# 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) ?

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#### gabbagabbahey

Homework Helper
Gold Member
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.