Jellium Model: Finite Confinement & Coulomb Interactions

[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?

 

[alxm]

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)

 

[Morberticus]

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.

 

[t!m]

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.

 

[Morberticus]

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.