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:Finds hydrogen like wave-functions (i.e. no electron-election interaction)Works out density function via$n(z) = \sum_i \left | \phi_i(z)  \right |^2$Works out Hartree potential$ V_H(r) = \int_0^\infty \frac{n(\vec{s})}{\left | \vec{r}-\vec{s} \right |} d\vec{s} $Uses Kohn-Sham equation to find new individual wavefuncitons$\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$Go back to step 2. and repeatI am having a number of difficulties and would be very grateful for some help with any of my questions.Firstly to compute the Hartree potential I need to turn it into a 1D problem, so we assume even spherical distribution of charge:$ 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.Any help would be great.
Implementing simple atom model using density functional theory (DFT)
computational chemistry;hartree fock;density functional theory
Use the Poisson equation in its differential form to obtain the Hartree potential. It has a nice form in spherical coordinates. See for example the seminal paper of Becke, in which this problem is solved for the general polyatomic case without relying on spherical symmetry. 