals. Further, the kinetic energy functional of such a system is known exactly. The exchangecorrelation part of the totalenergy 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 orbitalfree density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the noninteracting system.
Note: Recently, another foundation to construct the DFT without the HohenbergKohn theorems is getting popular, that is, as a Legendre transformation from external potential to electron density. See, e.g., Density Functional Theory  an introduction, Rev. Mod. Phys. 78, 865–951 (2006), and references there in. A book, 'The Fundamentals of Density Functional Theory' written by H.Eschrig, contains detailed mathematical discussions on the DFT; there is a difficulty for Nparticle system with infinite volume; however, we have no mathematical problems in finite periodic system (torus).
As usual in manybody electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the BornOppenheimer approximation), generating a static external potential V in which the electrons are moving. A stationary electronic state is then described by a wavefunction $\backslash Psi(\backslash vec\; r\_1,\backslash dots,\backslash vec\; r\_N)$ satisfying the manyelectron timeindependent Schrödinger equation
where, for the $\backslash \; N$electron system, $\backslash hat\; H$ is the Hamiltonian, $\backslash \; E$ is the total energy, $\backslash hat\; T$ is the kinetic energy, $\backslash hat\; V$ is the potential energy from the external field due to positively charged nuclei, and $\backslash hat\; U$ is the electronelectron interaction energy. The operators $\backslash hat\; T$ and $\backslash hat\; U$ are called universal operators as they are the same for any $\backslash \; N$electron system, while $\backslash hat\; V$ is system dependent. This complicated manyparticle equation is not separable into simpler singleparticle equations because of the interaction term $\backslash hat\; U$.
There are many sophisticated methods for solving the manybody Schrödinger equation based on the expansion of the wavefunction in Slater determinants. While the simplest one is the HartreeFock method, more sophisticated approaches are usually categorized as postHartreeFock 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.
Here DFT provides an appealing alternative, being much more versatile as it provides a way to systematically map the manybody problem, with $\backslash hat\; U$, onto a singlebody problem without $\backslash hat\; U$. In DFT the key variable is the particle density $n(\backslash vec\; r),$ which for a normalized $\backslash ,\backslash !\backslash Psi$ is given by
This relation can be reversed, i.e. for a given groundstate density $n\_0(\backslash vec\; r)$ it is possible, in principle, to calculate the corresponding groundstate wavefunction $\backslash Psi\_0(\backslash vec\; r\_1,\backslash dots,\backslash vec\; r\_N)$. In other words, $\backslash ,\backslash !\backslash Psi\_0$ is a unique functional of $\backslash ,\backslash !n\_0$,^{[1]}
and consequently the groundstate expectation value of an observable $\backslash ,\backslash hat\; O$ is also a functional of $\backslash ,\backslash !n\_0$
In particular, the groundstate energy is a functional of $\backslash ,\backslash !n\_0$
where the contribution of the external potential $\backslash left\backslash langle\; \backslash Psi[n\_0]\; \backslash left\backslash hat\; V\; \backslash right\; \backslash Psi[n\_0]\; \backslash right\backslash rangle$ can be written explicitly in terms of the groundstate density $\backslash ,\backslash !n\_0$
More generally, the contribution of the external potential $\backslash left\backslash langle\; \backslash Psi\; \backslash left\backslash hat\; V\; \backslash right\; \backslash Psi\; \backslash right\backslash rangle$ can be written explicitly in terms of the density $\backslash ,\backslash !n$,
The functionals $\backslash ,\backslash !T[n]$ and $\backslash ,\backslash !U[n]$ are called universal functionals, while $\backslash ,\backslash !V[n]$ is called a nonuniversal functional, as it depends on the system under study. Having specified a system, i.e., having specified $\backslash hat\; V$, one then has to minimize the functional
with respect to $n(\backslash vec\; r)$, assuming one has got reliable expressions for $\backslash ,\backslash !T[n]$ and $\backslash ,\backslash !U[n]$. A successful minimization of the energy functional will yield the groundstate density $\backslash ,\backslash !n\_0$ and thus all other groundstate observables.
The variational problems of minimizing the energy functional $\backslash ,\backslash !E[n]$ can be solved by applying the Lagrangian method of undetermined multipliers.^{[4]} First, one considers an energy functional that doesn't explicitly have an electronelectron interaction energy term,
where $\backslash hat\; T\_s$ denotes the noninteracting kinetic energy and $\backslash hat\; V\_s$ is an external effective potential in which the particles are moving. Obviously, $n\_s(\backslash vec\; r)\backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; n(\backslash vec\; r)$ if $\backslash hat\; V\_s$ is chosen to be
Thus, one can solve the socalled KohnSham equations of this auxiliary noninteracting system,
which yields the orbitals $\backslash ,\backslash !\backslash phi\_i$ that reproduce the density $n(\backslash vec\; r)$ of the original manybody system
The effective singleparticle potential can be written in more detail as
where the second term denotes the socalled Hartree term describing the electronelectron Coulomb repulsion, while the last term $\backslash ,\backslash !V\_\{\backslash rm\; XC\}$ is called the exchangecorrelation potential. Here, $\backslash ,\backslash !V\_\{\backslash rm\; XC\}$ includes all the manyparticle interactions. Since the Hartree term and $\backslash ,\backslash !V\_\{\backslash rm\; XC\}$ depend on $n(\backslash vec\; r\; )$, which depends on the $\backslash ,\backslash !\backslash phi\_i$, which in turn depend on $\backslash ,\backslash !V\_s$, the problem of solving the KohnSham equation has to be done in a selfconsistent (i.e., iterative) way. Usually one starts with an initial guess for $n(\backslash vec\; r)$, then calculates the corresponding $\backslash ,\backslash !V\_s$ and solves the KohnSham equations for the $\backslash ,\backslash !\backslash phi\_i$. 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.
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:
The local spindensity approximation (LSDA) is a straightforward generalization of the LDA to include electron spin:
Highly accurate formulae for the exchangecorrelation energy density $\backslash epsilon\_\{\backslash rm\; XC\}(n\_\backslash uparrow,n\_\backslash downarrow)$ have been constructed from quantum Monte Carlo simulations of a free electron model.^{[5]}
Generalized gradient approximations (GGA) are still local but also take into account the gradient of the density at the same coordinate:
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. 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 HartreeFock theory. Functionals of this type are known as hybrid functionals.
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 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,^{[3]} 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, 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.
with isosurface of groundstate electron density as calculated with DFT.]]
In practice, KohnSham 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 PerdewBurkeErnzerhof 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 HartreeFock 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.
For molecular applications, in particular for hybrid functionals, KohnSham DFT methods are usually implemented just like HartreeFock itself.
The predecessor to density functional theory was the Thomas–Fermi model, developed by 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.^{[6]} For each element of coordinate space volume $d^\{3\}r$ we can fill out a sphere of momentum space up to the Fermi momentum $p\_f$ ^{[7]}
Equating the number of electrons in coordinate space to that in phase space gives:
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:
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:^{[8]}^{[9]}
1.If two systems of electrons, one trapped in a potential $v\_1(\backslash vec\; r)$ and the other in $v\_2(\backslash vec\; r)$ have the same groundstate density $n(\backslash vec\; r)$ then necessarily $v\_1(\backslash vec\; r)v\_2(\backslash vec\; r)\; =\; const$.
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 $v(\backslash vec\; r)$ the density functional $E\_\{(v,N)\}[n]\; =\; F[n]+\backslash int\{v(\backslash vec\; r)n(\backslash vec\; r)d^3r\}$ obtains its minimal value at the groundstate density of $N$ electrons in the potential $v(\backslash vec\; r)$. The minimal value of $E\_\{(v,N)\}[n]$ is then the ground state energy of this system.
DFT is supported by many Quantum chemistry and solid state physics codes, often along with other methods.
