# Jellium Model: Finite Confinement & Coulomb Interactions

• Morberticus
In summary, the Jellium model is only suitable for an electron gas of infinite volume. If you confined a gas to a finite volume using an infinite potential well, the infinities in the coulomb interactions between electrons would still be present.
Morberticus
Is the Jellium model only suitable for an electron gas of infinite volume? If I confined a gas to a finite volume using an infinite potential well, is there still a way to cancel out the infinities in the coulomb interactions between electrons?

Sure, it's the basis for the LDA/LSDA. Do you have Parr and Yang's book? It's probably the best for this stuff. (Specifically, Appendix E deals with the uniform electron gas)

alxm said:
Sure, it's the basis for the LDA/LSDA. Do you have Parr and Yang's book? It's probably the best for this stuff. (Specifically, Appendix E deals with the uniform electron gas)

Thanks, will give the book a look. I have a set of coulomb integrals <a b|V(r-r')|c d> where a-d are plane-wave orbitals with periodic boundary conditions, and they are all diverging due to the singularity when r = r'. I am trying to figure out how these infinities are normally dealt with.

This Coulomb integral shouldn't be a problem ... unless I'm missing something, it will be identical to the Fourier transform of the Coulomb potential. While that integral looks to be problematic at a first-go, the trick of inserting a slowly decaying exp(-u*r) term in the limit u going to zero shows the Fourier transform to be 1/k^2. In your case, k will be some linear combination of the wavevectors of your plane-waves a-d.

t!m said:
This Coulomb integral shouldn't be a problem ... unless I'm missing something, it will be identical to the Fourier transform of the Coulomb potential. While that integral looks to be problematic at a first-go, the trick of inserting a slowly decaying exp(-u*r) term in the limit u going to zero shows the Fourier transform to be 1/k^2. In your case, k will be some linear combination of the wavevectors of your plane-waves a-d.

I think what is confusing me is some of the phrases I am finding in books.

I have come across the following hamiltonian a few times.

$$\hat{H} = \sum\frac{ \hat{p}_i^2}{2m} + \frac{1}{L_d }\sum_{q \neq 0} \left[ \hat{n}_{-q}\hat{n}_{q} - \hat{N}\right]$$

But it is always qualified by phrases like "This Hamiltonian is well defined within the thermodynamic limit" [Quantum theory of the electron liquid By Gabriele Giuliani, Giovanni Vignale]. Which implies it is not valid for small volumes. My volume will be on the order of 10-1000 nm^3 with only a handful of electrons so not sure if it's valid.

## 1. What is the Jellium Model?

The Jellium Model is a simplified theoretical model used to describe the behavior of electrons in a metal. It assumes that the positively charged atomic nuclei are uniformly distributed in a sea of electrons, similar to a jellium or jelly-like substance.

## 2. How does finite confinement affect the Jellium Model?

In the Jellium Model, finite confinement refers to the limitation of the space in which the electrons are allowed to move. This can greatly affect the behavior of the electrons, leading to quantization of energy levels and changes in the electron density distribution. These effects can be observed in small metallic particles or clusters.

## 3. What role do Coulomb interactions play in the Jellium Model?

Coulomb interactions refer to the electrostatic repulsion between particles with opposite charges. In the Jellium Model, these interactions between the negatively charged electrons and positively charged nuclei are considered to be the dominant force that determines the behavior and properties of the system.

## 4. How does the Jellium Model explain the properties of metals?

The Jellium Model provides a simplified, but accurate, explanation for the properties of metals such as their high electrical and thermal conductivity. It also explains the phenomenon of screening, where the negatively charged electrons shield the positively charged nuclei from each other, reducing the strength of the Coulomb interactions.

## 5. What are the limitations of the Jellium Model?

The Jellium Model is a simplified theoretical model and does not take into account the complexities of real-world metal systems. It does not consider the effects of electron-electron interactions, lattice vibrations, or impurities, which can all play significant roles in the behavior of metals. Therefore, while the Jellium Model provides valuable insights, it is not always accurate in predicting the properties of real materials.

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