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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.
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.
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.
Introduction to Density Functional Theory and Exchange-Correlation Energy Functionals R. Jones Institute for Solid State Research Forschungszentrum Ju¨lich 52425 Ju¨lich, Germany E-mail: [email protected] Density functional calculations of cohesive and structural properties of molecules and solids. The exchange correlation functional is virtually “the core of DFT” because it eventually affects the predictive strength and efficiency of the theory. This work does not claim to give a full description of the evolution of the exchange correlation functionals.
- 1Overview of method
- 9Pseudo-potentials
Overview of method[edit]
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]
Origins[edit]
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 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.
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[edit]
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 wavefunctionΨ(r1→,…,rN→) satisfying the many-electron time-independent Schrödinger equation
where, for the N-electron system, Ĥ is the Hamiltonian, E is the total energy, T̂ is the kinetic energy, V̂ is the potential energy from the external field due to positively charged nuclei, and Û is the electron–electron interaction energy. The operators T̂ and Û are called universal operators as they are the same for any N-electron system, while V̂ 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 normalizedΨ is given by
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]
and consequently the ground-state expectation value of an observable Ô is also a functional of n0
In particular, the ground-state energy is a functional of n0
where the contribution of the external potential ⟨ Ψ[n0] | V̂ | Ψ[n0] ⟩ can be written explicitly in terms of the ground-state density n0
More generally, the contribution of the external potential ⟨ Ψ | V̂ | Ψ ⟩ can be written explicitly in terms of the density n,
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 V̂, one then has to minimize the functional
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,
where T̂ denotes the kinetic energy operator and V̂s 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,
which yields the orbitalsφi that reproduce the density n(r→) of the original many-body system
The effective single-particle potential can be written in more detail as
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
- 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.
- It is possible to extend the DFT idea to the case of the Green functionG 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.
- There is no one-to-one correspondence between one-body density matrixn(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)[edit]
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):
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:
To find out the eigenfunctions and corresponding energies, one solves the eigenfunction equation
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:
and analogously, the virial theorem applied to the eigenfunction equation with the square of the Hamiltonian[15] yields
- .
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 reasonablywide 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:
- ,
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 weshould allocate a linear part of functional increment when argument function is altered.
Deploying written above equation it is easy to find the following formula for functional derivative
- ,
where A and B stay for mc2∫ ne dτ and √m2c4+emc2∫Vnedτ respectively. And finally V(τ0) is a value of potential insome 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.
Apparently this equation could have solution only if A is equal to B. This last condition provides us with Lagrangeequation for functional F, which could be finally written down in the following form.
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 as well Kohn-Sham equation in this particular case. Looking back onto the definition of the functional F we clearly see that the functional produces energy ofthe system for appropriate density, because the first term amounts to zero for such density and thesecond one delivers the energy value.
Approximations (exchange–correlation functionals)[edit]
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] In physics the most widely used approximation is the local-density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:
The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electron spin:
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 εX ∝ n1⁄3. 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:
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[edit]
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[edit]
C60 with isosurface of ground-state electron density as calculated with DFT.
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[edit]
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]
Equating the number of electrons in coordinate space to that in phase space gives:
Solving for pf and substituting into the classicalkinetic 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 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]
Hohenberg–Kohn theorems[edit]
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
- 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.
- For any positive integer N and potential v(r→), a density functional F[n] exists such that
Pseudo-potentials[edit]
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[edit]
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
where Rl(r) is the radial part of the wavefunction with angular momentuml; 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[edit]
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[edit]
DFT is supported by many quantum chemistry and solid state physics software packages, often along with other methods.
See also[edit]
Lists[edit]
References[edit]
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- ^Langreth, David C; Mehl, M J (1983). 'Beyond the local-density approximation in calculations of ground-state electronic properties'. Physical Review B. 28 (4): 1809. Bibcode:1983PhRvB..28.1809L. doi:10.1103/physrevb.28.1809.
- ^Grayce, Christopher; Harris, Robert (1994). 'Magnetic-field density-functional theory'. Physical Review A. 50 (4): 3089–3095. Bibcode:1994PhRvA..50.3089G. doi:10.1103/PhysRevA.50.3089. PMID9911249.
- ^Viraht, Xiao-Yin (2012). 'Hohenberg–Kohn theorem including electron spin'. Physical Review A. 86 (4): 042502. Bibcode:2012PhRvA..86d2502P. doi:10.1103/physreva.86.042502.
- ^Segall, M. D.; Lindan, P. J. (2002). 'First-principles simulation: ideas, illustrations and the CASTEP code'. Journal of Physics: Condensed Matter. 14 (11): 2717. Bibcode:2002JPCM...14.2717S. CiteSeerX10.1.1.467.6857. doi:10.1088/0953-8984/14/11/301.
- ^Soleymanabadi, Hamed; Rastegar, Somayeh F. (2014-01-01). 'Theoretical investigation on the selective detection of SO2 molecule by AlN nanosheets'. Journal of Molecular Modeling. 20 (9): 2439. doi:10.1007/s00894-014-2439-6. PMID25201451.
- ^Soleymanabadi, Hamed; Rastegar, Somayeh F. (2013-01-01). 'DFT studies of acrolein molecule adsorption on pristine and Al-doped graphenes'. Journal of Molecular Modeling. 19 (9): 3733–3740. doi:10.1007/s00894-013-1898-5. PMID23793719.
- ^Music, D.; Geyer, R. W.; Schneider, J. M. (2016). 'Recent progress and new directions in density functional theory based design of hard coatings'. Surface & Coatings Technology. 286: 178–190. doi:10.1016/j.surfcoat.2015.12.021.
- ^(Parr & Yang 1989, p. 47)
- ^March, N. H. (1992). Electron Density Theory of Atoms and Molecules. Academic Press. p. 24. ISBN978-0-12-470525-8.
- ^Weizsäcker, C. F. v. (1935). 'Zur Theorie der Kernmassen' [On the theory of nuclear masses]. Zeitschrift für Physik. 96 (7–8): 431–458. Bibcode:1935ZPhy...96..431W. doi:10.1007/BF01337700.
- ^(Parr & Yang 1989, p. 127)
- ^Topp, William C.; Hopfield, John J. (1973-02-15). 'Chemically Motivated Pseudopotential for Sodium'. Physical Review B. 7 (4): 1295–1303. Bibcode:1973PhRvB...7.1295T. doi:10.1103/PhysRevB.7.1295.
- ^Michelini, M. C.; Pis Diez, R.; Jubert, A. H. (25 June 1998). 'A Density Functional Study of Small Nickel Clusters'. International Journal of Quantum Chemistry. 70 (4–5): 694. doi:10.1002/(SICI)1097-461X(1998)70:4/5<693::AID-QUA15>3.0.CO;2-3.
- ^'Finite temperature approaches – smearing methods'. VASP the GUIDE. Retrieved 21 October 2016.
- ^Tong, Lianheng. 'Methfessel–Paxton Approximation to Step Function'. Metal CONQUEST. Retrieved 21 October 2016.
Key papers[edit]
- Parr, R. G.; Yang, W. (1989). Density-Functional Theory of Atoms and Molecules. New York: Oxford University Press. ISBN978-0-19-504279-5.
- 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.; Kohn, W. (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). 'Self-Consistent 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). 'Density-functional 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 Colle–Salvetti correlation-energy 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). 'Time-dependent density functional theory: Past, present, and future'. The Journal of Chemical Physics. 123 (6): 062206. arXiv:cond-mat/0410362. Bibcode:2005JChPh.123f2206B. doi:10.1063/1.1904586. PMID16122292.
- Lejaeghere, K.; Bihlmayer, G.; Bjorkman, T.; Blaha, P.; Blugel, S.; Blum, V.; Caliste, D.; Castelli, I. E.; Clark, S. J.; Dal Corso, A.; de Gironcoli, S.; Deutsch, T.; Dewhurst, J. K.; Di Marco, I.; Draxl, C.; Du ak, M.; Eriksson, O.; Flores-Livas, J. A.; Garrity, K. F.; Genovese, L.; Giannozzi, P.; Giantomassi, M.; Goedecker, S.; Gonze, X.; Granas, O.; Gross, E. K. U.; Gulans, A.; Gygi, F.; Hamann, D. R.; Hasnip, P. J.; Holzwarth, N. A. W.; Iu an, D.; Jochym, D. B.; Jollet, F.; Jones, D.; Kresse, G.; Koepernik, K.; Kucukbenli, E.; Kvashnin, Y. O.; Locht, I. L. M.; Lubeck, S.; Marsman, M.; Marzari, N.; Nitzsche, U.; Nordstrom, L.; Ozaki, T.; Paulatto, L.; Pickard, C. J.; Poelmans, W.; Probert, M. I. J.; Refson, K.; Richter, M.; Rignanese, G.-M.; Saha, S.; Scheffler, M.; Schlipf, M.; Schwarz, K.; Sharma, S.; Tavazza, F.; Thunstrom, P.; Tkatchenko, A.; Torrent, M.; Vanderbilt, D.; van Setten, M. J.; Van Speybroeck, V.; Wills, J. M.; Yates, J. R.; Zhang, G.-X.; Cottenier, S. (2016). 'Reproducibility in density functional theory calculations of solids'. Science. 351 (6280): aad3000. Bibcode:2016Sci...351.....L. doi:10.1126/science.aad3000. ISSN0036-8075. PMID27013736.
External links[edit]
- Walter Kohn, Nobel Laureate Freeview video interview with Walter on his work developing density functional theory by the Vega Science Trust.
- Capelle, Klaus (2002). 'A bird's-eye view of density-functional theory'. arXiv:cond-mat/0211443.
- Walter Kohn, Nobel Lecture
- Argaman, Nathan; Makov, Guy (2000). 'Density Functional Theory -- an introduction'. American Journal of Physics. 68 (2000): 69–79. arXiv:physics/9806013. Bibcode:2000AmJPh..68...69A. doi:10.1119/1.19375.
- Burke, Kieron. 'The ABC of DFT'(PDF).
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Density_functional_theory&oldid=901225082'
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 (and not, for example, derivatives of the density or the Kohn–Sham orbitals). 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 (molecules and solids).
In general, for a spin-unpolarized system, a local-density approximation for the exchange-correlation energy is written as
where ρ is the electronic density and εxc is the exchange-correlation energy per particle of a homogeneous electron gas of charge density ρ. The exchange-correlation energy is decomposed into exchange and correlation terms linearly,
so that separate expressions for Ex and Ec are sought. The exchange term takes on a simple analytic form for the HEG. Only limiting expressions for the correlation density are known exactly, leading to numerous different approximations for εc.
Local-density approximations are important in the construction of more sophisticated approximations to the exchange-correlation energy, such as generalized gradient approximations or hybrid functionals, as a desirable property of any approximate exchange-correlation functional is that it reproduce the exact results of the HEG for non-varying densities. As such, LDA's are often an explicit component of such functionals.
Applications[edit]
Local density approximations, as with Generalised Gradient Approximations (GGA) are employed extensively by solid state physicists in ab-initio DFT studies to interpret electronic and magnetic interactions in semiconductor materials including semiconducting oxides and spintronics. The importance of these computational studies stems from the system complexities which bring about high sensitivity to synthesis parameters necessitating first-principles based analysis. The prediction of Fermi level and band structure in doped semiconducting oxides is often carried out using LDA incorporated into simulation packages such as CASTEP and DMol3.[1] However an underestimation in Band gap values often associated with LDA and GGA approximations may lead to false predictions of impurity mediated conductivity and/or carrier mediated magnetism in such systems.[2]
Homogeneous electron gas[edit]
Approximation for εxc depending only upon the density can be developed in numerous ways. The most successful approach is based on the homogeneous electron gas. This is constructed by placing N interacting electrons in to a volume, V, with a positive background charge keeping the system neutral. N and V are then taken to infinity in the manner that keeps the density (ρ = N / V) finite. This is a useful approximation as the total energy consists of contributions only from the kinetic energy and exchange-correlation energy, and that the wavefunction is expressible in terms of planewaves. In particular, for a constant density ρ, the exchange energy density is proportional to ρ⅓.
Exchange functional[edit]
The exchange-energy density of a HEG is known analytically. The LDA for exchange employs this expression under the approximation that the exchange-energy in a system where the density is not homogeneous, is obtained by applying the HEG results pointwise, yielding the expression[3][4]
Correlation functional[edit]
Analytic expressions for the correlation energy of the HEG are available in the high- and low-density limits corresponding to infinitely-weak and infinitely-strong correlation. For a HEG with density ρ, the high-density limit of the correlation energy density is[3]
and the low limit
where the Wigner-Seitz parameter is dimensionless.[5] It is defined as the radius of a sphere which encompasses exactly one electron, divided by the Bohr radius. The Wigner-Seitz parameter is related to the density as
An analytical expression for the full range of densities has been proposed based on the many-body perturbation theory. The calculated correlation energies are in agreement with the results from quantum Monte Carlo simulation to within 2 milli-Hartree.
- The Chachiyo correlation functional
- [6]
The parameters and are not from empirical fitting to the Monte Carlo data, but from the theoretical constraint that the functional approaches high-density limit. The Chachiyo's formula is more accurate than the standard VWN fit function.[7] In the atomic unit, . The closed-form expression for does exist; but it is more convenient to use the numerical value: . Here, has been evaluated exactly using a closed-form integral and a zeta function (Eq. 21, G.Hoffman 1992).[8] Keeping the same functional form,[9] the parameter has also been fitted to the Monte Carlo simulation, providing a better agreement. Also in this case, the must either be in the atomic unit or be divided by the Bohr radius, making it a dimensionless parameter.[5]
As such, the Chachiyo formula is a simple (also accurate) first-principle correlation functional for DFT (uniform electron density). Tests on phonon dispersion curves [10] yield sufficient accuracy compared to the experimental data. Its simplicity is also suitable for introductory density functional theory courses.[11][12]
Comparison between several LDA correlation energy functionals and the quantum Monte Carlo simulation
Accurate quantum Monte Carlo simulations for the energy of the HEG have been performed for several intermediate values of the density, in turn providing accurate values of the correlation energy density.[13] The most popular LDA's to the correlation energy density interpolate these accurate values obtained from simulation while reproducing the exactly known limiting behavior. Various approaches, using different analytic forms for εc, have generated several LDA's for the correlation functional, including
- Vosko-Wilk-Nusair (VWN) [14]
- Perdew-Zunger (PZ81) [15]
- Cole-Perdew (CP) [16]
- Perdew-Wang (PW92) [17]
Predating these, and even the formal foundations of DFT itself, is the Wigner correlation functional obtained perturbatively from the HEG model.[18]
Spin polarization[edit]
The extension of density functionals to spin-polarized systems is straightforward for exchange, where the exact spin-scaling is known, but for correlation further approximations must be employed. A spin polarized system in DFT employs two spin-densities, ρα and ρβ with ρ = ρα + ρβ, and the form of the local-spin-density approximation (LSDA) is
For the exchange energy, the exact result (not just for local density approximations) is known in terms of the spin-unpolarized functional:[19]
The spin-dependence of the correlation energy density is approached by introducing the relative spin-polarization:
corresponds to the paramagnetic spin-unpolarized situation with equal and spin densities whereas corresponds to the ferromagnetic situation where one spin density vanishes. The spin correlation energy density for a given values of the total density and relative polarization, εc(ρ,ς), is constructed so to interpolate the extreme values. Several forms have been developed in conjunction with LDA correlation functionals.[14][20]
Illustrative calculations[edit]
LDA calculations are in reasonable agreement with experimental values.
LSD | LDA | HF | Exp. | |
---|---|---|---|---|
H | 13.4 | 12.0 | 13.6 | |
He | 24.5 | 26.4 | 24.6 | |
Li | 5.7 | 5.4 | 5.3 | 5.4 |
Be | 9.1 | 8.0 | 9.3 | |
B | 8.8 | 7.9 | 8.3 | |
C | 12.1 | 10.8 | 11.3 | |
N | 15.3 | 14.0 | 14.5 | |
O | 14.2 | 16.5 | 11.9 | 13.6 |
F | 18.4 | 16.2 | 17.4 | |
Ne | 22.6 | 22.5 | 19.8 | 21.6 |
Exp. | LSD | Error | |
---|---|---|---|
H2 | 0.74 | 0.77 | 0.03 |
Li2 | 2.67 | 2.71 | 0.04 |
B2 | 1.59 | 1.60 | 0.02 |
C2 | 1.24 | 1.24 | 0.00 |
N2 | 1.10 | 1.10 | 0.00 |
O2 | 1.21 | 1.20 | 0.01 |
F2 | 1.42 | 1.38 | 0.04 |
Na2 | 3.08 | 3.00 | 0.08 |
Al2 | 2.47 | 2.46 | 0.01 |
Si2 | 2.24 | 2.27 | 0.03 |
P2 | 1.89 | 1.89 | 0.01 |
S2 | 1.89 | 1.89 | 0.00 |
Cl2 | 1.99 | 1.98 | 0.01 |
Average | 0.02 |
Exchange-correlation potential[edit]
The exchange-correlation potential corresponding to the exchange-correlation energy for a local density approximation is given by[3]
In finite systems, the LDA potential decays asymptotically with an exponential form. This is in error; the true exchange-correlation potential decays much slower in a Coulombic manner. The artificially rapid decay manifests itself in the number of Kohn–Sham orbitals the potential can bind (that is, how many orbitals have energy less than zero). The LDA potential can not support a Rydberg series and those states it does bind are too high in energy. This results in the HOMO energy being too high in energy, so that any predictions for the ionization potential based on Koopmans' theorem are poor. Further, the LDA provides a poor description of electron-rich species such as anions where it is often unable to bind an additional electron, erroneously predicating species to be unstable.[15][21]
References[edit]
- ^Segall, M.D.; Lindan, P.J (2002). 'First-principles simulation: ideas, illustrations and the CASTEP code'. Journal of Physics: Condensed Matter. 14 (11): 2717. Bibcode:2002JPCM...14.2717S. doi:10.1088/0953-8984/14/11/301.
- ^Assadi, M.H.N; et al. (2013). 'Theoretical study on copper's energetics and magnetism in TiO2 polymorphs'. Journal of Applied Physics. 113 (23): 233913–233913–5. arXiv:1304.1854. Bibcode:2013JAP...113w3913A. doi:10.1063/1.4811539.
- ^ abcdeParr, Robert G; Yang, Weitao (1994). Density-Functional Theory of Atoms and Molecules. Oxford: Oxford University Press. ISBN978-0-19-509276-9.
- ^Dirac, P. A. M. (1930). 'Note on exchange phenomena in the Thomas-Fermi atom'. Proc. Camb. Phil. Soc. 26 (3): 376–385. Bibcode:1930PCPS...26..376D. doi:10.1017/S0305004100016108.
- ^ abMurray Gell-Mann and Keith A. Brueckner (1957). 'Correlation Energy of an Electron Gas at High Density'. Phys. Rev. 106 (2): 364–368. Bibcode:1957PhRv..106..364G. doi:10.1103/PhysRev.106.364.
- ^Teepanis Chachiyo (2016). 'Communication: Simple and accurate uniform electron gas correlation energy for the full range of densities'. J. Chem. Phys. 145 (2): 021101. Bibcode:2016JChPh.145b1101C. doi:10.1063/1.4958669.
- ^Richard J. Fitzgerald (2016). 'A simpler ingredient for a complex calculation'. Physics Today. 69 (9): 20. Bibcode:2016PhT....69i..20F. doi:10.1063/PT.3.3288.
- ^Gary G. Hoffman (1992). 'Correlation energy of a spin-polarized electron gas at high density'. Phys. Rev. B. 45 (15): 8730–8733. Bibcode:1992PhRvB..45.8730H. doi:10.1103/PhysRevB.45.8730.
- ^Valentin V. Karasiev (2016). 'Comment on 'Communication: Simple and accurate uniform electron gas correlation energy for the full range of densities' [J. Chem. Phys. 145, 021101 (2016)]'. J. Chem. Phys. 145 (2): 157101. arXiv:1609.05408. Bibcode:2016JChPh.145o7101K. doi:10.1063/1.4964758.
- ^Ukrit Jitropas and Chung-Hao Hsu (2017). 'Study of the first-principles correlation functional in the calculation of silicon phonon dispersion curves'. Japanese Journal of Applied Physics. 56 (7): 070313. Bibcode:2017JaJAP..56g0313J. doi:10.7567/JJAP.56.070313.
- ^Boudreau, Joseph; Swanson, Eric (2017). Applied Computational Physics. Oxford University Press. p. 829. ISBN978-0-198-70863-6.
- ^Roman, Adrian (November 26, 2017). 'DFT for a Quantum Dot'. Computational Physics Blog. Retrieved December 7, 2017.
- ^D. M. Ceperley and B. J. Alder (1980). 'Ground State of the Electron Gas by a Stochastic Method'. Phys. Rev. Lett. 45 (7): 566–569. Bibcode:1980PhRvL..45..566C. doi:10.1103/PhysRevLett.45.566.
- ^ abS. H. Vosko, L. Wilk and M. Nusair (1980). 'Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis'. Can. J. Phys. 58 (8): 1200–1211. Bibcode:1980CaJPh..58.1200V. doi:10.1139/p80-159.
- ^ abJ. P. Perdew and A. Zunger (1981). 'Self-interaction correction to density-functional approximations for many-electron systems'. Phys. Rev. B. 23 (10): 5048–5079. Bibcode:1981PhRvB..23.5048P. doi:10.1103/PhysRevB.23.5048.
- ^L. A. Cole and J. P. Perdew (1982). 'Calculated electron affinities of the elements'. Phys. Rev. A. 25 (3): 1265–1271. Bibcode:1982PhRvA..25.1265C. doi:10.1103/PhysRevA.25.1265.
- ^John P. Perdew and Yue Wang (1992). 'Accurate and simple analytic representation of the electron-gas correlation energy'. Phys. Rev. B. 45 (23): 13244–13249. Bibcode:1992PhRvB..4513244P. doi:10.1103/PhysRevB.45.13244.
- ^E. Wigner (1934). 'On the Interaction of Electrons in Metals'. Phys. Rev. 46 (11): 1002–1011. Bibcode:1934PhRv...46.1002W. doi:10.1103/PhysRev.46.1002.
- ^Oliver, G. L.; Perdew, J. P. (1979). 'Spin-density gradient expansion for the kinetic energy'. Phys. Rev. A. 20 (2): 397–403. Bibcode:1979PhRvA..20..397O. doi:10.1103/PhysRevA.20.397.
- ^von Barth, U.; Hedin, L. (1972). 'A local exchange-correlation potential for the spin polarized case'. J. Phys. C: Solid State Phys. 5 (13): 1629–1642. Bibcode:1972JPhC....5.1629V. doi:10.1088/0022-3719/5/13/012.
- ^Fiolhais, Carlos; Nogueira, Fernando; Marques Miguel (2003). A Primer in Density Functional Theory. Springer. p. 60. ISBN978-3-540-03083-6.
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Local-density_approximation&oldid=901213475'