Solid spherical harmonics

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates. There are two kinds: the regular solid harmonics $R^m_\ell\left(\mathbf\left\{r\right\}\right)$, which vanish at the origin and the irregular solid harmonics $I^m_\left\{\ell\right\}\left(\mathbf\left\{r\right\}\right)$, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:



R^m_{\ell}(\mathbf{r}) \equiv \sqrt{\frac{4\pi}{2\ell+1}}\; r^\ell Y^m_{\ell}(\theta,\varphi)



I^m_{\ell}(\mathbf{r}) \equiv \sqrt{\frac{4\pi}{2\ell+1}} \; \frac{ Y^m_{\ell}(\theta,\varphi)}{r^{\ell+1}}

Derivation, relation to spherical harmonics

Introducing r, θ, and φ for the spherical polar coordinates of the 3-vector r, we can write the Laplace equation in the following form

$\nabla^2\Phi\left(\mathbf\left\{r\right\}\right) = \left\left(\frac\left\{1\right\}\left\{r\right\} \frac\left\{\partial^2\right\}\left\{\partial r^2\right\}r - \frac\left\{L^2\right\}\left\{\hbar^2 r^2\right\}\right\right)\Phi\left(\mathbf\left\{r\right\}\right) = 0 , \qquad \mathbf\left\{r\right\} \ne \mathbf\left\{0\right\},$

where L2 is the square of the angular momentum operator,

$\mathbf\left\{L\right\} = -i\hbar\, \left(\mathbf\left\{r\right\} \times \mathbf\left\{\nabla\right\}\right) .$

It is known that spherical harmonics Yml are eigenfunctions of L2:



L^2 Y^m_{\ell}\equiv \left[ L^2_x +L^2_y+L^2_z\right]Y^m_{\ell} = \hbar^2 \ell(\ell+1) Y^m_{\ell}.

Substitution of Φ(r) = F(r) Yml into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution,



\frac{1}{r}\frac{\partial^2}{\partial r^2}r F(r) = \frac{\ell(\ell+1)}{r^2} F(r) \Longrightarrow F(r) = A r^\ell + B r^{-\ell-1}.

The particular solutions of the total Laplace equation are regular solid harmonics:



R^m_{\ell}(\mathbf{r}) \equiv \sqrt{\frac{4\pi}{2\ell+1}}\; r^\ell Y^m_{\ell}(\theta,\varphi), and irregular solid harmonics:



I^m_{\ell}(\mathbf{r}) \equiv \sqrt{\frac{4\pi}{2\ell+1}} \; \frac{ Y^m_{\ell}(\theta,\varphi)}{r^{\ell+1}} .

Racah's normalization

Racah's normalization (also known as Schmidt's semi-normalization) is applied to both functions



\int_{0}^{\pi}\sin\theta\, d\theta \int_0^{2\pi} d\varphi\; R^m_{\ell}(\mathbf{r})^*\; R^m_{\ell}(\mathbf{r}) = \frac{4\pi}{2\ell+1} r^{2\ell} (and analogously for the irregular solid harmonic) instead of normalization to unity. This is convenient because in many applications the Racah normalization factor appears unchanged throughout the derivations.

The translation of the regular solid harmonic gives a finite expansion,

$R^m_\ell\left(\mathbf\left\{r\right\}+\mathbf\left\{a\right\}\right) = \sum_\left\{\lambda=0\right\}^\ell\binom\left\{2\ell\right\}\left\{2\lambda\right\}^\left\{1/2\right\} \sum_\left\{\mu=-\lambda\right\}^\lambda R^\mu_\left\{\lambda\right\}\left(\mathbf\left\{r\right\}\right) R^\left\{m-\mu\right\}_\left\{\ell-\lambda\right\}\left(\mathbf\left\{a\right\}\right)\;$

\langle \lambda, \mu; \ell-\lambda, m-\mu| \ell m \rangle, where the Clebsch-Gordan coefficient is given by



\langle \lambda, \mu; \ell-\lambda, m-\mu| \ell m \rangle = \binom{\ell+m}{\lambda+\mu}^{1/2} \binom{\ell-m}{\lambda-\mu}^{1/2} \binom{2\ell}{2\lambda}^{-1/2}.

The similar expansion for irregular solid harmonics gives an infinite series,

$I^m_\ell\left(\mathbf\left\{r\right\}+\mathbf\left\{a\right\}\right) = \sum_\left\{\lambda=0\right\}^\infty\binom\left\{2\ell+2\lambda+1\right\}\left\{2\lambda\right\}^\left\{1/2\right\} \sum_\left\{\mu=-\lambda\right\}^\lambda R^\mu_\left\{\lambda\right\}\left(\mathbf\left\{r\right\}\right) I^\left\{m-\mu\right\}_\left\{\ell+\lambda\right\}\left(\mathbf\left\{a\right\}\right)\;$

\langle \lambda, \mu; \ell+\lambda, m-\mu| \ell m \rangle with $|r| \le |a|\,$. The quantity between pointed brackets is again a Clebsch-Gordan coefficient,



\langle \lambda, \mu; \ell+\lambda, m-\mu| \ell m \rangle = (-1)^{\lambda+\mu}\binom{\ell+\lambda-m+\mu}{\lambda+\mu}^{1/2} \binom{\ell+\lambda+m-\mu}{\lambda-\mu}^{1/2} \binom{2\ell+2\lambda+1}{2\lambda}^{-1/2}.