multipoles is a Python package for multipole expansions of the solutions of the Poisson equation (e.g. electrostatic or gravitational potentials). It can handle discrete and continuous charge or mass distributions.

Simply use pip:

pip install --upgrade multipoles 

The documentation is available here.

For a given function $\rho(x,y,z)$, the solution $\Phi(x,y,z)$ of the Poisson equation $\nabla^2\Phi=-4\pi \rho$ with vanishing Dirichlet boundary conditions at infinity is

$$\Phi(x,y,z)=\int d^3r'\frac{\rho(r')}{|r-r'|}$$

Examples of this are the electrostatic and Newtonian gravitational potential. If you need to evaluate $\Phi(x,y,z)$ at many points, calculating the integral for each point is computationally expensive. As a faster alternative, we can express $\Phi(x,y,z)$ in terms of the multipole moments $q_{lm}$ or $I_ {lm}$ (note some literature uses the subscripts $(\cdot)_{nm}$):

$$\Phi(x,y,z)=\sum_{l=0}^\infty\underbrace{\sqrt{\frac{4\pi}{2l+1}}\sum_{m=-l}^lY_{lm}(\theta, \varphi)\frac{q_ {lm}}{r^{l+1}}}_{\Phi^{(l)}}$$

for a exterior expansion, or

$$\Phi(x,y,z)=\sum_{l=0}^\infty\underbrace{\sqrt{\frac{4\pi}{2l+1}}\sum_{m=-l}^lY_{lm}(\theta, \varphi)I_{lm}r^{l}}_ {\Phi^{(l)}}$$

for an interior expansion; where $r, \theta, \varphi$ are the usual spherical coordinates corresponding to the cartesian coordinates $x, y, z$ and $Y_{lm}(\theta, \varphi)$ are the spherical harmonics.

The multipole moments for the exterior expansion are:

$$q_{lm} = \sqrt{\frac{4\pi}{2l+1}}\int d^3 r' \rho(r')r'^l Y^*_{lm}(\theta', \varphi')$$

and the multipole moments for the interior expansion are:

$$I_{lm} = \sqrt{\frac{4\pi}{2l+1}}\int d^3 r' \frac{\rho(r')}{r'^{l+1}} Y^*_{lm}(\theta', \varphi')$$

This approach is usually much faster because the contributions $\Phi^{(l)}$ are getting smaller with increasing l. So we just have to calculate a few integrals for obtaining some $q_{lm}$ or $I_{lm}$.

Some literature considers the $\sqrt{\frac{4\pi}{2l+1}}$ as part of the definition of $Y_{lm}(\theta, \varphi)$.

As example for a discrete charge distribution we model two point charges with positive and negative unit charge located on the z-axis:

from multipoles import MultipoleExpansion # Prepare the charge distribution dict for the MultipoleExpansion object: charge_dist = { 'discrete': True, # point charges are discrete charge distributions 'charges': [ {'q': 1, 'xyz': (0, 0, 1)}, {'q': -1, 'xyz': (0, 0, -1)}, ] } l_max = 2 # where to stop the infinite multipole sum; here we expand up to the quadrupole (l=2) Phi = MultipoleExpansion(charge_dist, l_max) # We can evaluate the multipole expanded potential at a given point like this: x, y, z = 30.5, 30.6, 30.7 value = Phi(x, y, z) # The multipole moments are stored in a dict, where the keys are (l, m) and the values q_lm: Phi.multipole_moments

As an example for a continuous charge distribution, we smear out the point charges from the previous example:

from multipoles import MultipoleExpansion import numpy as np # First we set up our grid, a cube of length 10 centered at the origin: npoints = 101 edge = 10 x, y, z = [np.linspace(-edge / 2., edge / 2., npoints)] * 3 XYZ = np.meshgrid(x, y, z, indexing='ij') # We model our smeared out charges as gaussian functions: def gaussian(XYZ, xyz0, sigma): g = np.ones_like(XYZ[0]) for k in range(3): g *= np.exp(-(XYZ[k] - xyz0[k]) ** 2 / sigma ** 2) g *= (sigma ** 2 * np.pi) ** -1.5 return g sigma = 1.5 # the width of our gaussians # Initialize the charge density rho, which is a 3D numpy array: rho = gaussian(XYZ, (0, 0, 1), sigma) - gaussian(XYZ, (0, 0, -1), sigma) # Prepare the charge distribution dict for the MultipoleExpansion object: charge_dist = { 'discrete': False, # we have a continuous charge distribution here 'rho': rho, 'xyz': XYZ } # The rest is the same as for the discrete case: l_max = 2 # where to stop the infinite multipole sum; here we expand up to the quadrupole (l=2) Phi = MultipoleExpansion(charge_dist, l_max) x, y, z = 30.5, 30.6, 30.7 value = Phi(x, y, z)