Tight binding - evaluate integral

In summary, the conversation discusses the energy of an electron within a band, given by the tight-binding expression in one dimension. The participants also consider typical expressions for integrals \alpha and \gamma, and evaluate the integral \gamma for a specific wavefunction, assuming it is an eigenstate of the Hamiltonian. They also discuss the energy of an electron in a linear chain of atoms with a spacing a and make a graph for different cases. The conversation ends with a question about the variables \phi_{m}, \phi_{n}, and \rho, and the importance of \rho in the given scenario.
  • #1
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Homework Statement



The energy of an electron within a band as a function of its wavevector is given by the
tight-binding expression (in one dimension),

E(k)=-[itex]\alpha[/itex]-[itex]\gamma[/itex][itex]\Sigma[/itex][itex]_{m}[/itex] exp (-ik[itex]\rho[/itex][itex]_{m}[/itex])

(a)What are typical expressions for integrals [itex]\alpha[/itex] and [itex]\gamma[/itex]?
(b) Evaluate the integral [itex]\gamma[/itex] for the following wavefunction, assuming it is an eigenstate of the Hamiltonian, being careful to distinguish the cases 2x[itex]_{0}[/itex][itex]\leq[/itex][itex]\rho[/itex] and 2x[itex]_{0}[/itex]>[itex]\rho[/itex]:

[itex]\phi[/itex](x)=[itex]\sqrt{\frac{1}{2x_{0}}}[/itex] |x|[itex]\leq[/itex]x[itex]_{0}[/itex]

[itex]\phi[/itex](x)=0 |x|>x[itex]_{0}[/itex]

(c) Hence evaluate the energy of an electron in a linear chain of these atoms with a
spacing a and make a graph of the result for the two cases 2x0 a and 2x0 > a.

The Attempt at a Solution



[itex]\alpha[/itex]=-<[itex]\phi[/itex][itex]_{n}[/itex]|H|[itex]\phi[/itex][itex]_{n}[/itex]>
[itex]\gamma[/itex]=-<[itex]\phi[/itex][itex]_{m}[/itex]|H|[itex]\phi[/itex][itex]_{n}[/itex]>

But I cannot do part b) because I do not know what [itex]\phi[/itex][itex]_{m}[/itex] and [itex]\phi[/itex][itex]_{n}[/itex] are. All that I know is that sometimes n=m and sometimes it does not. Please help.
 
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  • #2
What is [itex]\rho[/itex]? Why does it matter whether [itex]\rho[/itex] is bigger or smaller than x[itex]_{0}[/itex]?
 

Related to Tight binding - evaluate integral

What is tight binding?

Tight binding is a method used in quantum mechanics to calculate the electronic structure of a material. It involves approximating the wavefunction of an electron as a linear combination of atomic orbitals, and then solving for the energy levels and wavefunctions of the entire material.

Why is tight binding used?

Tight binding is used because it is a relatively simple and efficient way to calculate the electronic structure of a material. It can provide insight into the electronic properties of a material, such as its band structure and density of states, without requiring extensive computational resources.

What is the integral in tight binding?

The integral in tight binding refers to the mathematical process of evaluating the overlap between atomic orbitals in a material. This overlap determines the strength of the bonding between atoms, and is used to calculate the energy levels and wavefunctions of the material.

How is the integral evaluated in tight binding?

The integral in tight binding is typically evaluated using a numerical method, such as the extended Hückel method or the orthogonalized plane wave method. These methods involve solving a set of equations to determine the coefficients of the atomic orbitals, which are then used to calculate the overlap integral.

What are the limitations of tight binding?

Tight binding is a simplification of the more accurate ab initio methods used in quantum mechanics. Therefore, it is limited in its ability to accurately predict the electronic properties of a material, especially in complex systems. It also cannot account for the effect of electron-electron interactions, which can be significant in some materials.

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