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Tight binding model (Read 2562 times)
Gerrit-Jan Linker
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Tight binding model
30.05.10 at 21:59:57
Tight binding model
The tight binding model (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 linear combination of atomic orbitals molecular orbital method used for molecules.  
Tight binding calculates the ground state electronic energy and position of bandgaps for a molecule.
Atomic orbitals φm( r ), which are eigenfunctions of the atomic Hamiltonian Hat 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 ΔU required to obtain the true Hamiltonian H of the system, are assumed small:
    H(r) = ∑Rn Hat(r + Rn) +  ∆U(r)  
H(r) is the crystal Hamiltonian in dependence of the position vector r
The sum runs over the translation vector Rn
Hat(r + Rn) is the atomic hamiltonian, dependent on the position vector and the translation vector.
∆U(r) is the deviation from the atomic potential to obtain the true Hamiltonian of the system.
A solution ψ(r) to the time-independent single electron Schrödinger equation is then approximated as a linear combination of atomic orbitals φm( r − Rn ):
    ψ(r) = ∑m,Rn bm(Rnm(r-Rn)  
where m refers to the m-th atomic energy level and Rn locates an atomic site in the crystal lattice.
The opposite extreme to the tight binding model is the nearly-free electron approximation: Bloch's theorem.
See also:
Bloch's theorem
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« Last Edit: 16.01.12 at 13:24:39 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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