# Tight binding

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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

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.

## Historical background

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.

## Mathematical formulation

We introduce the atomic orbitals ${\displaystyle \varphi _{m}(\mathbf {r} )}$, which are eigenfunctions of the Hamiltonian ${\displaystyle H_{\rm {at}}}$ 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 ${\displaystyle \Delta U}$ required to obtain the true Hamiltonian ${\displaystyle H}$ of the system, are assumed small:

${\displaystyle H(\mathbf {r} )=H_{\mathrm {at} }(\mathbf {r} )+\sum _{\mathbf {R_{n}} \neq \mathbf {0} }V(\mathbf {r} -\mathbf {R_{n}} )=H_{\mathrm {at} }(\mathbf {r} )+\Delta U(\mathbf {r} )\ ,}$

where ${\displaystyle V(\mathbf {r} -\mathbf {R_{n}} )}$ denotes the atomic potential of one atom located at site ${\displaystyle \mathbf {R} _{n}}$ in the crystal lattice. A solution ${\displaystyle \psi _{m}}$ to the time-independent single electron Schrödinger equation is then approximated as a linear combination of atomic orbitals ${\displaystyle \varphi _{m}(\mathbf {r-R_{n}} )}$:

${\displaystyle \psi _{m}(\mathbf {r} )=\sum _{\mathbf {R_{n}} }b_{m}(\mathbf {R_{n}} )\ \varphi _{m}(\mathbf {r-R_{n}} )}$,

where ${\displaystyle m}$ refers to the m-th atomic energy level.

### Translational symmetry and normalization

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

${\displaystyle \psi (\mathbf {r+R_{\ell }} )=e^{i\mathbf {k\cdot R_{\ell }} }\psi (\mathbf {r} )\ ,}$

where ${\displaystyle \mathbf {k} }$ is the wave vector of the wave function. Consequently, the coefficients satisfy

${\displaystyle \sum _{\mathbf {R_{n}} }b_{m}(\mathbf {R_{n}} )\ \varphi _{m}(\mathbf {r-R_{n}+R_{\ell }} )=e^{i\mathbf {k\cdot R_{\ell }} }\sum _{\mathbf {R_{n}} }b_{m}(\mathbf {R_{n}} )\ \varphi _{m}(\mathbf {r-R_{n}} )\ .}$

By substituting ${\displaystyle \mathbf {R_{p}} =\mathbf {R_{n}} -\mathbf {R_{\ell }} }$, we find

${\displaystyle b_{m}(\mathbf {R_{p}+R_{\ell }} )=e^{i\mathbf {k\cdot R_{\ell }} }b_{m}(\mathbf {R_{p}} )\ ,}$ (where in RHS we have replaced the dummy index ${\displaystyle \mathbf {R_{n}} }$ with ${\displaystyle \mathbf {R_{p}} }$)

or

${\displaystyle b_{m}(\mathbf {R_{l}} )=e^{i\mathbf {k\cdot R_{l}} }b_{m}(\mathbf {0} )\ .}$

Normalizing the wave function to unity:

${\displaystyle \int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )\psi _{m}(\mathbf {r} )=1}$
${\displaystyle =\sum _{\mathbf {R_{n}} }b_{m}^{*}(\mathbf {R_{n}} )\sum _{\mathbf {R_{\ell }} }b_{m}(\mathbf {R_{\ell }} )\int d^{3}r\ \varphi _{m}^{*}(\mathbf {r-R_{n}} )\varphi _{m}(\mathbf {r-R_{\ell }} )}$
${\displaystyle =b_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R_{n}} }e^{-i\mathbf {k\cdot R_{n}} }\sum _{\mathbf {R_{\ell }} }e^{i\mathbf {k\cdot R_{\ell }} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r-R_{n}} )\varphi _{m}(\mathbf {r-R_{\ell }} )}$
${\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R_{p}} }e^{-i\mathbf {k\cdot R_{p}} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r-R_{p}} )\varphi _{m}(\mathbf {r} )\ }$
${\displaystyle =Nb_{m}^{*}(0)b_{m}(0)\sum _{\mathbf {R_{p}} }e^{i\mathbf {k\cdot R_{p}} }\ \int d^{3}r\ \varphi _{m}^{*}(\mathbf {r} )\varphi _{m}(\mathbf {r-R_{p}} )\ ,}$

so the normalization sets ${\displaystyle b_{m}(0)}$ as

${\displaystyle b_{m}^{*}(0)b_{m}(0)={\frac {1}{N}}\ \cdot \ {\frac {1}{1+\sum _{\mathbf {R_{p}\neq 0} }e^{i\mathbf {k\cdot R_{p}} }\alpha _{m}(\mathbf {R_{p}} )}}\ ,}$

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

${\displaystyle b_{m}(0)\approx {\frac {1}{\sqrt {N}}}\ ,}$

and

${\displaystyle \psi _{m}(\mathbf {r} )\approx {\frac {1}{\sqrt {N}}}\sum _{\mathbf {R_{n}} }e^{i\mathbf {k\cdot R_{n}} }\ \varphi _{m}(\mathbf {r-R_{n}} )\ .}$

### The tight binding Hamiltonian

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 ${\displaystyle \varepsilon _{m}}$ are of the form

${\displaystyle \varepsilon _{m}=\int d^{3}r\ \psi _{m}^{*}(\mathbf {r} )H(\mathbf {r} )\psi (\mathbf {r} )}$
${\displaystyle =\sum _{\mathbf {R_{n}} }b^{*}(\mathbf {R_{n}} )\ \int d^{3}r\ \varphi ^{*}(\mathbf {r-R_{n}} )H(\mathbf {r} )\psi (\mathbf {r} )\ }$
${\displaystyle =\sum _{\mathbf {R_{\ell }} }\ \sum _{\mathbf {R_{n}} }b^{*}(\mathbf {R_{n}} )\ \int d^{3}r\ \varphi ^{*}(\mathbf {r-R_{n}} )H_{\mathrm {at} }(\mathbf {r-R_{\ell }} )\psi (\mathbf {r} )\ +\sum _{\mathbf {R_{n}} }b^{*}(\mathbf {R_{n}} )\ \int d^{3}r\ \varphi ^{*}(\mathbf {r-R_{n}} )\Delta U(\mathbf {r} )\psi (\mathbf {r} )\ .}$
${\displaystyle \approx E_{m}+b^{*}(0)\sum _{\mathbf {R_{n}} }e^{-i\mathbf {k\cdot R_{n}} }\ \int d^{3}r\ \varphi ^{*}(\mathbf {r-R_{n}} )\Delta U(\mathbf {r} )\psi (\mathbf {r} )\ .}$

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

${\displaystyle \varepsilon _{m}(\mathbf {k} )=E_{m}-N\ |b(0)|^{2}\left(\beta _{m}+\sum _{\mathbf {R_{n}} \neq 0}\sum _{l}\gamma _{m,l}(\mathbf {R_{n}} )e^{i\mathbf {k} \cdot \mathbf {R_{n}} }\right)\ ,}$
${\displaystyle =E_{m}-\ {\frac {\beta _{m}+\sum _{\mathbf {R_{n}} \neq 0}\sum _{l}e^{i\mathbf {k} \cdot \mathbf {R_{n}} }\gamma _{m,l}(\mathbf {R_{n}} )}{\ \ 1+\sum _{\mathbf {R_{n}\neq 0} }\sum _{l}e^{i\mathbf {k\cdot R_{n}} }\alpha _{m,l}(\mathbf {R_{n}} )}}\ ,}$

where Em is the energy of the m-th atomic level, and ${\displaystyle \alpha _{m,l}}$, ${\displaystyle \beta _{m}}$ and ${\displaystyle \gamma _{m,l}}$ are the tight binding matrix elements.

### The tight binding matrix elements

The element

${\displaystyle \beta _{m}=-\int \varphi _{m}^{*}(\mathbf {r} )\Delta U(\mathbf {r} )\varphi _{m}(\mathbf {r} )\,d^{3}r\ }$,

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

${\displaystyle \gamma _{m,l}(\mathbf {R_{n}} )=-\int \varphi _{m}^{*}(\mathbf {r} )\Delta U(\mathbf {r} )\varphi _{l}(\mathbf {r-R_{n}} )\,d^{3}r\ ,}$

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

${\displaystyle \alpha _{m,l}(\mathbf {R_{n}} )=\int \varphi _{m}^{*}(\mathbf {r} )\varphi _{l}(\mathbf {r-R_{n}} )\,d^{3}r\ }$,

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

## Evaluation of the matrix elements

As mentioned before the values of the ${\displaystyle \beta _{m}}$-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 ${\displaystyle \beta _{m}}$ 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 ${\displaystyle \gamma _{m,l}}$ 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 ${\displaystyle \alpha _{m,l}}$ 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]

## Connection to Wannier functions

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

${\displaystyle \psi _{m}\mathbf {(k,r)} ={\frac {1}{\sqrt {N}}}\sum _{n}{a_{m}\mathbf {(R_{n},r)} }e^{\mathbf {ik\cdot R_{n}} }\ ,}$

where Rn denotes an atomic site in a periodic crystal lattice, k is the wave vector of the Bloch's function, r is the electron position, m is the band index, and the sum is over all N atomic sites. The Bloch's function is an exact eigensolution for the wave function of an electron in a periodic crystal potential corresponding to an energy Em (k), and is spread over the entire crystal volume.

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

${\displaystyle a_{m}\mathbf {(R_{n},r)} ={\frac {1}{\sqrt {N}}}\sum _{\mathbf {k} }{e^{\mathbf {-ik\cdot R_{n}} }\psi _{m}\mathbf {(k,r)} }={\frac {1}{\sqrt {N}}}\sum _{\mathbf {k} }{e^{\mathbf {ik\cdot (r-R_{n})} }u_{m}\mathbf {(k,r)} }.}$

These real space wave functions ${\displaystyle {a_{m}\mathbf {(R_{n},r)} }}$ are called Wannier functions, and are fairly closely localized to the atomic site Rn. 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.

## Second quantization

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:

${\displaystyle H=-t\sum _{\langle i,j\rangle ,\sigma }(c_{i,\sigma }^{\dagger }c_{j,\sigma }^{}+h.c.)}$,
${\displaystyle c_{i\sigma }^{\dagger },c_{j\sigma }}$ - creation and annihilation operators
${\displaystyle \displaystyle \sigma }$ - spin polarization
${\displaystyle \displaystyle t}$ - hopping integral
${\displaystyle \displaystyle \langle i,j\rangle }$ - nearest neighbor index
${\displaystyle \displaystyle h.c.}$ - the hermitian conjugate of the other term(s)

Here, hopping integral ${\displaystyle \displaystyle t}$ corresponds to the transfer integral ${\displaystyle \displaystyle \gamma }$ in tight binding model. Considering extreme cases of ${\displaystyle t\rightarrow 0}$, 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 (${\displaystyle \displaystyle t>0}$) 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

${\displaystyle \displaystyle H_{ee}={\frac {1}{2}}\sum _{n,m,\sigma }\langle n_{1}m_{1},n_{2}m_{2}|{\frac {e^{2}}{|r_{1}-r_{2}|}}|n_{3}m_{3},n_{4}m_{4}\rangle c_{n_{1}m_{1}\sigma _{1}}^{\dagger }c_{n_{2}m_{2}\sigma _{2}}^{\dagger }c_{n_{4}m_{4}\sigma _{2}}c_{n_{3}m_{3}\sigma _{1}}}$

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.

## Example: one-dimensional s-band

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

${\displaystyle |k\rangle ={\frac {1}{\sqrt {N}}}\sum _{n=1}^{N}e^{inka}|n\rangle }$

where N = total number of sites and ${\displaystyle k}$ is a real parameter with ${\displaystyle -{\frac {\pi }{a}}\leqq k\leqq {\frac {\pi }{a}}}$. (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

${\displaystyle \langle n|H|n\rangle =E_{0}=E_{i}-U\ .}$
${\displaystyle \langle n\pm 1|H|n\rangle =-\Delta \ }$
${\displaystyle \langle n|n\rangle =1\$ ;}${\displaystyle \langle n\pm 1|n\rangle =S\ .}$

The energy Ei 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 ${\displaystyle \langle n\pm 1|H|n\rangle =-\Delta }$ elements, which are the Slater and Koster interatomic matrix elements, are the bond energies ${\displaystyle E_{i,j}}$. In this one dimensional s-band model we only have ${\displaystyle \sigma }$-bonds between the s-orbitals with bond energy ${\displaystyle E_{s,s}=V_{ss\sigma }}$. The overlap between states on neighboring atoms is S. We can derive the energy of the state ${\displaystyle |k\rangle }$ using the above equation:

${\displaystyle H|k\rangle ={\frac {1}{\sqrt {N}}}\sum _{n}e^{inka}H|n\rangle }$
${\displaystyle \langle k|H|k\rangle ={\frac {1}{N}}\sum _{n,\ m}e^{i(n-m)ka}\langle m|H|n\rangle }$${\displaystyle ={\frac {1}{N}}\sum _{n}\langle n|H|n\rangle +{\frac {1}{N}}\sum _{n}\langle n-1|H|n\rangle e^{+ika}+{\frac {1}{N}}\sum _{n}\langle n+1|H|n\rangle e^{-ika}}$${\displaystyle =E_{0}-2\Delta \,\cos(ka)\ ,}$

where, for example,

${\displaystyle {\frac {1}{N}}\sum _{n}\langle n|H|n\rangle =E_{0}{\frac {1}{N}}\sum _{n}1=E_{0}\ ,}$

and

${\displaystyle {\frac {1}{N}}\sum _{n}\langle n-1|H|n\rangle e^{+ika}=-\Delta e^{ika}{\frac {1}{N}}\sum _{n}1=-\Delta e^{ika}\ .}$
${\displaystyle {\frac {1}{N}}\sum _{n}\langle n-1|n\rangle e^{+ika}=Se^{ika}{\frac {1}{N}}\sum _{n}1=Se^{ika}\ .}$

Thus the energy of this state ${\displaystyle |k\rangle }$ can be represented in the familiar form of the energy dispersion:

${\displaystyle E(k)={\frac {E_{0}-2\Delta \,\cos(ka)}{1+2S\,\cos(ka)}}}$.
• For ${\displaystyle k=0}$ the energy is ${\displaystyle E=(E_{0}-2\Delta )/(1+2S)}$ and the state consists of a sum of all atomic orbitals. This state can be viewed as a chain of bonding orbitals.
• For ${\displaystyle k=\pi /(2a)}$ the energy is ${\displaystyle E=E_{0}}$ and the state consists of a sum of atomic orbitals which are a factor ${\displaystyle e^{i\pi /2}}$ out of phase. This state can be viewed as a chain of non-bonding orbitals.
• Finally for ${\displaystyle k=\pi /a}$ the energy is ${\displaystyle E=(E_{0}+2\Delta )/(1-2S)}$ 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.

## Table of interatomic matrix elements

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]

${\displaystyle E_{i,j}({\vec {\mathbf {r} }}_{n,n'})=\langle n,i|H|n',j\rangle }$

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 ${\displaystyle V_{ss\sigma }}$, ${\displaystyle V_{pp\pi }}$ and ${\displaystyle V_{dd\delta }}$ 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 ${\displaystyle (l,m,n)}$, even though it is not explicitly stated every time.).

The interatomic vector is expressed as

${\displaystyle {\vec {\mathbf {r} }}_{n,n'}=(r_{x},r_{y},r_{z})=d(l,m,n)}$

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

${\displaystyle E_{s,s}=V_{ss\sigma }}$
${\displaystyle E_{s,x}=lV_{sp\sigma }}$
${\displaystyle E_{x,x}=l^{2}V_{pp\sigma }+(1-l^{2})V_{pp\pi }}$
${\displaystyle E_{x,y}=lmV_{pp\sigma }-lmV_{pp\pi }}$
${\displaystyle E_{x,z}=lnV_{pp\sigma }-lnV_{pp\pi }}$
${\displaystyle E_{s,xy}={\sqrt {3}}lmV_{sd\sigma }}$
${\displaystyle E_{s,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}(l^{2}-m^{2})V_{sd\sigma }}$
${\displaystyle E_{s,3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]V_{sd\sigma }}$
${\displaystyle E_{x,xy}={\sqrt {3}}l^{2}mV_{pd\sigma }+m(1-2l^{2})V_{pd\pi }}$
${\displaystyle E_{x,yz}={\sqrt {3}}lmnV_{pd\sigma }-2lmnV_{pd\pi }}$
${\displaystyle E_{x,zx}={\sqrt {3}}l^{2}nV_{pd\sigma }+n(1-2l^{2})V_{pd\pi }}$
${\displaystyle E_{x,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}l(l^{2}-m^{2})V_{pd\sigma }+l(1-l^{2}+m^{2})V_{pd\pi }}$
${\displaystyle E_{y,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}m(l^{2}-m^{2})V_{pd\sigma }-m(1+l^{2}-m^{2})V_{pd\pi }}$
${\displaystyle E_{z,x^{2}-y^{2}}={\frac {\sqrt {3}}{2}}n(l^{2}-m^{2})V_{pd\sigma }-n(l^{2}-m^{2})V_{pd\pi }}$
${\displaystyle E_{x,3z^{2}-r^{2}}=l[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}ln^{2}V_{pd\pi }}$
${\displaystyle E_{y,3z^{2}-r^{2}}=m[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }-{\sqrt {3}}mn^{2}V_{pd\pi }}$
${\displaystyle E_{z,3z^{2}-r^{2}}=n[n^{2}-(l^{2}+m^{2})/2]V_{pd\sigma }+{\sqrt {3}}n(l^{2}+m^{2})V_{pd\pi }}$
${\displaystyle E_{xy,xy}=3l^{2}m^{2}V_{dd\sigma }+(l^{2}+m^{2}-4l^{2}m^{2})V_{dd\pi }+(n^{2}+l^{2}m^{2})V_{dd\delta }}$
${\displaystyle E_{xy,yz}=3lm^{2}nV_{dd\sigma }+ln(1-4m^{2})V_{dd\pi }+ln(m^{2}-1)V_{dd\delta }}$
${\displaystyle E_{xy,zx}=3l^{2}mnV_{dd\sigma }+mn(1-4l^{2})V_{dd\pi }+mn(l^{2}-1)V_{dd\delta }}$
${\displaystyle E_{xy,x^{2}-y^{2}}={\frac {3}{2}}lm(l^{2}-m^{2})V_{dd\sigma }+2lm(m^{2}-l^{2})V_{dd\pi }+[lm(l^{2}-m^{2})/2]V_{dd\delta }}$
${\displaystyle E_{yz,x^{2}-y^{2}}={\frac {3}{2}}mn(l^{2}-m^{2})V_{dd\sigma }-mn[1+2(l^{2}-m^{2})]V_{dd\pi }+mn[1+(l^{2}-m^{2})/2]V_{dd\delta }}$
${\displaystyle E_{zx,x^{2}-y^{2}}={\frac {3}{2}}nl(l^{2}-m^{2})V_{dd\sigma }+nl[1-2(l^{2}-m^{2})]V_{dd\pi }-nl[1-(l^{2}-m^{2})/2]V_{dd\delta }}$
${\displaystyle E_{xy,3z^{2}-r^{2}}={\sqrt {3}}\left[lm(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }-2lmn^{2}V_{dd\pi }+[lm(1+n^{2})/2]V_{dd\delta }\right]}$
${\displaystyle E_{yz,3z^{2}-r^{2}}={\sqrt {3}}\left[mn(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+mn(l^{2}+m^{2}-n^{2})V_{dd\pi }-[mn(l^{2}+m^{2})/2]V_{dd\delta }\right]}$
${\displaystyle E_{zx,3z^{2}-r^{2}}={\sqrt {3}}\left[ln(n^{2}-(l^{2}+m^{2})/2)V_{dd\sigma }+ln(l^{2}+m^{2}-n^{2})V_{dd\pi }-[ln(l^{2}+m^{2})/2]V_{dd\delta }\right]}$
${\displaystyle E_{x^{2}-y^{2},x^{2}-y^{2}}={\frac {3}{4}}(l^{2}-m^{2})^{2}V_{dd\sigma }+[l^{2}+m^{2}-(l^{2}-m^{2})^{2}]V_{dd\pi }+[n^{2}+(l^{2}-m^{2})^{2}/4]V_{dd\delta }}$
${\displaystyle E_{x^{2}-y^{2},3z^{2}-r^{2}}={\sqrt {3}}\left[(l^{2}-m^{2})[n^{2}-(l^{2}+m^{2})/2]V_{dd\sigma }/2+n^{2}(m^{2}-l^{2})V_{dd\pi }+[(1+n^{2})(l^{2}-m^{2})/4]V_{dd\delta }\right]}$
${\displaystyle E_{3z^{2}-r^{2},3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]^{2}V_{dd\sigma }+3n^{2}(l^{2}+m^{2})V_{dd\pi }+{\frac {3}{4}}(l^{2}+m^{2})^{2}V_{dd\delta }}$

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. Note that swapping orbital indices amounts to taking ${\displaystyle (l,m,n)\rightarrow (-l,-m,-n)}$, i.e. ${\displaystyle E_{\alpha ,\beta }(l,m,n)=E_{\beta ,\alpha }(-l,-m,-n)}$. For example, ${\displaystyle E_{x,s}=-lV_{sp\sigma }}$.

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In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy 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.

In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. Mathematically, the theorem states

In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918. 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 physics, a Langevin equation is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. One application is to Brownian motion, which models the fluctuating motion of a small particle in a fluid.

The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

Stellar dynamics is the branch of astrophysics which describes in a statistical way the collective motions of stars subject to their mutual gravity. The essential difference from celestial mechanics is that the number of body

In physics, the reciprocal lattice represents the Fourier transform of another lattice. In normal usage, the initial lattice is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space. The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively.

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 mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

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.

Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions in periodic systems. It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system in order to accurately calculate the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.

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.

A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.

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 machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

## References

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3. 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.
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5. 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.
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