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Tight binding

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Electronic structuremethods

Valence bond theory

Generalized valence bond

Modern valence bond

Molecular orbital theory

HartreeFock method

MllerPlesset perturbation theory

Configuration interaction

Coupled cluster

Multi-configurational self-consistent field

Quantum chemistry composite methods

Quantum Monte Carlo

Linear combination of atomic orbitals

Electronic band structure

Nearly free electron model

Tight binding

Muffin-tin approximation

Density functional theory

kp perturbation theory

Empty Lattice Approximation

Book

v t
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e

Insolid-state physics, the tight-binding model (orTB model) is an approach to the calculation

ofelectronic band structureusing an approximate set ofwave functionsbased uponsuperpositionof wave functions for isolatedatomslocated at each atomic site. The method is

closely related to theLCAO methodused in chemistry. Tight-binding models are applied to a

wide variety of solids. The model gives good qualitative results in many cases and can becombined 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 ofsurface statesand application to various kinds ofmany-body problemandquasiparticlecalculations.

Contents

1 Introduction 2 Historical background 3 Mathematical formulation

o 3.1 Translational symmetry and normalizationo 3.2 The tight binding Hamiltoniano 3.3 The tight binding matrix elements

4 Evaluation of the matrix elements 5 Connection to Wannier functions 6 Second quantization 7 Example: one-dimensional s-band 8 Table of interatomic matrix elements 9 See also 10 References 11 Further reading 12 External links

Introduction

The name "tight binding" of thiselectronic band structure modelsuggests that thisquantum

mechanical modeldescribes the properties of tightly bound electrons in solids. Theelectronsin

this model should be tightly bound to theatomto which they belong and they should havelimited interaction withstatesand potentials on surrounding atoms of the solid. As a result the

wave functionof the electron will be rather similar to theatomic orbitalof the free atom to which

it belongs. The energy of the electron will also be rather close to theionization energyof theelectron in the free atom or ion because the interaction with potentials and states on neighboring

atoms is limited.
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Though the mathematical formulation[1]

of the one-particle tight-bindingHamiltonianmay look

complicated at first glance, the model is not complicated at all and can be understood intuitively

quite easily. There are onlythree kinds of elementsthat play a significant role in the theory. Twoof 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 thebond energiesby a chemist.

In general there are a number ofatomic energy levelsand atomic orbitals involved in the model.

This can lead to complicated band structures because the orbitals belong to differentpoint-grouprepresentations. Thereciprocal latticeand theBrillouin zoneoften belong to a differentspace

groupthan thecrystalof 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 aboutgroup theory.

The tight-binding model has a long history and has been applied in many ways and with manydifferent 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 thenearly-freeelectron model. The model itself, or parts of it, can serve as the basis for other calculations.

[2]In

the study ofconductive polymers,organic semiconductorsandmolecular electronics, for

example, tight-binding-like models are applied in which the role of the atoms in the original

concept is replaced by themolecular orbitalsofconjugated systemsand where the interatomicmatrix elements are replaced by inter- or intramolecular hopping andtunnelingparameters.

These conductors nearly all have very anisotropic properties and sometimes are almost perfectly

one-dimensional.

Historical background

By 1928, the idea of a molecular orbital had been advanced byRobert Mulliken, who wasinfluenced considerably by the work ofFriedrich 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 byFelix Bloch, as part of his doctoral dissertation in1928, 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

oftransition metals, is the parameterized tight-binding method conceived in 1954 byJohn ClarkeSlaterand George Fred Koster,

[1]sometimes referred to as theSK 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 originalBloch's theorembut, rather, first-principles calculations arecarried out only at high-symmetry points and the band structure is interpolated over theremainder of theBrillouin zonebetween these points.

In this approach, interactions between different atomic sites are considered asperturbations.There exist several kinds of interactions we must consider. The crystalHamiltonianis 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
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wave function. There are further explanations in the next section with some mathematical

expressions.

Recently, in the research aboutstrongly correlated material, the tight binding approach is basic

approximation because highly localized electrons like 3-dtransition metalelectrons sometimes

display strongly correlated behaviors. In this case, the role of electron-electron interaction mustbe considered using themany-body physicsdescription.

The tight-binding model is typically used for calculations ofelectronic band structureandbandgapsin the static regime. However, in combination with other methods such as therandom phase

approximation(RPA) model, the dynamic response of systems may also be studied.

Mathematical formulation

We introduce theatomic orbitals , which areeigenfunctionsof theHamiltonian of asingle 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:

A solution to the time-independent single electronSchrdinger equationis then approximated

as alinear combination of atomic orbitals :

,

where refers to the m-th atomic energy level and locates an atomic site in thecrystallattice.

Translational symmetry and normalization

Thetranslational symmetryof the crystal implies the wave function under translation can change

only by a phase factor:

where is thewave vectorof the wave function. Consequently, the coefficients satisfy
https://en.wikipedia.org/wiki/Strongly_correlated_materialhttps://en.wikipedia.org/wiki/Strongly_correlated_materialhttps://en.wikipedia.org/wiki/Strongly_correlated_materialhttps://en.wikipedia.org/wiki/Transition_metalhttps://en.wikipedia.org/wiki/Transition_metalhttps://en.wikipedia.org/wiki/Transition_metalhttps://en.wikipedia.org/wiki/Many-body_theoryhttps://en.wikipedia.org/wiki/Many-body_theoryhttps://en.wikipedia.org/wiki/Many-body_theoryhttps://en.wikipedia.org/wiki/Electronic_band_structurehttps://en.wikipedia.org/wiki/Electronic_band_structurehttps://en.wikipedia.org/wiki/Electronic_band_structurehttps://en.wikipedia.org/wiki/Band_gaphttps://en.wikipedia.org/wiki/Band_gaphttps://en.wikipedia.org/wiki/Band_gaphttps://en.wikipedia.org/wiki/Band_gaphttps://en.wikipedia.org/wiki/Random_phase_approximationhttps://en.wikipedia.org/wiki/Random_phase_approximationhttps://en.wikipedia.org/wiki/Random_phase_approximationhttps://en.wikipedia.org/wiki/Random_phase_approximationhttps://en.wikipedia.org/wiki/Atomic_orbitalhttps://en.wikipedia.org/wiki/Atomic_orbitalhttps://en.wikipedia.org/wiki/Atomic_orbitalhttps://en.wikipedia.org/wiki/Eigenfunctionhttps://en.wikipedia.org/wiki/Eigenfunctionhttps://en.wikipedia.org/wiki/Eigenfunctionhttps://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29https://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29https://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttps://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttps://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Crystal_structurehttps://en.wikipedia.org/wiki/Crystal_structurehttps://en.wikipedia.org/wiki/Crystal_structurehttps://en.wikipedia.org/wiki/Crystal_structurehttps://en.wikipedia.org/wiki/Translational_symmetryhttps://en.wikipedia.org/wiki/Translational_symmetryhttps://en.wikipedia.org/wiki/Translational_symmetryhttps://en.wikipedia.org/wiki/Wave_vectorhttps://en.wikipedia.org/wiki/Wave_vectorhttps://en.wikipedia.org/wiki/Wave_vectorhttps://en.wikipedia.org/wiki/Wave_vectorhttps://en.wikipedia.org/wiki/Translational_symmetryhttps://en.wikipedia.org/wiki/Crystal_structurehttps://en.wikipedia.org/wiki/Crystal_structurehttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttps://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29https://en.wikipedia.org/wiki/Eigenfunctionhttps://en.wikipedia.org/wiki/Atomic_orbitalhttps://en.wikipedia.org/wiki/Random_phase_approximationhttps://en.wikipedia.org/wiki/Random_phase_approximationhttps://en.wikipedia.org/wiki/Band_gaphttps://en.wikipedia.org/wiki/Band_gaphttps://en.wikipedia.org/wiki/Electronic_band_structurehttps://en.wikipedia.org/wiki/Many-body_theoryhttps://en.wikipedia.org/wiki/Transition_metalhttps://en.wikipedia.org/wiki/Strongly_correlated_material


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By substituting , we find

or

Normalizingthe wave function to unity:

so the normalization sets b(0) as

where (Rp ) are the atomic overlap integrals, which frequently are neglected resulting in[3]

and
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The tight binding Hamiltonian

Using the tight binding form for the wave function, and assuming only the m-th atomicenergy

levelis 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

whereEm is the energy of the m-th atomic level, and , and are the tight bindingmatrix elements.

The tight binding matrix elements

The element

,

is the atomic energy shift due to the potential on neighboring atoms. This term is relatively smallin 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
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is theinter atomic matrix elementbetween the atomic orbitals m and lon 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 theoverlap integralsbetween the atomic orbitals m and lon adjacent atoms.

Evaluation of the matrix elements

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 thecentral 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 inter atomic

distances can be too small or the charges on the atoms or ions in the lattice is wrong for example.

The inter atomic matrix elements can be calculated directly if the atomic wave functionsand 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 fromchemical bond

energy data. Energies and eigenstates on some high symmetry points in theBrillouin zonecan be

evaluated and values integrals in the matrix elements can be matched with band structure datafrom other sources.

The inter atomic overlap matrix elements should be rather small or neglectable. If they arelarge 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 inter atomic distance for example. In metals andtransition 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 anearly free electron model.

The tight binding model works particularly well in cases where the band width is small and theelectrons 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]
https://en.wikipedia.org/wiki/Tight_binding#Table_of_inter_atomic_matrix_elementshttps://en.wikipedia.org/wiki/Tight_binding#Table_of_inter_atomic_matrix_elementshttps://en.wikipedia.org/wiki/Tight_binding#Table_of_inter_atomic_matrix_elementshttps://en.wikipedia.org/wiki/Overlap_matrixhttps://en.wikipedia.org/wiki/Overlap_matrixhttps://en.wikipedia.org/wiki/Overlap_matrixhttps://en.wikipedia.org/wiki/Bond_energy#External_linkshttps://en.wikipedia.org/wiki/Bond_energy#External_linkshttps://en.wikipedia.org/wiki/Bond_energy#External_linkshttps://en.wikipedia.org/wiki/Bond_energy#External_linkshttps://en.wikipedia.org/wiki/Brillouin_zonehttps://en.wikipedia.org/wiki/Brillouin_zonehttps://en.wikipedia.org/wiki/Brillouin_zonehttps://en.wikipedia.org/wiki/Nearly_free_electron_modelhttps://en.wikipedia.org/wiki/Nearly_free_electron_modelhttps://en.wikipedia.org/wiki/Nearly_free_electron_modelhttps://en.wikipedia.org/wiki/Tight_binding#cite_note-Harrison-2https://en.wikipedia.org/wiki/Tight_binding#cite_note-Harrison-2https://en.wikipedia.org/wiki/Tight_binding#cite_note-Harrison-2https://en.wikipedia.org/wiki/Tight_binding#cite_note-Harrison-2https://en.wikipedia.org/wiki/Nearly_free_electron_modelhttps://en.wikipedia.org/wiki/Brillouin_zonehttps://en.wikipedia.org/wiki/Bond_energy#External_linkshttps://en.wikipedia.org/wiki/Bond_energy#External_linkshttps://en.wikipedia.org/wiki/Overlap_matrixhttps://en.wikipedia.org/wiki/Tight_binding#Table_of_inter_atomic_matrix_elements


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Connection to Wannier functions

Bloch wavefunctions describe the electronic states in a periodiccrystal lattice. Bloch functions

can be represented as aFourier series[4]

where Rn denotes an atomic site in a periodic crystal lattice, kis thewave vectorof the Bloch

wave, ris the electron position, m is the band index, and the sum is over allNatomic sites. The

Bloch wave is an exact eigensolution for the wave function of an electron in a periodic crystalpotential corresponding to an energyEm (k), and is spread over the entire crystal volume.

Using theFourier transformanalysis, a spatially localized wave function for the m-th energyband can be constructed from multiple Bloch waves:

These real space wave functions are calledWannier functions, and are fairly closelylocalized to the atomic site Rn. Of course, if we have exactWannier functions, the exact Bloch

functions can be derived using the inverse Fourier transform.

However it is not easy to calculate directly eitherBloch functionsorWannier functions. Anapproximate approach is necessary in the calculation ofelectronic structuresof solids. If we

consider the extreme case of isolated atoms, the Wannier function would become an isolatedatomic orbital. That limit suggests the choice of an atomic wave function as an approximate formfor the Wannier function, the so-called tight binding approximation.

Second quantization

Modern explanations of electronic structure liket-J modelandHubbard modelare based on tight

binding model.[5]

If we introducesecond quantizationformalism, it is clear to understand theconcept of tight binding model.

Using the atomic orbital as a basis state, we can establish the second quantization Hamiltonian

operator in tight binding model.

,

- creation and annihilation operators- spin polarization
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- hopping integral

-nearest neighbor index

Here, hopping integral corresponds to the transfer integral in tight binding model.

Considering extreme cases of , it is impossible for electron to hop into neighboring sites.

This case is the isolated atomic system. If the hopping term is turned on ( ) electrons canstay in both sites lowering theirkinetic 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 directCoulombinteraction energy and exchange

interaction energy between electrons. There are several novel physics induced from this electron-electron interaction energy, such asmetal-insulator transitions(MIT),high-temperature

superconductivity, and severalquantum phase transitions.

Example: one-dimensional s-band

Here the tight binding model is illustrated with a s-band model for a string of atoms with a

singles-orbitalin a straight line with spacing a and bondsbetween atomic sites.

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

atomic orbitals

whereN= total number of sites and is a real parameter with . (This wavefunction is normalized to unity by the leading factor 1/N provided overlap of atomic wavefunctions is ignored.) Assuming only nearest neighbor overlap, the only non-zero matrix

elements of the Hamiltonian can be expressed as

The energyEi is the ionization energy corresponding to the chosen atomic orbital and Uis theenergy shift of the orbital as a result of the potential of neighboring atoms. The

elements, which are theSlater and Koster interatomic matrix elements,
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are thebond 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 sumof all atomic orbitals. This state can be viewed as a chain ofbonding orbitals.

For the energy is and the state consists of a sum of atomicorbitals which are a factor out of phase. This state can be viewed as a chain ofnon-bonding orbitals.

Finally for the energy is and the state consistsof an alternating sum of atomic orbitals. This state can be viewed as a chain ofanti-

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
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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.

Table of interatomic matrix elements

In 1954 J.C. Slater and F.G. Koster published, mainly for the calculation oftransition metald-

bands, a table of interatomic matrix elements[1]

which, with a little patience and effort, can also be derived from thecubic harmonic orbitalsstraightforwardly. The table expresses the matrix elements as functions ofLCAOtwo-centre

bond integralsbetween twocubic harmonicorbitals, i andj, on adjacent atoms. The bond

integrals are for example the , and forsigma,pianddeltabonds.

The interatomic vector is expressed as

where dis the distance between the atoms and l, m and n are thedirection cosinesto theneighboring atom.
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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 matrixelements in the table.

See also

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)

References
https://en.wikipedia.org/wiki/Electronic_band_structurehttps://en.wikipedia.org/wiki/Electronic_band_structurehttps://en.wikipedia.org/wiki/Nearly-free_electron_modelhttps://en.wikipedia.org/wiki/Nearly-free_electron_modelhttps://en.wikipedia.org/wiki/Nearly-free_electron_modelhttps://en.wikipedia.org/wiki/Nearly-free_electron_modelhttps://en.wikipedia.org/wiki/Nearly-free_electron_modelhttps://en.wikipedia.org/wiki/Bloch_wavehttps://en.wikipedia.org/wiki/Bloch_wavehttps://en.wikipedia.org/wiki/Kronig-Penney_modelhttps://en.wikipedia.org/wiki/Kronig-Penney_modelhttps://en.wikipedia.org/wiki/Fermi_surfacehttps://en.wikipedia.org/wiki/Fermi_surfacehttps://en.wikipedia.org/wiki/Wannier_functionhttps://en.wikipedia.org/wiki/Wannier_functionhttps://en.wikipedia.org/wiki/Hubbard_modelhttps://en.wikipedia.org/wiki/Hubbard_modelhttps://en.wikipedia.org/wiki/T-J_modelhttps://en.wikipedia.org/wiki/T-J_modelhttps://en.wikipedia.org/wiki/Effective_mass_%28solid-state_physics%29https://en.wikipedia.org/wiki/Effective_mass_%28solid-state_physics%29https://en.wikipedia.org/wiki/Anderson%27s_rulehttps://en.wikipedia.org/wiki/Anderson%27s_rulehttps://en.wikipedia.org/wiki/Dynamical_theory_of_diffractionhttps://en.wikipedia.org/wiki/Dynamical_theory_of_diffractionhttps://en.wikipedia.org/wiki/Solid_state_physicshttps://en.wikipedia.org/wiki/Solid_state_physicshttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_methodhttps://en.wikipedia.org/wiki/Solid_state_physicshttps://en.wikipedia.org/wiki/Dynamical_theory_of_diffractionhttps://en.wikipedia.org/wiki/Anderson%27s_rulehttps://en.wikipedia.org/wiki/Effective_mass_%28solid-state_physics%29https://en.wikipedia.org/wiki/T-J_modelhttps://en.wikipedia.org/wiki/Hubbard_modelhttps://en.wikipedia.org/wiki/Wannier_functionhttps://en.wikipedia.org/wiki/Fermi_surfacehttps://en.wikipedia.org/wiki/Kronig-Penney_modelhttps://en.wikipedia.org/wiki/Bloch_wavehttps://en.wikipedia.org/wiki/Nearly-free_electron_modelhttps://en.wikipedia.org/wiki/Nearly-free_electron_modelhttps://en.wikipedia.org/wiki/Electronic_band_structure


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Wikimedia Commons has media related to:Dispersion r elati ons of electrons

1. âbcJ. C. Slater, G. F. Koster (1954). "Simplified LCAO method for the PeriodicPotential Problem".Physical Review94 (6): 14981524.Bibcode:1954PhRv...94.1498S.doi:10.1103/PhysRev.94.1498.

2. âbWalter Ashley Harrison (1989).Electronic Structure and the Properties of Solids.Dover Publications.ISBN0-486-66021-4.

3. Âs an alternative to neglecting overlap, one may choose as a basis instead of atomicorbitals a set of orbitals based upon atomic orbitals but arranged to be orthogonal toorbitals on other atomic sites, the so-calledLwdin 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.ISBN3-540-25470-6.

4. Ôrfried Madelung,Introduction to Solid-State Theory (Springer-Verlag, BerlinHeidelberg, 1978).

5. Âlexander Altland and Ben Simons (2006)."Interaction effects in the tight-bindingsystem".Condensed Matter Field Theory. Cambridge University Press. pp. 58ff.ISBN978-0-521-84508-3.

6. ^Sir Nevill F Mott & H Jones (1958)."II 4 Motion of electrons in a periodic field".Thetheory of the properties of metals and alloys (Reprint of Clarendon Press (1936) ed.).Courier Dover Publications. pp. 56ff.ISBN0-486-60456-X.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, Toronto,1976).

Stephen BlundellMagnetism in Condensed Matter(Oxford, 2001). S.Maekawa et al.Physics of Transition Metal Oxides (Spinger-Verlag Berlin Heidelberg,

2004).

John SingletonBand Theory and Electronic Properties of Solids (Oxford, 2001).Further reading

Walter Ashley Harrison (1989).Electronic Structure and the Properties of Solids. DoverPublications.ISBN0-486-66021-4.

N. W. Ashcroft and N. D. Mermin (1976). Solid State Physics. Toronto: ThomsonLearning.

Davies, John H. (1998). The physics of low-dimensional semiconductors: Anintroduction. Cambridge, United Kingdom: Cambridge University Press.ISBN0-521-

48491-X. Goringe, C M; Bowler, D R; Hernndez, E (1997). "Tight-binding modelling of

materials".Reports on Progress in Physics60 (12): 14471512.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 PotentialProblem".Physical Review94 (6): 14981524.Bibcode:1954PhRv...94.1498S.doi:10.1103/PhysRev.94.1498.
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External links

Crystal-field Theory, Tight-binding Method, and Jahn-Teller Effectin E. Pavarini, E.Koch, F. Anders, and M. Jarrell (eds.): Correlated Electrons: From Models to Materials,Jlich 2012,ISBN 978-3-89336-796-2
http://www.cond-mat.de/events/correl12/manuscripts/pavarini.pdfhttp://www.cond-mat.de/events/correl12/manuscripts/pavarini.pdfhttps://en.wikipedia.org/wiki/Special:BookSources/9783893367962https://en.wikipedia.org/wiki/Special:BookSources/9783893367962https://en.wikipedia.org/wiki/Special:BookSources/9783893367962https://en.wikipedia.org/wiki/Special:BookSources/9783893367962http://www.cond-mat.de/events/correl12/manuscripts/pavarini.pdf

wave function summation of bloch electron.docx

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