Expansion of a point-charge potential

[Morberticus]

I'm currently trying to manipulate the expansion

$$\frac{1}{|x_1-x_2|} = \sum_{l=0} \frac{r_{<}^l}{r_{>}^{l+1}}P_{l}\left(cos\left(\gamma\right)\right)$$

Where r< is the lesser of |x1| and |x2| and P_l are the legendre polynomials.

I have included up to 4 terms and the resultant potential isn't resembling the expected potential curve. Does anyone have any experience with such expansions, or would you recommend a different expansion (I ultimately need it to make a function integrable) ?

 

[gabbagabbahey]

I'm currently trying to manipulate the expansion

$$\frac{1}{|x_1-x_2|} = \sum_{l=0} \frac{r_{<}^l}{r_{>}^{l+1}}P_{l}\left(cos\left(\gamma\right)\right)$$

Where r< is the lesser of |x1| and |x2| and P_l are the legendre polynomials.

I have included up to 4 terms and the resultant potential isn't resembling the expected potential curve. Does anyone have any experience with such expansions, or would you recommend a different expansion (I ultimately need it to make a function integrable) ?

What is the resultant potential you are getting, and what were you expecting? What exactly are you trying to make integrable?

 

[alxm]

Here you go:
Carlson and Rushbrooke, "On the expansion of a coulomb potential in spherical harmonics"
Mathematical Proceedings of the Cambridge Philosophical Society (1950), 46: 626-633
doi:10.1017/S0305004100026190

Equations Ia and Ib give you the correct expansion.

 

[Morberticus]

Here you go:
Carlson and Rushbrooke, "On the expansion of a coulomb potential in spherical harmonics"
Mathematical Proceedings of the Cambridge Philosophical Society (1950), 46: 626-633
doi:10.1017/S0305004100026190

Equations Ia and Ib give you the correct expansion.

Hi, thanks. I have implemented Ib and it seems to have made an improvement.

gabbagabbahey: When considering |x1-X|^-1: If X is close to the origin and on the z axis, I get a predictable approximation to |x1-X|^-1 (i.e. If I plot the approximate potential across the singularity, I get a finite peak). However, if X is placed further away from the origin, the potential changes, and a caldera forms. I ultimately am constructing plane-wave molecular integrals, so perhaps this isn't even the best approach.

 

[sardar]

I would recommend exploring other methods for expanding the point-charge potential, such as using Taylor series or Fourier series. These methods may provide a more accurate representation of the potential curve and make it easier to integrate the function. It is also important to carefully consider the terms included in the expansion and their impact on the overall shape of the potential curve. Additionally, consulting with other scientists or experts in the field may provide valuable insights and recommendations for improving the expansion.