Electrostatic energy per e- for an alkali metal

[VivaLaFisica]

I'm trying to think about what it is I should be considering before I start throwing up some maths.

I'm thinking to assume that the positively charged core is a point charge +e and the electron is uniformly distributed over a sphere of radius R_s.

The self energy of the electron distribution, which is repulsive, should be included along with the attraction to the positive core, right?

Now how to go about calculating the electron's self energy...?

 

[Kholdstare]

First of all you can assume a pseudopotential (instead of coulomb potential) for combined core and non-valence electrons. That way you don't have to worry about other electrons and repulsive self-energy (as only one electron is present).

You can download various pseudopotentials from the internet.