 Density functional theory

Density functional theory (DFT) is a quantum mechanical modelling method 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 solidstate 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. In many cases the results of DFT calculations for solidstate systems agree quite satisfactorily with experimental data. Computational costs are relatively low when compared to traditional methods, such as Hartree–Fock theory and its descendants based on the complex manyelectron wavefunction.
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)^{[1]} or where dispersion competes significantly with other effects (e.g. in biomolecules).^{[2]} The development of new DFT methods designed to overcome this problem, by alterations to the functional^{[3]} or by the inclusion of additive terms,^{[4]}^{[5]}^{[6]} is a current research topic.
Contents
 1 Overview of method
 2 Derivation and formalism
 3 Approximations (Exchangecorrelation functionals)
 4 Generalizations to include magnetic fields
 5 Applications
 6 Thomas–Fermi model
 7 HohenbergKohn Theorems
 8 Pseudopotentials
 9 Software supporting DFT
 10 See also
 11 References
 12 Bibliography
 13 Key papers
 14 External links
Overview of method
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).^{[7]} 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.^{[8]}^{[9]}
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 therein. 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).
Derivation and formalism
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 timeindependent Schrödinger equation
where, for the electron system, is the Hamiltonian, is the total energy, is the kinetic energy, is the potential energy from the external field due to positively charged nuclei, and is the electronelectron interaction energy. The operators and are called universal operators as they are the same for any electron system, while is system dependent. This complicated manyparticle equation is not separable into simpler singleparticle equations because of 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 ,^{[7]}
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.^{[10]} First, one considers an energy functional that doesn't explicitly have an electronelectron interaction energy term,
where denotes the kinetic energy operator and is an external effective potential in which the particles are moving, so that .
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. A noniterative approximate formulation called Harris functional DFT is an alternative approach to this.
Approximations (Exchangecorrelation functionals)
The major problem with DFT is that the exact functionals for exchange and correlation are not known except for the free electron gas. However, approximations exist which permit the calculation of certain physical quantities quite accurately. In physics the most widely used approximation is the localdensity approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:
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 jellium.^{[11]}
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 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.
Generalizations to include magnetic fields
The DFT formalism described above breaks down, to various degrees, in the presence of a vector potential, i.e. a magnetic field. In such a situation, the onetoone mapping between the groundstate electron density and wavefunction is lost. Generalizations to include the effects of magnetic fields have led to two different theories: current density functional theory (CDFT) and magnetic field 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,^{[9]} 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,^{[12]} 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.
Applications
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.
Thomas–Fermi model
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.^{[13]} 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} ^{[14]}
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:
 where
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:^{[15]}^{[16]}
HohenbergKohn Theorems
1.If two systems of electrons, one trapped in a potential and the other in , have the same groundstate density then necessarily .
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 the density functional obtains its minimal value at the groundstate density of N electrons in the potential . The minimal value of E_{(v,N)}[n] is then the ground state energy of this system.
Pseudopotentials
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, 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 pseudopotentials introduce in calculations and remained forgotten until the late 50’s
Abinitio Pseudopotentials
A crucial step toward more realistic pseudopotentials was given by Topp and Hopfield and more recently Cronin, who suggested that the pseudopotential should be adjusted such that they describe the valence charge density accurately. Based on that idea, modern pseudopotentials 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:
Where R_{l}(r). is the radial part of the wavefunction with angular momentum l_{.}, and pp_{.} and AE_{.} denote, respectively, the pseudo wavefunction and the true (allelectron) wavefunction. The index n in the true wavefunctions denotes the valence level. The distance beyond which the true and the pseudo wavefunctions are equal, rl_{.}, is also l_{.}dependent.
Software supporting DFT
DFT is supported by many Quantum chemistry and solid state physics codes, often along with other methods.
See also
 Harris functional
 Basis set (chemistry)
 Gas in a box
 Helium atom
 Kohn–Sham equations
 Local density approximation
 Molecule
 Molecular design software
 Molecular modelling
 Quantum chemistry
 List of quantum chemistry and solid state physics software
 List of software for molecular mechanics modeling
 Thomas–Fermi model
 Timedependent density functional theory
 Dynamical Mean Field Theory
References
 ^ Van Mourik, Tanja; Gdanitz, Robert J. (2002). "A critical note on density functional theory studies on raregas dimers". Journal of Chemical Physics 116 (22): 9620–9623. Bibcode 2002JChPh.116.9620V. doi:10.1063/1.1476010.
 ^ Vondrášek, Jiří; Bendová, Lada; Klusák, Vojtěch; and Hobza, Pavel (2005). "Unexpectedly strong energy stabilization inside the hydrophobic core of small protein rubredoxin mediated by aromatic residues: correlated ab initio quantum chemical calculations". Journal of the American Chemical Society 127 (8): 2615–2619. doi:10.1021/ja044607h. PMID 15725017.
 ^ Grimme, Stefan (2006). "Semiempirical hybrid density functional with perturbative secondorder correlation". Journal of Chemical Physics 124 (3): 034108. Bibcode 2006JChPh.124c4108G. doi:10.1063/1.2148954. PMID 16438568.
 ^ Zimmerli, Urs; Parrinello, Michele; and Koumoutsakos, Petros (2004). "Dispersion corrections to density functionals for water aromatic interactions". Journal of Chemical Physics 120 (6): 2693–2699. Bibcode 2004JChPh.120.2693Z. doi:10.1063/1.1637034. PMID 15268413.
 ^ Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". Journal of Computational Chemistry 25 (12): 1463–1473. doi:10.1002/jcc.20078. PMID 15224390.
 ^ Von Lilienfeld, O. Anatole; Tavernelli, Ivano; Rothlisberger, Ursula; and Sebastiani, Daniel (2004). "Optimization of effective atom centered potentials for London dispersion forces in density functional theory". Physical Review Letters 93 (15): 153004. Bibcode 2004PhRvL..93o3004V. doi:10.1103/PhysRevLett.93.153004. PMID 15524874.
 ^ ^{a} ^{b} Hohenberg, Pierre; Walter Kohn (1964). "Inhomogeneous electron gas". Physical Review 136 (3B): B864–B871. Bibcode 1964PhRv..136..864H. doi:10.1103/PhysRev.136.B864.
 ^ Levy, Mel (1979). "Universal variational functionals of electron densities, firstorder density matrices, and natural spinorbitals and solution of the vrepresentability problem". Proceedings of the National Academy of Sciences (United States National Academy of Sciences) 76 (12): 6062–6065. Bibcode 1979PNAS...76.6062L. doi:10.1073/pnas.76.12.6062.
 ^ ^{a} ^{b} Vignale, G.; Mark Rasolt (1987). "Densityfunctional theory in strong magnetic fields". Physical Review Letters (American Physical Society) 59 (20): 2360–2363. Bibcode 1987PhRvL..59.2360V. doi:10.1103/PhysRevLett.59.2360. PMID 10035523.
 ^ Kohn, W.; Sham, L. J. (1965). "Selfconsistent equations including exchange and correlation effects". Physical Review 140 (4A): A1133–A1138. Bibcode 1965PhRv..140.1133K. doi:10.1103/PhysRev.140.A1133.
 ^ John P. Perdew, Adrienn Ruzsinszky, Jianmin Tao, Viktor N. Staroverov, Gustavo Scuseria and Gábor I. Csonka (2005). "Prescriptions for the design and selection of density functional approximations: More constraint satisfaction with fewer fits". Journal of Chemical Physics 123 (6): 062201. Bibcode 2005JChPh.123f2201P. doi:10.1063/1.1904565. PMID 16122287.
 ^ Grayce, Christopher; Robert Harris (1994). "Magneticfield densityfunctional theory". Physical Review A 50 (4): 3089–3095. Bibcode 1994PhRvA..50.3089G. doi:10.1103/PhysRevA.50.3089. PMID 9911249.
 ^ (Parr & Yang 1989, p. 47)
 ^ March, N. H. (1992). Electron Density Theory of Atoms and Molecules. Academic Press. p. 24. ISBN 0124705251.
 ^ Weizsäcker, C. F. v. (1935). "Zur Theorie der Kernmassen". Zeitschrift für Physik 96 (7–8): 431–58. Bibcode 1935ZPhy...96..431W. doi:10.1007/BF01337700.
 ^ (Parr & Yang 1989, p. 127)
Bibliography
 Parr, R. G.; Yang, W. (1989). DensityFunctional Theory of Atoms and Molecules. New York: Oxford University Press. ISBN 0195042794. http://books.google.com/?id=mGOpScSIwU4C&printsec=frontcover&dq=DensityFunctional+Theory+of+Atoms+and+Molecules&cd=1#v=onepage&q.. ISBN 0195092767 (paperback).
Key papers
 Thomas, L. H. (1927). "The calculation of atomic fields". Proc. Camb. Phil. Soc 23 (5): 542–548. Bibcode 1927PCPS...23..542T. doi:10.1017/S0305004100011683.
 Hohenberg, P. (1964). "Inhomogeneous Electron Gas". Physical Review 136 (3B): B864. Bibcode 1964PhRv..136..864H. doi:10.1103/PhysRev.136.B864.
 Kohn, W.; Sham, L. J. (1965). "SelfConsistent Equations Including Exchange and Correlation Effects". Physical Review 140 (4A): A1133. Bibcode 1965PhRv..140.1133K. doi:10.1103/PhysRev.140.A1133.
 Becke, Axel D. (1993). "Densityfunctional thermochemistry. III. The role of exact exchange". The Journal of Chemical Physics 98 (7): 5648. Bibcode 1993JChPh..98.5648B. doi:10.1063/1.464913.
 Lee, Chengteh; Yang, Weitao; Parr, Robert G. (1988). "Development of the ColleSalvetti correlationenergy formula into a functional of the electron density". Physical Review B 37 (2): 785. Bibcode 1988PhRvB..37..785L. doi:10.1103/PhysRevB.37.785.
 Burke, Kieron; Werschnik, Jan; Gross, E. K. U. (2005). "Timedependent density functional theory: Past, present, and future". The Journal of Chemical Physics 123 (6): 062206. arXiv:condmat/0410362. Bibcode 2005JChPh.123f2206B. doi:10.1063/1.1904586.
External links
 Walter Kohn, Nobel Laureate Freeview video interview with Walter on his work developing density functional theory by the Vega Science Trust.
 Klaus Capelle, A bird'seye view of densityfunctional theory
 Walter Kohn, Nobel Lecture
 Density functional theory on arxiv.org
 FreeScience Library > Density Functional Theory
 Density Functional Theory – an introduction
 Electron Density Functional Theory – Lecture Notes
 Density Functional Theory through Legendre Transformationpdf
 Kieron Burke : Book On DFT : " THE ABC OF DFT " http://dft.uci.edu/materials/bookABCDFT/gamma/g1.pdf
Categories: Density functional theory
 Electronic structure methods
 Physics theorems
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