Theory

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 \(Y_{lm}(\theta, \varphi)\).