# Unusual coulomb potential

1. Sep 25, 2010

### Morberticus

I have seen the fourier transform of the coulomb potential quite often.

However, I have come across a sum expression for an electrostatic potential

$$V_{cb}(r-r') = \frac{1}{V}\sum_{q \neq 0} \frac{4\pi}{q^2}e^{iq(r-r')}$$

It is equation (2.6) here: http://people.web.psi.ch/mudry/FALL01/lecture03.pdf

I have assumed this is the coulomb integral. Is it? Has anyone come across such an expression before? Is it valid to take such an expansion for the coulomb integral in a box of finite volume, under the Jellium model?

2. Sep 27, 2010

### DrDu

Yes, it's a Fourier representation of the Coulomb interaction. Note that excluding the term with q=0 is due a subtraction of the energy of the interaction with the homogeneous positive background.
I think the main problem in solving your jellium in a box model is that the single particle wavefunctions are now of the form sin and cos and have an explicit dependence on x, while the eigenfunctions with periodic boundary conditions depend on x only via a rather trivial phase factor. I.e. you loose translation symmetry, which will probably kill you.
As a poster in your other thread already remarked, you probably will have to resort to semi-empirical methods like LDA density functional theory.

3. Sep 27, 2010

### Morberticus

Thanks for the confirmation.

If I used periodic boundary conditions (which I am guessing would correspond to an 'open' box), would I still be integrating over just the volume of the box for <ab|V|cd>? With periodic functions, if I integrate over infinity, I stumble across products of dirac functions and dirac-orthogonality stops me. If I integrate over the box, I get a sum of well-behaved functions that look like sinc functions that are simple to work with.

So if I were to, say, interpret the box as an 'open' box, can I treat the periodic functions as usual (i.e. Normalise them over the volume of the box and build coulomb integrals by integrating over just the box)?

If I can't, I have written a numerical program that generates the integrals using sin and cos functions, and a finite number of terms for the coulomb representation (from -100 to 100) and certainly kills me in terms of computational time (but it is easily parallelised so I'm not too worried about that). When you said it will kill me, do you mean it in another way? If the physics is sound, I can live with inefficiency.

4. Sep 27, 2010

### DrDu

What do you want to calculate, exactly?

5. Sep 27, 2010

### Morberticus

Essentially the ground-state energy of N electrons in a box.