# Density functional theory

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Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron 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 condensed-matter physics, computational physics, and computational chemistry.

Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Chemistry is the scientific discipline involved with elements and compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo during a reaction with other substances.

## Contents

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 and correlation interactions. Computational costs are relatively low when compared to traditional methods, such as exchange only Hartree–Fock theory and its descendants that include electron correlation.

Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic-scale properties. Thus, solid-state physics forms a theoretical basis of materials science. It also has direct applications, for example in the technology of transistors and semiconductors.

Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems. It is also called molecular quantum mechanics.

In chemistry and physics, the exchange interaction is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical force, it is not a true force as it lacks a force carrier.

Despite recent improvements, there are still difficulties in using density functional theory to properly describe: intermolecular interactions (of critical importance to understanding chemical reactions), especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces, dopant interactions and some strongly correlated systems; and in calculations of the band gap and ferromagnetism in semiconductors. [1] The 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) [2] or where dispersion competes significantly with other effects (e.g. in biomolecules). [3] The development of new DFT methods designed to overcome this problem, by alterations to the functional [4] or by the inclusion of additive terms, [5] [6] [7] [8] is a current research topic.

Intermolecular forces (IMF) are the forces which mediate interaction between molecules, including forces of attraction or repulsion which act between molecules and other types of neighboring particles, e.g., atoms or ions. Intermolecular forces are weak relative to intramolecular forces – the forces which hold a molecule together. For example, the covalent bond, involving sharing electron pairs between atoms, is much stronger than the forces present between neighboring molecules. Both sets of forces are essential parts of force fields frequently used in molecular mechanics.

In molecular physics, the van der Waals force, named after Dutch scientist Johannes Diderik van der Waals, is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and therefore more susceptible to disturbance. The van der Waals force quickly vanishes at longer distances between interacting molecules.

Strongly correlated materials are a wide class of heavy fermion compounds that include insulators and electronic materials, and show unusual electronic and magnetic properties, such as metal-insulator transitions, half-metallicity, and spin-charge separation. The essential feature that defines these materials is that the behavior of their electrons or spinons cannot be described effectively in terms of non-interacting entities. Theoretical models of the electronic (fermionic) structure of strongly correlated materials must include electronic (fermionic) correlation to be accurate. As of recently, the label Quantum Materials is also used to refer to Strongly Correlated Materials, among others.

## Overview of method

In the context of computational materials science, ab initio (from first principles) DFT calculations allow the prediction and calculation of material behaviour on the basis of quantum mechanical considerations, without requiring higher order parameters such as fundamental material properties. In contemporary DFT techniques the electronic structure is evaluated using a potential acting on the system’s electrons. This DFT potential is constructed as the sum of external potentials Vext, which is determined solely by the structure and the elemental composition of the system, and an effective potential Veff, which represents interelectronic interactions. Thus, a problem for a representative supercell of a material with n electrons can be studied as a set of n one-electron Schrödinger-like equations, which are also known as Kohn–Sham equations. [9]

The interdisciplinary field of materials science, also commonly termed materials science and engineering is the design and discovery of new materials, particularly solids. The intellectual origins of materials science stem from the Enlightenment, when researchers began to use analytical thinking from chemistry, physics, and engineering to understand ancient, phenomenological observations in metallurgy and mineralogy. Materials science still incorporates elements of physics, chemistry, and engineering. As such, the field was long considered by academic institutions as a sub-field of these related fields. Beginning in the 1940s, materials science began to be more widely recognized as a specific and distinct field of science and engineering, and major technical universities around the world created dedicated schools of the study, within either the Science or Engineering schools, hence the naming.

Ab initio is a Latin term meaning "from the beginning" and is derived from the Latin ab ("from") + initio, ablative singular of initium ("beginning").

The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

### 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 (H–K). [10] 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. [11] [12]

The Thomas–Fermi (TF) model, named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equation. It stands separate from wave function theory as being formulated in terms of the electronic density alone and as such is viewed as a precursor to modern density functional theory. The TF model is correct only in the limit of an infinite nuclear charge. Using the approximation for realistic systems yields poor quantitative predictions, even failing to reproduce some general features of the density such as shell structure in atoms and Friedel oscillations in solids. It has, however, found modern applications in many fields through the ability to extract qualitative trends analytically and with the ease at which the model can be solved. The kinetic energy expression of Thomas–Fermi theory is also used as a component in more sophisticated density approximation to the kinetic energy within modern orbital-free density functional theory.

Walter Kohn was an American theoretical physicist and theoretical chemist. He was awarded, with John Pople, the Nobel Prize in Chemistry in 1998. The award recognized their contributions to the understandings of the electronic properties of materials. In particular, Kohn played the leading role in the development of density functional theory, which made it possible to calculate quantum mechanical electronic structure by equations involving the electronic density. This computational simplification led to more accurate calculations on complex systems as well as many new insights, and it has become an essential tool for materials science, condensed-phase physics, and the chemical physics of atoms and molecules.

Pierre C. Hohenberg was a French-American theoretical physicist, who worked primarily on statistical mechanics. Hohenberg studied at Harvard, where he earned his bachelor's degree in 1956 and a master's degree in 1958, and his doctorate in 1962. From 1962-1963, he was at the Institute for Physical Problems in Moscow, followed by a stay at the École Normale Supérieure in Paris. From 1964 to 1995 he was at Bell Laboratories in Murray Hill. From 1985 to 1989, he was director of the department of theoretical physics and from 1989 to 1995 he was "Distinguished Member of Technical Staff". From 1974 to 1977, he was also professor of theoretical physics at the TU München, where he had previously been a 1972-1973 guest professor. From 1995 to 2003 he was "Deputy Provost of Science and Technology" at Yale University. Subsequently, he was the Yale "Eugene Higgins Adjunct Professor of Physics and Applied Physics". Hohenberg was additionally from 1963–1964 and again in 1988 guest professor in Paris and 1990-1991 as Lorentz-Professor in Leiden. In 2004 he became Senior Vice Provost of Research at New York University, a position held until 2011, when he stepped down to join the Physics Department as Professor. In 2012 he became Emeritus Professor of Physics at NYU.

The first H–K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates. It set down the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to three spatial coordinates, through the use of 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.

In quantum mechanics, and in particular quantum chemistry, the electronic density is a measure of the probability of an electron occupying an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either ρ(r) or n(r). The density is determined, through definition, by the normalized N-electron wavefunction which itself depends upon 4N variables. Conversely, the density determines the wave function modulo a phase factor, providing the formal foundation of density functional theory.

In mathematics, the term functional has at least three meanings.

Time-dependent density functional theory (TDDFT) is a quantum mechanical theory used in physics and chemistry to investigate the properties and dynamics of many-body systems in the presence of time-dependent potentials, such as electric or magnetic fields. The effect of such fields on molecules and solids can be studied with TDDFT to extract features like excitation energies, frequency-dependent response properties, and photoabsorption spectra.

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.

In work that later won them the Nobel prize in chemistry, the H–K theorem was further developed by Walter Kohn and Lu Jeu Sham to produce 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 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 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 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 H–K theorems, is orbital-free density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the noninteracting system.

## Derivation and formalism

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 electronic state is then described by a wavefunction Ψ(r1,…,rN) satisfying the many-electron time-independent Schrödinger equation

${\displaystyle {\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\left({\vec {r}}_{i}\right)+\sum _{i

where, for the N-electron system, Ĥ is the Hamiltonian, E is the total energy, is the kinetic energy, is the potential energy from the external field due to positively charged nuclei, and Û is the electron–electron interaction energy. The operators and Û are called universal operators as they are the same for any N-electron system, while is system-dependent. This complicated many-particle equation is not separable into simpler single-particle equations because of the interaction term Û.

There are many sophisticated methods for solving the many-body Schrödinger equation based on the expansion of the wavefunction in Slater determinants. 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.

Here DFT provides an appealing alternative, being much more versatile as it provides a way to systematically map the many-body problem, with Û, onto a single-body problem without Û. In DFT the key variable is the electron density n(r), which for a normalized Ψ is given by

${\displaystyle n\left({\vec {r}}\right)=N\int {\mathrm {d} }^{3}r_{2}\cdots \int {\mathrm {d} }^{3}r_{N}\,\Psi ^{*}\left({\vec {r}},{\vec {r}}_{2},\dots ,{\vec {r}}_{N}\right)\Psi \left({\vec {r}},{\vec {r}}_{2},\dots ,{\vec {r}}_{N}\right).}$

This relation can be reversed, i.e., for a given ground-state density n0(r) it is possible, in principle, to calculate the corresponding ground-state wavefunction Ψ0(r1,…,rN). In other words, Ψ is a unique functional of n0, [10]

${\displaystyle \Psi _{0}=\Psi [n_{0}]}$

and consequently the ground-state expectation value of an observable Ô is also a functional of n0

${\displaystyle O[n_{0}]=\left\langle \Psi [n_{0}]\left|{\hat {O}}\right|\Psi [n_{0}]\right\rangle .}$

In particular, the ground-state energy is a functional of n0

${\displaystyle E_{0}=E[n_{0}]=\left\langle \Psi [n_{0}]\left|{\hat {T}}+{\hat {V}}+{\hat {U}}\right|\Psi [n_{0}]\right\rangle }$

where the contribution of the external potential ⟨ Ψ[n0] | | Ψ[n0] ⟩ can be written explicitly in terms of the ground-state density n0

${\displaystyle V[n_{0}]=\int V\left({\vec {r}}\right)n_{0}\left({\vec {r}}\right)\,{\mathrm {d} }^{3}r.}$

More generally, the contribution of the external potential ⟨ Ψ | | Ψ ⟩ can be written explicitly in terms of the density n,

${\displaystyle V[n]=\int V({\vec {r}})n({\vec {r}})\,{\mathrm {d} }^{3}r.}$

The functionals T[n] and U[n] are called universal functionals, while V[n] is called a non-universal functional, as it depends on the system under study. Having specified a system, i.e., having specified , one then has to minimize the functional

${\displaystyle E[n]=T[n]+U[n]+\int V({\vec {r}})n({\vec {r}})\,{\mathrm {d} }^{3}r}$

with respect to n(r), assuming one has reliable expressions for T[n] and U[n]. A successful minimization of the energy functional will yield the ground-state density n0 and thus all other ground-state observables.

The variational problems of minimizing the energy functional E[n] can be solved by applying the Lagrangian method of undetermined multipliers. [13] First, one considers an energy functional that does not explicitly have an electron–electron interaction energy term,

${\displaystyle E_{s}[n]=\left\langle \Psi _{\mathrm {s} }[n]\left|{\hat {T}}+{\hat {V}}_{\mathrm {s} }\right|\Psi _{\mathrm {s} }[n]\right\rangle }$

where denotes the kinetic energy operator and s is an external effective potential in which the particles are moving, so that ns(r) ≝ n(r).

Thus, one can solve the so-called Kohn–Sham equations of this auxiliary noninteracting system,

${\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{\mathrm {s} }({\vec {r}})\right]\varphi _{i}({\vec {r}})=\varepsilon _{i}\varphi _{i}({\vec {r}})}$

which yields the orbitals φi that reproduce the density n(r) of the original many-body system

${\displaystyle n({\vec {r}})\ {\stackrel {\mathrm {def} }{=}}\ n_{\mathrm {s} }({\vec {r}})=\sum _{i}^{N}\left|\varphi _{i}({\vec {r}})\right|^{2}.}$

The effective single-particle potential can be written in more detail as

${\displaystyle V_{\mathrm {s} }({\vec {r}})=V({\vec {r}})+\int {\frac {e^{2}n_{\mathrm {s} }\left({\vec {r}}'\right)}{\left|{\vec {r}}-{\vec {r}}'\right|}}\,{\mathrm {d} }^{3}r'+V_{\mathrm {XC} }[n_{\mathrm {s} }({\vec {r}})]}$

where the second term denotes the so-called Hartree term describing the electron–electron Coulomb repulsion, while the last term VXC is called the exchange–correlation potential. Here, VXC includes all the many-particle interactions. Since the Hartree term and VXC depend on n(r), which depends on the φi, which in turn depend on Vs, the problem of solving the Kohn–Sham equation has to be done in a self-consistent (i.e., iterative) way. Usually one starts with an initial guess for n(r), then calculates the corresponding Vs and solves the Kohn–Sham equations for the φi. 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.

Notes

1. The one-to-one correspondence between electron density and single-particle potential is not so smooth. It contains kinds of non-analytic structure. Es[n] contains kinds of singularities, cuts and branches. This may indicate a limitation of our hope for representing exchange–correlation functional in a simple analytic form.
2. It is possible to extend the DFT idea to the case of the Green function G instead of the density n. It is called as Luttinger–Ward 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.
3. There is no one-to-one correspondence between one-body density matrix n(r,r) and the one-body potential V(r,r). (Remember that all the eigenvalues of n(r,r) are 1.) In other words, it ends up with a theory similar to the Hartree–Fock (or hybrid) theory.

## Relativistic density functional theory (ab initio functional forms)

The same theorems can be proven in the case of relativistic electrons, thereby providing generalization of DFT for the relativistic case. Unlike the nonrelativistic theory, in the relativistic case it is possible to derive a few exact and explicit formulas for the relativistic density functional.

Let one consider an electron in a hydrogen-like ion obeying the relativistic Dirac equation. The Hamiltonian H for a relativistic electron moving in the Coulomb potential can be chosen in the following form (atomic units are used):

${\displaystyle H=c({\vec {\alpha }}\cdot {\vec {p}})+eV+mc^{2}\beta ,}$

where V = −eZ/r is the Coulomb potential of a pointlike nucleus, p is a momentum operator of the electron, and e, m and c are the elementary charge, electron mass and the speed of light respectively, and finally α and β are a set of Dirac 2 × 2 matrices:

{\displaystyle {\begin{aligned}{\vec {\alpha }}&={\begin{pmatrix}0&{\vec {\sigma }}\\{\vec {\sigma }}&0\end{pmatrix}},\\\beta &={\begin{pmatrix}I&0\\0&-I\end{pmatrix}}.\end{aligned}}}

To find out the eigenfunctions and corresponding energies, one solves the eigenfunction equation

${\displaystyle H\Psi =E\Psi ,}$

where Ψ = (Ψ(1), Ψ(2), Ψ(3), Ψ(4))T is a four-component wavefunction and E is the associated eigenenergy. It is demonstrated in Brack (1983) [14] that application of the virial theorem to the eigenfunction equation produces the following formula for the eigenenergy of any bound state:

${\displaystyle E=mc^{2}\left\langle \Psi \left|\beta \right|\Psi \right\rangle =mc^{2}\int \left|\Psi (1)\right|^{2}+\left|\Psi (2)\right|^{2}-\left|\Psi (3)\right|^{2}-\left|\Psi (4)\right|^{2}\,\mathrm {d} \tau ,}$

and analogously, the virial theorem applied to the eigenfunction equation with the square of the Hamiltonian [15] yields

${\displaystyle E^{2}=m^{2}c^{4}+emc^{2}\left\langle \Psi \left|V\beta \right|\Psi \right\rangle }$.

It is easy to see that both of the above formulae represent density functionals. The former formula can be easily generalized for the multi-electron case. [16]

One may observe that both of the functionals written above don't have extremals, of course if reasonably wide set of functions is allowed for variation. Nevertheless it is possible to design a density functional with desired extremal properties out of those ones. Let us make it in the following way:

${\displaystyle F[n]={1 \over {mc^{2}}}\left(mc^{2}\int nd\tau -{\sqrt {m^{2}c^{4}+emc^{2}\int Vnd\tau }}\right)^{2}+\delta _{n,n_{e}}mc^{2}\int nd\tau }$,

where ne in Kronecker delta symbol of the second term denotes any extremal for the functional represented by the first term of the functional F. The second term amounts to zero for any function which is not an extremal for the first term of functional F. To proceed further we'd like to find Lagrange equation for this functional. In order to do this we should allocate a linear part of functional increment when argument function is altered.

${\displaystyle F[n_{e}+\delta n]={1 \over {mc^{2}}}\left(mc^{2}\int (n_{e}+\delta n)d\tau -{\sqrt {m^{2}c^{4}+emc^{2}\int V(n_{e}+\delta n)d\tau }}\right)^{2}}$

Deploying written above equation it is easy to find the following formula for functional derivative

${\displaystyle {\frac {\delta F[n_{e}]}{\delta n}}=2A-{\frac {2B^{2}+AeV(\tau _{0})}{B}}+eV(\tau _{0})}$,

where A and B stay for mc2∫ ne and √m2c4+emc2∫Vne respectively. And finally V(τ0) is a value of potential in some point, specified by support of variation function δn which is supposed to be infinitesimal. To advance toward Lagrange equation we equate functional derivative to zero and after simple algebraic manipulations arrive to the following equation.

${\displaystyle 2B(A-B)=eV(\tau _{0})(A-B)}$

Apparently this equation could have solution only if A is equal to B. This last condition provides us with Lagrange equation for functional F, which could be finally written down in the following form.

${\displaystyle \left(mc^{2}\int nd\tau \right)^{2}={m^{2}c^{4}+emc^{2}\int Vnd\tau }}$

Solutions of this equation represent extremals for functional F. It's easy to see that all real densities, that is densities corresponding to the bound states of the system in question, are solutions of written above equation, which could be called the Kohn–Sham equation in this particular case. Looking back onto the definition of the functional F we clearly see that the functional produces energy of the system for appropriate density, because the first term amounts to zero for such density and the second one delivers the energy value.

## Approximations (exchange–correlation 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. [17] One of the simplest approximations is the local-density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:

${\displaystyle E_{\mathrm {XC} }^{\mathrm {LDA} }[n]=\int \varepsilon _{\mathrm {XC} }(n)n({\vec {r}})\,{\mathrm {d} }^{3}r.}$

The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electron spin:

${\displaystyle E_{\mathrm {XC} }^{\mathrm {LSDA} }\left[n_{\uparrow },n_{\downarrow }\right]=\int \varepsilon _{\mathrm {XC} }\left(n_{\uparrow },n_{\downarrow }\right)n({\vec {r}})\,{\mathrm {d} }^{3}r.}$

In LDA, the exchange–correlation energy is typically separated into the exchange part and the correlation part: εXC = εX + εC. The exchange part is called the Dirac (or sometimes Slater) exchange which takes the form εXn13. There are, however, many mathematical forms for the correlation part. Highly accurate formulae for the correlation energy density εC(n,n) have been constructed from quantum Monte Carlo simulations of jellium. [18] A simple first-principles correlation functional has been recently proposed as well. [19] [20] Although unrelated to the Monte Carlo simulation, the two variants provide comparable accuracy. [21]

The LDA assumes that the density is the same everywhere. Because of this, the LDA has a tendency to underestimate the exchange energy and over-estimate the correlation energy. [22] The errors due to the exchange and correlation parts tend to compensate each other to a certain degree. To correct for this tendency, it is common to expand in terms of the gradient of the density in order to account for the non-homogeneity of the true electron density. This allows for corrections based on the changes in density away from the coordinate. These expansions are referred to as generalized gradient approximations (GGA) [23] [24] [25] and have the following form:

${\displaystyle E_{\mathrm {XC} }^{\mathrm {GGA} }\left[n_{\uparrow },n_{\downarrow }\right]=\int \varepsilon _{\mathrm {XC} }\left(n_{\uparrow },n_{\downarrow },\nabla n_{\uparrow },\nabla n_{\downarrow }\right)n({\vec {r}})\,{\mathrm {d} }^{3}r.}$

Using the latter (GGA), very good results for molecular geometries and ground-state energies have been achieved.

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.

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 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, [12] 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, [26] 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 [27] extended the Hohenberg–Kohn theorem for varying magnetic fields using the density and the current density as fundamental variables.

## Applications

In general, density functional theory finds increasingly broad application in chemistry and materials science for the interpretation and prediction of complex system behavior at an atomic scale. Specifically, DFT computational methods are applied for synthesis-related systems and processing parameters. In such systems, experimental studies are often encumbered by inconsistent results and non-equilibrium conditions. Examples of contemporary DFT applications include studying the effects of dopants on phase transformation behavior in oxides, magnetic behavior in dilute magnetic semiconductor materials, and the study of magnetic and electronic behavior in ferroelectrics and dilute magnetic semiconductors. [1] [28] It has also been shown that DFT gives good results in the prediction of sensitivity of some nanostructures to environmental pollutants like sulfur dioxide [29] or acrolein [30] as well as prediction of mechanical properties. [31]

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 which are 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, one of the most widely used functionals is the revised Perdew–Burke–Ernzerhof exchange model (a direct generalized gradient parameterization 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. 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). 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 h3 of volume. [32] For each element of coordinate space volume d3r we can fill out a sphere of momentum space up to the Fermi momentum pf [33]

${\displaystyle {\tfrac {4}{3}}\pi p_{\mathrm {f} }^{3}({\vec {r}}).}$

Equating the number of electrons in coordinate space to that in phase space gives:

${\displaystyle n({\vec {r}})={\frac {8\pi }{3h^{3}}}p_{\mathrm {f} }^{3}({\vec {r}}).}$

Solving for pf and substituting into the classical kinetic energy formula then leads directly to a kinetic energy represented as a functional of the electron density:

{\displaystyle {\begin{aligned}t_{\mathrm {TF} }[n]&={\frac {p^{2}}{2m_{\mathrm {e} }}}\propto {\frac {\left(n^{\frac {1}{3}}\right)^{2}}{2m_{\mathrm {e} }}}\propto n^{\frac {2}{3}}({\vec {r}}),\\[6pt]T_{\mathrm {TF} }[n]&=C_{\mathrm {F} }\int n({\vec {r}})n^{\frac {2}{3}}({\vec {r}})\,\mathrm {d} ^{3}r=C_{\mathrm {F} }\int n^{\frac {5}{3}}({\vec {r}})\,\mathrm {d} ^{3}r,\end{aligned}}}

where

${\displaystyle C_{\mathrm {F} }={\frac {3h^{2}}{10m_{\mathrm {e} }}}\left({\frac {3}{8\pi }}\right)^{\frac {2}{3}}.}$

As such, they were able to calculate the energy of an atom using this kinetic energy functional combined with the classical expressions for the nucleus–electron and electron–electron 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: [34] [35]

${\displaystyle T_{\mathrm {W} }[n]={\frac {\hbar ^{2}}{8m}}\int {\frac {|\nabla n({\vec {r}})|^{2}}{n({\vec {r}})}}\,\mathrm {d} ^{3}r.}$

## Hohenberg–Kohn theorems

The Hohenberg–Kohn theorems relate to any system consisting of electrons moving under the influence of an external potential.

Theorem 1. The external potential (and hence the total energy), is a unique functional of the electron density.
If two systems of electrons, one trapped in a potential v1(r) and the other in v2(r), have the same ground-state density n(r) then v1(r) − v2(r) is necessarily a constant.
Corollary: the ground state density uniquely determines the potential and thus all properties of the system, including the many-body wavefunction. In particular, the H–K functional, defined as F[n] = T[n] + U[n], is a universal functional of the density (not depending explicitly on the external potential).
Theorem 2. The functional that delivers the ground state energy of the system gives the lowest energy if and only if the input density is the true ground state density.
For any positive integer N and potential v(r), a density functional F[n] exists such that
${\displaystyle E_{(v,N)}[n]=F[n]+\int v({\vec {r}})n({\vec {r}})\,\mathrm {d} ^{3}r}$
obtains its minimal value at the ground-state density of N electrons in the potential v(r). The minimal value of E(v,N)[n] is then the ground state energy of this system.

## Pseudo-potentials

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 screen the nucleus, thus forming with the nucleus an almost inert core. Binding properties are almost completely due to the valence electrons, especially in metals and semiconductors. 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 1950s.

### Ab initio pseudo-potentials

A crucial step toward more realistic pseudo-potentials was given by Topp and Hopfield [36] 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-wavefunctions to coincide with the true valence wave functions beyond a certain distance rl. The pseudo-wavefunctions are also forced to have the same norm as the true valence wavefunctions and can be written as

{\displaystyle {\begin{aligned}R_{l}^{\mathrm {PP} }(r)&=R_{nl}^{\mathrm {AE} }(r),\\[6pt]\int _{0}^{rl}\left|R_{l}^{\mathrm {PP} }(r)\right|^{2}r^{2}\,\mathrm {d} r&=\int _{0}^{rl}\left|R_{nl}^{\mathrm {AE} }(r)\right|^{2}r^{2}\,\mathrm {d} r,\end{aligned}}}

where Rl(r) is the radial part of the wavefunction with angular momentum l; and PP and AE denote, respectively, the pseudo-wavefunction and the true (all-electron) 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 dependent on l.

## Electron smearing

The electrons of a system will occupy the lowest Kohn–Sham eigenstates up to a given energy level according to the Aufbau principle. This corresponds to the steplike Fermi–Dirac distribution at absolute zero. If there are several degenerate or close to degenerate eigenstates at the Fermi level, it is possible to get convergence problems, since very small perturbations may change the electron occupation. One way of damping these oscillations is to smear the electrons, i.e. allowing fractional occupancies. [37] One approach of doing this is to assign a finite temperature to the electron Fermi–Dirac distribution. Other ways is to assign a cumulative Gaussian distribution of the electrons or using a Methfessel–Paxton method. [38] [39]

## Software supporting DFT

DFT is supported by many quantum chemistry and solid state physics software packages, often along with other methods.

## Related Research Articles

A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen constitutes about 75% of the baryonic mass of the universe.

A Bloch wave, named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal. A wavefunction ψ is a Bloch wave if it has the form:

In solid-state physics, the electronic band structure of a solid describes the range of energies an electron within the solid may have and ranges of energy that it may not have.

In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

In physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering. The pseudopotential approximation was first introduced by Hans Hellmann in 1934.

Koopmans' theorem states that in closed-shell Hartree–Fock theory (HF), the first ionization energy of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO). This theorem is named after Tjalling Koopmans, who published this result in 1934.

Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space. Many approaches can yield local approximations to the XC energy. However, overwhelmingly successful local approximations are those that have been derived from the homogeneous electron gas (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, which are then applied to realistic systems.

In physics and quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the one electron Schrödinger equation of a fictitious system of non-interacting particles that generate the same density as any given system of interacting particles. The Kohn–Sham equation is defined by a local effective (fictitious) external potential in which the non-interacting particles move, typically denoted as vs(r) or veff(r), called the Kohn–Sham potential. As the particles in the Kohn–Sham system are non-interacting fermions, the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals that are the lowest energy solutions to

Car–Parrinello molecular dynamics or CPMD refers to either a method used in molecular dynamics or the computational chemistry software package used to implement this method.

Hybrid functionals are a class of approximations to the exchange–correlation energy functional in density functional theory (DFT) that incorporate a portion of exact exchange from Hartree–Fock theory with the rest of the exchange–correlation energy from other sources. The exact exchange energy functional is expressed in terms of the Kohn–Sham orbitals rather than the density, so is termed an implicit density functional. One of the most commonly used versions is B3LYP, which stands for "Becke, 3-parameter, Lee–Yang–Parr".

In density functional theory (DFT), the Harris energy functional is a non-self-consistent approximation to the Kohn-Sham density functional theory. It gives the energy of a combined system as a function of the electronic densities of the isolated parts. The energy of the Harris functional varies much less than the energy of the Kohn-Sham functional as the density moves away from the converged density.

In computational chemistry, spin contamination is the artificial mixing of different electronic spin-states. This can occur when an approximate orbital-based wave function is represented in an unrestricted form – that is, when the spatial parts of α and β spin-orbitals are permitted to differ. Approximate wave functions with a high degree of spin contamination are undesirable. In particular, they are not eigenfunctions of the total spin-squared operator, Ŝ2, but can formally be expanded in terms of pure spin states of higher multiplicities.

A helium atom is an atom of the chemical element helium. Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with either one or two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom.

In quantum chemistry, Brillouin's theorem, proposed by the French physicist Léon Brillouin in 1934, states that given a self-consistent optimized Hartree-Fock wavefunction , the matrix element of the Hamiltonian between the ground state and a single excited determinant must be zero.

In quantum mechanics, specifically time-dependent density functional theory, the Runge–Gross theorem shows that for a many-body system evolving from a given initial wavefunction, there exists a one-to-one mapping between the potential in which the system evolves and the density of the system. The potentials under which the theorem holds are defined up to an additive purely time-dependent function: such functions only change the phase of the wavefunction and leave the density invariant. Most often the RG theorem is applied to molecular systems where the electronic density, ρ(r,t) changes in response to an external scalar potential, v(r,t), such as a time-varying electric field.

In computational chemistry, orbital-free density functional theory is a quantum mechanical approach to electronic structure determination which is based on functionals of the electronic density. It is most closely related to the Thomas–Fermi model. Orbital-free density functional theory is, at present, less accurate than Kohn–Sham density functional theory models, but has the advantage of being fast, so that it can be applied to large systems.

The projector augmented wave method (PAW) is a technique used in ab initio electronic structure calculations. It is a generalization of the pseudopotential and linear augmented-plane-wave methods, and allows for density functional theory calculations to be performed with greater computational efficiency.

The Strictly-Correlated-Electrons (SCE) density functional theory approach, originally proposed by Michael Seidl [1], is a formulation of density functional theory, alternative to the widely used Kohn-Sham DFT, especially aimed at the study of strongly-correlated systems. The essential difference between the two approaches is the choice of the auxiliary system. In Kohn-Sham DFT this system is composed by non-interacting electrons, for which the kinetic energy can be calculated exactly and the interaction term has to be approximated. In SCE DFT, instead, the starting point is totally the opposite one: the auxiliary system has infinite electronic correlation and zero kinetic energy.

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