kelly0303
- 573
- 33
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!
$$\int_{r_e=0}^{r_e=\infty}\int_{r_n=0}^{r_n=R}\frac{\rho(r_n)}{|r_e-r_n|}d^3r_ed^3r_n$$
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!