Implementing simple atom model using density functional theory (DFT)

Question

I am trying to write computer code which will find the energy and density function for a atom with Z protons and N electrons. I am working in 1D for simplicity and would like to make the overall code as simple as possible (I am using Hartree units aswell). I am using also using the Hartree approximation. This is my overall algorithm:

1. Finds hydrogen like wave-functions (i.e. no electron-election interaction)

2. Works out density function via $n(z) = \sum_i \left | \phi_i(z) \right |^2$

3. Works out Hartree potential $ V_H(r) = \int_0^\infty \frac{n(\vec{s})}{\left | \vec{r}-\vec{s} \right |} d\vec{s} $

4. Uses Kohn-Sham equation to find new individual wavefuncitons $\left (-\tfrac{1}{2}\bigtriangledown^2 + \frac{Z}{\left |r \right |} + V_H(\vec{r}) + V_{EX}(\vec{r}) \right )\phi_i = \varepsilon_i \phi_i$

5. Go back to step 2. and repeat

I am having a number of difficulty and would be very grateful for some help with any of my questions.

Firstly to compute the Hartree potential I need to turn it into a 1D problem, so we assume even spherical distribution of charge:

$ V_H(r) = \int_0^\infty 4\pi s^2 \frac{ n(s)}{\left |r-s\right |} ds $

However this blows up when r = s! I have been looking into Gauss's law but had no luck.

Additionally, I am unsure how to compute the exchange-correlation term. I am trying to find a function which I can easily implement.

Any help would be great.

Answer 1

The Hartree potential is typically calculated not through the integral you're giving, but by solving Poisson's equation. Look here or there for details about this calculation applied to DFT, and here for what I found to be a good lower-level introduction to Poisson solvers in general. Good luck for your quest, it's really a nontrivial thing to do!

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As an aside: what basis set are you going to employ?