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In the tight binding model, it is assumed
that the full Hamiltonian <math>H</math> of the system may be approximated by the
Hamiltonian of an isolated atom centred at each lattice point. The
atomic orbitals <math>\psi_n</math>, which are eigenfunctions of the single atom
Hamiltonian <math>H_{at}</math>, are assumed to be very small at distances exceeding the
lattice constant. This is what is meant by tight-binding. It is
further assumed that any corrections to the atomic potential <math>\Delta U</math>, which are
required to obtain the full Hamiltonian <math>H</math> of the system, are appreciable
only when the atomic orbitals are small. The solution to the
time-independent single electron Schrödinger equation <math>\phi</math> is then
assumed to be a linear combination of atomic orbitals

<math>\phi(\vec{r}) = \sum_n b_n \psi_n(\vec{r})</math>.

This leads to a matrix equation for the coefficients <math>b_n</math> and Bloch energies <math>\varepsilon</math> of the form

<math>\varepsilon(\vec{k}) = E_m - {\beta_m + \sum_{\vec{R}\neq 0} \gamma_m(\vec{R}) e^{i \vec{k} \cdot \vec{R}}\over b_m + \sum_{\vec{R}\neq 0} \alpha_m(\vec{R}) e^{i \vec{k} \cdot \vec{R}}}</math>,

where <math>E_m</math> is the energy of the <math>m</math>th atomic level,

<math> \beta_m = -\int \psi_m^*(\vec{r})\Delta U(\vec{r}) \phi(\vec{r}) d\vec{r}</math>,

<math> \alpha_m(\vec{R}) = \int \psi_m^*(\vec{r}) \phi(\vec{r}-\vec{R}) d\vec{r}</math>,

and

<math> \gamma_m(\vec{R}) = -\int \psi_m^*(\vec{r}) \Delta U(\vec{r}) \phi(\vec{r}-\vec{R}) d\vec{r}</math>

are the overlap integrals.

The tight binding model is typically used for calculations of electronic band structure and energy 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.

References

• J.C. Slater and G.F. Koster, Phys. Rev. 94, 1498 (1954).
• C.M. Goringe, D.R. Bowler and E. Hernández, Rep. Prog. Phys. 60, 1447 (1997).
• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, Toronto, 1976).