In solid-state physics, the **tight-binding model** (or **TB model**) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method (linear combination of atomic orbitals method) used in chemistry. Tight-binding models are applied to a wide variety of solids. The model gives good qualitative results in many cases and can be combined with other models that give better results where the tight-binding model fails. Though the tight-binding model is a one-electron model, the model also provides a basis for more advanced calculations like the calculation of surface states and application to various kinds of many-body problem and quasiparticle calculations.

- Introduction
- Historical background
- Mathematical formulation
- Translational symmetry and normalization
- The tight binding Hamiltonian
- The tight binding matrix elements
- Evaluation of the matrix elements
- Connection to Wannier functions
- Second quantization
- Example: one-dimensional s-band
- Table of interatomic matrix elements
- See also
- References
- Further reading
- External links

The name "tight binding" of this electronic band structure model suggests that this quantum mechanical model describes the properties of tightly bound electrons in solids. The electrons in this model should be tightly bound to the atom to which they belong and they should have limited interaction with states and potentials on surrounding atoms of the solid. As a result, the wave function of the electron will be rather similar to the atomic orbital of the free atom to which it belongs. The energy of the electron will also be rather close to the ionization energy of the electron in the free atom or ion because the interaction with potentials and states on neighboring atoms is limited.

Though the mathematical formulation^{ [1] } of the one-particle tight-binding Hamiltonian may look complicated at first glance, the model is not complicated at all and can be understood intuitively quite easily. There are only three kinds of matrix elements that play a significant role in the theory. Two of those three kinds of elements should be close to zero and can often be neglected. The most important elements in the model are the interatomic matrix elements, which would simply be called the bond energies by a chemist.

In general there are a number of atomic energy levels and atomic orbitals involved in the model. This can lead to complicated band structures because the orbitals belong to different point-group representations. The reciprocal lattice and the Brillouin zone often belong to a different space group than the crystal of the solid. High-symmetry points in the Brillouin zone belong to different point-group representations. When simple systems like the lattices of elements or simple compounds are studied it is often not very difficult to calculate eigenstates in high-symmetry points analytically. So the tight-binding model can provide nice examples for those who want to learn more about group theory.

The tight-binding model has a long history and has been applied in many ways and with many different purposes and different outcomes. The model doesn't stand on its own. Parts of the model can be filled in or extended by other kinds of calculations and models like the nearly-free electron model. The model itself, or parts of it, can serve as the basis for other calculations.^{ [2] } In the study of conductive polymers, organic semiconductors and molecular electronics, for example, tight-binding-like models are applied in which the role of the atoms in the original concept is replaced by the molecular orbitals of conjugated systems and where the interatomic matrix elements are replaced by inter- or intramolecular hopping and tunneling parameters. These conductors nearly all have very anisotropic properties and sometimes are almost perfectly one-dimensional.

By 1928, the idea of a molecular orbital had been advanced by Robert Mulliken, who was influenced considerably by the work of Friedrich Hund. The LCAO method for approximating molecular orbitals was introduced in 1928 by B. N. Finklestein and G. E. Horowitz, while the LCAO method for solids was developed by Felix Bloch, as part of his doctoral dissertation in 1928, concurrently with and independent of the LCAO-MO approach. A much simpler interpolation scheme for approximating the electronic band structure, especially for the d-bands of transition metals, is the parameterized tight-binding method conceived in 1954 by John Clarke Slater and George Fred Koster,^{ [1] } sometimes referred to as the SK tight-binding method. With the SK tight-binding method, electronic band structure calculations on a solid need not be carried out with full rigor as in the original Bloch's theorem but, rather, first-principles calculations are carried out only at high-symmetry points and the band structure is interpolated over the remainder of the Brillouin zone between these points.

In this approach, interactions between different atomic sites are considered as perturbations. There exist several kinds of interactions we must consider. The crystal Hamiltonian is only approximately a sum of atomic Hamiltonians located at different sites and atomic wave functions overlap adjacent atomic sites in the crystal, and so are not accurate representations of the exact wave function. There are further explanations in the next section with some mathematical expressions.

In the recent research about strongly correlated material the tight binding approach is basic approximation because highly localized electrons like 3-d transition metal electrons sometimes display strongly correlated behaviors. In this case, the role of electron-electron interaction must be considered using the many-body physics description.

The tight-binding model is typically used for calculations of electronic band structure and band gaps in the static regime. However, in combination with other methods such as the random phase approximation (RPA) model, the dynamic response of systems may also be studied.

We introduce the atomic orbitals , which are eigenfunctions of the Hamiltonian of a single isolated atom. When the atom is placed in a crystal, this atomic wave function overlaps adjacent atomic sites, and so are not true eigenfunctions of the crystal Hamiltonian. The overlap is less when electrons are tightly bound, which is the source of the descriptor "tight-binding". Any corrections to the atomic potential required to obtain the true Hamiltonian of the system, are assumed small:

where locates an atomic site in the crystal lattice. A solution to the time-independent single electron Schrödinger equation is then approximated as a linear combination of atomic orbitals :

- ,

where refers to the m-th atomic energy level.

The Bloch theorem states that the wave function in a crystal can change under translation only by a phase factor:

where is the wave vector of the wave function. Consequently, the coefficients satisfy

By substituting , we find

- (where in RHS we have replaced the dummy index with )

or

Normalizing the wave function to unity:

so the normalization sets * as*

where *α _{m}* (

and

Using the tight binding form for the wave function, and assuming only the *m-th* atomic energy level is important for the *m-th* energy band, the Bloch energies are of the form

Here terms involving the atomic Hamiltonian at sites other than where it is centered are neglected. The energy then becomes

where *E*_{m} is the energy of the *m*-th atomic level, and , and are the tight binding matrix elements.

The element

- ,

is the atomic energy shift due to the potential on neighboring atoms. This term is relatively small in most cases. If it is large it means that potentials on neighboring atoms have a large influence on the energy of the central atom.

The next term

is the interatomic matrix element between the atomic orbitals *m* and *l* on adjacent atoms. It is also called the bond energy or two center integral and it is the **most important element** in the tight binding model.

The last terms

- ,

denote the overlap integrals between the atomic orbitals *m* and *l* on adjacent atoms.

As mentioned before the values of the -matrix elements are not so large in comparison with the ionization energy because the potentials of neighboring atoms on the central atom are limited. If is not relatively small it means that the potential of the neighboring atom on the central atom is not small either. In that case it is an indication that the tight binding model is not a very good model for the description of the band structure for some reason. The interatomic distances can be too small or the charges on the atoms or ions in the lattice is wrong for example.

The interatomic matrix elements can be calculated directly if the atomic wave functions and the potentials are known in detail. Most often this is not the case. There are numerous ways to get parameters for these matrix elements. Parameters can be obtained from chemical bond energy data. Energies and eigenstates on some high symmetry points in the Brillouin zone can be evaluated and values integrals in the matrix elements can be matched with band structure data from other sources.

The interatomic overlap matrix elements should be rather small or neglectable. If they are large it is again an indication that the tight binding model is of limited value for some purposes. Large overlap is an indication for too short interatomic distance for example. In metals and transition metals the broad s-band or sp-band can be fitted better to an existing band structure calculation by the introduction of next-nearest-neighbor matrix elements and overlap integrals but fits like that don't yield a very useful model for the electronic wave function of a metal. Broad bands in dense materials are better described by a nearly free electron model.

The tight binding model works particularly well in cases where the band width is small and the electrons are strongly localized, like in the case of d-bands and f-bands. The model also gives good results in the case of open crystal structures, like diamond or silicon, where the number of neighbors is small. The model can easily be combined with a nearly free electron model in a hybrid NFE-TB model.^{ [2] }

Bloch wave functions describe the electronic states in a periodic crystal lattice. Bloch functions can be represented as a Fourier series ^{ [4] }

where **R**_{n} denotes an atomic site in a periodic crystal lattice, * k* is the wave vector of the Bloch wave,

Using the Fourier transform analysis, a spatially localized wave function for the *m*-th energy band can be constructed from multiple Bloch waves:

These real space wave functions are called Wannier functions, and are fairly closely localized to the atomic site **R**_{n}. Of course, if we have exact Wannier functions, the exact Bloch functions can be derived using the inverse Fourier transform.

However it is not easy to calculate directly either Bloch functions or Wannier functions. An approximate approach is necessary in the calculation of electronic structures of solids. If we consider the extreme case of isolated atoms, the Wannier function would become an isolated atomic orbital. That limit suggests the choice of an atomic wave function as an approximate form for the Wannier function, the so-called tight binding approximation.

Modern explanations of electronic structure like t-J model and Hubbard model are based on tight binding model.^{ [5] } Tight binding can be understood by working under a second quantization formalism.

Using the atomic orbital as a basis state, the second quantization Hamiltonian operator in the tight binding framework can be written as:

- ,
- - creation and annihilation operators

- - spin polarization

- - hopping integral

- - nearest neighbor index

- - the hermitian conjugate of the other term(s)

Here, hopping integral corresponds to the transfer integral in tight binding model. Considering extreme cases of , it is impossible for an electron to hop into neighboring sites. This case is the isolated atomic system. If the hopping term is turned on () electrons can stay in both sites lowering their kinetic energy.

In the strongly correlated electron system, it is necessary to consider the electron-electron interaction. This term can be written in

This interaction Hamiltonian includes direct Coulomb interaction energy and exchange interaction energy between electrons. There are several novel physics induced from this electron-electron interaction energy, such as metal-insulator transitions (MIT), high-temperature superconductivity, and several quantum phase transitions.

Here the tight binding model is illustrated with a **s-band model** for a string of atoms with a single s-orbital in a straight line with spacing *a* and σ bonds between atomic sites.

To find approximate eigenstates of the Hamiltonian, we can use a linear combination of the atomic orbitals

where *N* = total number of sites and is a real parameter with . (This wave function is normalized to unity by the leading factor 1/√N provided overlap of atomic wave functions is ignored.) Assuming only nearest neighbor overlap, the only non-zero matrix elements of the Hamiltonian can be expressed as

The energy *E*_{i} is the ionization energy corresponding to the chosen atomic orbital and *U* is the energy shift of the orbital as a result of the potential of neighboring atoms. The elements, which are the Slater and Koster interatomic matrix elements, are the bond energies . In this one dimensional s-band model we only have -bonds between the s-orbitals with bond energy . The overlap between states on neighboring atoms is *S*. We can derive the energy of the state using the above equation:

where, for example,

and

Thus the energy of this state can be represented in the familiar form of the energy dispersion:

- .

- For the energy is and the state consists of a sum of all atomic orbitals. This state can be viewed as a chain of bonding orbitals.
- For the energy is and the state consists of a sum of atomic orbitals which are a factor out of phase. This state can be viewed as a chain of non-bonding orbitals.
- Finally for the energy is and the state consists of an alternating sum of atomic orbitals. This state can be viewed as a chain of anti-bonding orbitals.

This example is readily extended to three dimensions, for example, to a body-centered cubic or face-centered cubic lattice by introducing the nearest neighbor vector locations in place of simply *n a*.^{ [6] } Likewise, the method can be extended to multiple bands using multiple different atomic orbitals at each site. The general formulation above shows how these extensions can be accomplished.

In 1954 J.C. Slater and G.F. Koster published, mainly for the calculation of transition metal d-bands, a table of interatomic matrix elements^{ [1] }

which can also be derived from the cubic harmonic orbitals straightforwardly. The table expresses the matrix elements as functions of LCAO two-centre bond integrals between two cubic harmonic orbitals, *i* and *j*, on adjacent atoms. The bond integrals are for example the , and for sigma, pi and delta bonds (Notice that these integrals should also depend on the distance between the atoms, i.e. are a function of , even though it is not explicitly stated every time.).

The interatomic vector is expressed as

where *d* is the distance between the atoms and *l*, *m* and *n* are the direction cosines to the neighboring atom.

Not all interatomic matrix elements are listed explicitly. Matrix elements that are not listed in this table can be constructed by permutation of indices and cosine directions of other matrix elements in the table.

- Electronic band structure
- Nearly-free electron model
- Bloch waves
- Kronig-Penney model
- Fermi surface
- Wannier function
- Hubbard model
- t-J model
- Effective mass
- Anderson's rule
- Dynamical theory of diffraction
- Solid state physics
- Linear combination of atomic orbitals molecular orbital method (LCAO)
- Holstein–Herring method
- Peierls substitution

In physics, the **cross section** is a measure of probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha-particle will be deflected by a given angle during a collision with an atomic nucleus. Cross section is typically denoted *σ* (sigma) and is expressed in terms of the transverse area that the incident particle must hit in order for the given process to occur.

In quantum mechanics, the **uncertainty principle** is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which the values for certain pairs of physical quantities of a particle, such as position, *x*, and momentum, *p*, can be predicted from initial conditions. Such variable pairs are known as complementary variables or canonically conjugate variables, and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.

In probability theory, a **log-normal distribution** is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then *Y* = ln(*X*) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, *X* = exp(*Y*), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences as well as medicine, economics and other fields, e.g. for energies, concentrations, lengths, financial returns and other amounts.

**Noether's theorem** or **Noether's first theorem** states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat and F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

In statistical mechanics, the **Fokker–Planck equation** is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck, and is also known as the **Kolmogorov forward equation**, after Andrey Kolmogorov, who independently discovered the concept in 1931. When applied to particle position distributions, it is better known as the **Smoluchowski equation**, and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is known in statistical mechanics as the Liouville equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion.

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In mathematics, **spectral theory** is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

In atomic physics, **hyperfine structure** is defined by small shifts and splittings in the energy levels of atoms, molecules, and ions, due to interaction between the state of the nucleus and the state of the electron clouds.

In functional analysis, a **reproducing kernel Hilbert space** (**RKHS**) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions and in the RKHS are close in norm, i.e., is small, then and are also pointwise close, i.e., is small for all . The reverse does not need to be true.

In physics, the **Lamb shift**, named after Willis Lamb, is a difference in energy between two energy levels ^{2}*S*_{1/2} and ^{2}*P*_{1/2} of the hydrogen atom which was not predicted by the Dirac equation, according to which these states should have the same energy.

In nuclear physics, the **chiral model**, introduced by Feza Gürsey in 1960, is a phenomenological model describing effective interactions of mesons in the chiral limit, but without necessarily mentioning quarks at all. It is a nonlinear sigma model with the principal homogeneous space of the Lie group SU(*N*) as its target manifold, where *N* is the number of quark flavors. The Riemannian metric of the target manifold is given by a positive constant multiplied by the Killing form acting upon the Maurer-Cartan form of SU(*N*).

In quantum field theory, the **LSZ reduction formula** is a method to calculate S-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.

In quantum physics, the **spin–orbit interaction** is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.

In electromagnetism, **charge density** is the amount of electric charge per unit length, surface area, or volume. **Volume charge density** is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m^{−3}), at any point in a volume. **Surface charge density** (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m^{−2}), at any point on a surface charge distribution on a two dimensional surface. **Linear charge density** (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m^{−1}), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

The **Jaynes–Cummings model** is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.

In applied mathematics, **discontinuous Galerkin methods ** form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications. DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, fluid mechanics and plasma physics.

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.

A flavor of the **k·p** perturbation theory used for calculating the structure of multiple, degenerate electronic bands in bulk and quantum well semiconductors. The method is a generalization of the single band **k·p** theory.

In continuum mechanics, **objective stress rates** are time derivatives of stress that do not depend on the frame of reference. Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate. The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame-indifferent (objective). If the stress and strain measures are material quantities then objectivity is automatically satisfied. However, if the quantities are spatial, then the objectivity of the stress-rate is not guaranteed even if the strain-rate is objective.

In machine learning, the **kernel embedding of distributions** comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space on which a sensible kernel function may be defined. For example, various kernels have been proposed for learning from data which are: vectors in , discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song , Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in.

Wikimedia Commons has media related to . Dispersion relations of electrons |

- 1 2 3 J. C. Slater, G. F. Koster (1954). "Simplified LCAO method for the Periodic Potential Problem".
*Physical Review*.**94**(6): 1498–1524. Bibcode:1954PhRv...94.1498S. doi:10.1103/PhysRev.94.1498. - 1 2 Walter Ashley Harrison (1989).
*Electronic Structure and the Properties of Solids*. Dover Publications. ISBN 0-486-66021-4. - ↑ As an alternative to neglecting overlap, one may choose as a basis instead of atomic orbitals a set of orbitals based upon atomic orbitals but arranged to be orthogonal to orbitals on other atomic sites, the so-called Löwdin orbitals. See PY Yu & M Cardona (2005). "Tight-binding or LCAO approach to the band structure of semiconductors".
*Fundamentals of Semiconductors*(3 ed.). Springrer. p. 87. ISBN 3-540-25470-6. - ↑ Orfried Madelung,
*Introduction to Solid-State Theory*(Springer-Verlag, Berlin Heidelberg, 1978). - ↑ Alexander Altland and Ben Simons (2006). "Interaction effects in the tight-binding system".
*Condensed Matter Field Theory*. Cambridge University Press. pp. 58*ff*. ISBN 978-0-521-84508-3. - ↑ Sir Nevill F Mott & H Jones (1958). "II §4 Motion of electrons in a periodic field".
*The theory of the properties of metals and alloys*(Reprint of Clarendon Press (1936) ed.). Courier Dover Publications. pp. 56*ff*. ISBN 0-486-60456-X.

- N. W. Ashcroft and N. D. Mermin,
*Solid State Physics*(Thomson Learning, Toronto, 1976). - Stephen Blundell
*Magnetism in Condensed Matter*(Oxford, 2001). - S.Maekawa
*et al.**Physics of Transition Metal Oxides*(Springer-Verlag Berlin Heidelberg, 2004). - John Singleton
*Band Theory and Electronic Properties of Solids*(Oxford, 2001).

- Walter Ashley Harrison (1989).
*Electronic Structure and the Properties of Solids*. Dover Publications. ISBN 0-486-66021-4. - N. W. Ashcroft and N. D. Mermin (1976).
*Solid State Physics*. Toronto: Thomson Learning. - Davies, John H. (1998).
*The physics of low-dimensional semiconductors: An introduction*. Cambridge, United Kingdom: Cambridge University Press. ISBN 0-521-48491-X. - Goringe, C M; Bowler, D R; Hernández, E (1997). "Tight-binding modelling of materials".
*Reports on Progress in Physics*.**60**(12): 1447–1512. Bibcode:1997RPPh...60.1447G. doi:10.1088/0034-4885/60/12/001. - Slater, J. C.; Koster, G. F. (1954). "Simplified LCAO Method for the Periodic Potential Problem".
*Physical Review*.**94**(6): 1498–1524. Bibcode:1954PhRv...94.1498S. doi:10.1103/PhysRev.94.1498.

- Crystal-field Theory, Tight-binding Method, and Jahn-Teller Effect in E. Pavarini, E. Koch, F. Anders, and M. Jarrell (eds.): Correlated Electrons: From Models to Materials, Jülich 2012, ISBN 978-3-89336-796-2
- Tight-Binding Studio: A Technical Software Package to Find the Parameters of Tight-Binding Hamiltonian

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.