# Tight binding - evaluate integral

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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.
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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)=-$\alpha$-$\gamma$$\Sigma$$_{m}$ exp (-ik$\rho$$_{m}$)

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

$\phi$(x)=$\sqrt{\frac{1}{2x_{0}}}$ |x|$\leq$x$_{0}$

$\phi$(x)=0 |x|>x$_{0}$

(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

$\alpha$=-<$\phi$$_{n}$|H|$\phi$$_{n}$>
$\gamma$=-<$\phi$$_{m}$|H|$\phi$$_{n}$>

But I cannot do part b) because I do not know what $\phi$$_{m}$ and $\phi$$_{n}$ are. All that I know is that sometimes n=m and sometimes it does not. Please help.

What is $\rho$? Why does it matter whether $\rho$ is bigger or smaller than x$_{0}$?

## 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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