# Computing a “best-fit” of discrete points from a multipole expansion, i.e. invert the multipole moments

Take a field $\phi(\bf{x})$ created from a charge distribution contained within a radius $R$. The multipole expansion in spherical harmonics $Y_{\ell,m}$ outside of $R$ is approximated by:

$$\phi({\bf x}) \approx \frac{1}{4\pi \epsilon_0} \sum_{\ell=0}^{\ell_{MAX}} \sum_{m=-\ell}^{\ell} \frac{4\pi}{2\ell +1} \alpha_{\ell, m} \frac{Y_{\ell,m}(\theta, \phi)}{r^{\ell +1}}$$

Given only the finite set of the multipole moments $\alpha_{\ell,m}$, is it possible to find a charge distribution of $N$ discrete charges $(q_i, r_i, \theta_i, \phi_i)$ that "best-fit" this potential? Right now I'm using a simple minima finder over the $3^N$ variables (see note 1), but I'm wondering if there is any prior work/observations on inverting the multipole moments.

Note 1

If the positions of the charges are fixed, the charge magnitudes $q_i$ are simply linear combinations:

$$\alpha_{\ell, m} = \sum_i^N q_i Y^*_{\ell,m}(\theta_i, \phi_i) r_i^\ell$$

thus simple linear algebra gives the best-fit charge magnitudes.

Note 2

It's easy to see that this charge distribution need not be unique, indeed if $\ell_{MAX}=0$ any combination such that $\sum_i^N q_i \propto \alpha_{0,0}$ should work. This is ok, I'm more concerned with finding good approximations than the absolute best fit.

Note 3

If the charge magnitudes are real (which is a given for a physical problem), the number of unique moments are reduced by roughly a factor of 2 since:

$$\alpha_{\ell,-m} = (-1)^m \alpha_{\ell,m}^*$$

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It's a nice problem but the solution is indeed very non-unique - for example, you may scale the distances from the origin $r$ and the charges in various inverse ways, and rotate the four $+-+-$ charges defining a quadrupole around the axis, and do many other things. And because the solution isn't unique, it seems unlikely that there is a "canonical formula" of any sort. – Luboš Motl Nov 2 '12 at 20:37
@LubošMotl I agree, but the problem is motivated by a practical concern, making (pseudo) accurate "toy models" of electrostatics for protein solutions. As such any thoughts on the matter, canonical or not, would suffice as solutions in this case. I would expect that the symmetry of the problem should suggest something better than brute force optimization. – Hooked Nov 2 '12 at 21:55

## 1 Answer

This problem has a unique solution if you only allow charges of one sign; it is known as the moment problem and is one of the central problems of measure theory. The wikipedia article on it should provide a good starting point for reading on it.

However, as Luboš points out, for a signed measure the moment problem is usually indeterminate. One way to phrase this is that there is a big set of charge distributions for which all moments vanish. (This includes, for instance, all bounded charge distributions with a conducting shell around them.) I don't know of results for finding solutions in the indeterminate case but phrasing the problem in these terms might help.

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