# Tight Binding Hamiltonian and Potential (U)

• Arya_
In summary, the question is whether the potential values (U) in the tight binding hamiltonian for a 3d solid can be directly plugged in from a potential profile in the x direction (U1, U2, U3...so on) or if any transformations are needed. The answer is that the potential function meets the superposition principle and the tight binding hamiltonian is an empirical effective hamiltonian that takes into account contributions from both the lowest eigenstates and excited atomic states, so there is no direct correspondence between the potential in the full hamiltonian and the U's in the tight binding approximation.
Arya_
Hi All,

Greetings !

Here is what I wish to know. Specifying a tight binding hamiltonian requires values of potential (U). Consider a 3d solid. If I have potential profile in x direction (U1, U2, U3...so on) can I directly plug in these U values into the tight binding hamiltonian or do I need to do some transformation (like change of space etc) before I can plug in 'potential values vs X' into Hamiltonian.

Thanks,
-Arya

Yes,I think you can directly plug in these V(x) values into the tight binding hamiltonian.Because potential function meets the superposition principle.

The tight binding hamiltonian is an empirical effective hamiltonian which is parametrized in such a way as to give correct energies for the lowest eigenstates of the hamiltonian. Excited atomic states also make a contribution, also to the potential. So there is no 1 to 1 correspondence between some potential in the full hamiltonian and the U's appearing in the tight binding approximation.

## 1. What is a Tight Binding Hamiltonian?

A Tight Binding Hamiltonian is a mathematical model used in solid state physics to describe the electronic structure of a material. It takes into account the interactions between the electrons in the material and the potential energy of the atoms that make up the material.

## 2. How is the Tight Binding Hamiltonian derived?

The Tight Binding Hamiltonian is derived from the Schrödinger equation, which describes the behavior of quantum particles. It involves approximating the wave function of the material's electrons using a linear combination of atomic orbitals. This allows for a more simplified and computationally efficient model of the electronic structure.

## 3. What is the significance of the potential energy term (U) in the Tight Binding Hamiltonian?

The potential energy term (U) in the Tight Binding Hamiltonian represents the interactions between the electrons and the atoms in the material. It takes into account the Coulomb interactions between the electrons and the nuclei, as well as the overlap between the atomic orbitals. This term is crucial in determining the electronic properties of the material.

## 4. How is the Tight Binding Hamiltonian used to calculate the electronic band structure of a material?

The Tight Binding Hamiltonian is used to calculate the electronic band structure of a material by solving the Schrödinger equation for the material's electrons. This involves diagonalizing the Hamiltonian matrix, which results in a set of energy values and corresponding wave functions. These energy values correspond to the electronic band structure of the material, which describes the allowed energy levels for the electrons in the material.

## 5. What are the limitations of the Tight Binding Hamiltonian?

The Tight Binding Hamiltonian is a simplified model and therefore has its limitations. It does not take into account all of the possible interactions between the electrons, such as electron-electron interactions or spin-orbit coupling. It also assumes a complete overlap between atomic orbitals, which may not always be the case. Additionally, it may not accurately describe materials with strong correlations between the electrons, such as in transition metal compounds.

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