Tight binding (physics)

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For the biological process, see Tight binding (biology).
Electronic structure methods
v • d • e
Tight binding
Hartree-Fock
Configuration interaction
Coupled cluster
Multi-configurational self-consistent field
Density functional theory
Quantum Monte Carlo
Density matrix renormalization group


In the tight binding model, it is assumed that the full Hamiltonian H of the system may be approximated by the Hamiltonian of an isolated atom centred at each lattice point. The atomic orbitals ψn, which are eigenfunctions of the single atom Hamiltonian Hat, 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 ΔU, which are required to obtain the full Hamiltonian H of the system, are appreciable only when the atomic orbitals are small. The solution to the time-independent single electron Schrödinger equation φ is then assumed to be a linear combination of atomic orbitals

\phi(\vec{r}) = \sum_n b_n \psi_n(\vec{r}).

This leads to a matrix equation for the coefficients bn and Bloch energies \varepsilon of the form

\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}}},

where Em is the energy of the mth atomic level,

\beta_m = -\int \psi_m^*(\vec{r})\Delta U(\vec{r}) \phi(\vec{r}) d\vec{r},
\alpha_m(\vec{R}) = \int \psi_m^*(\vec{r}) \phi(\vec{r}-\vec{R}) d\vec{r},

and

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

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.

[edit] 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).
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