Graduate Electrostatic interaction inside and outside the source

[kelly0303]

Hello! I want to get the electrostatic interaction (between and electron and a nucleus), while accounting for the fact that the electron can also be inside the nucleus (e.g. in an S$_{1/2}$ state). I ended up with this double integral:

$$\int_{r_e=0}^{r_e=\infty}\int_{r_n=0}^{r_n=R}\frac{\rho(r_n)}{|r_e-r_n|}d^3r_ed^3r_n$$

where $r_e$ and $r_n$ are the electron and nuclear coordinates and $R$ is the nuclear radius. Please note that we are not necessarily assuming that the nucleus is a perfect sphere (although it is usually very close to it). How can I expand the $\frac{1}{|r_e-r_n|}$ and get this into a simpler form that I can also truncate as needed? Thank you!

 

[Meir Achuz]

That function is expanded in Legendre polynomials as
$$\frac{1}{|{\bf r}-{\bf r'}|}=\sum_l\frac{r'^l}{r^{l+1}}P_l(\cos\theta)$$,
with the smaller r in the numerator.