Solid 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 $R^m_\ell(\mathbf{r})$, which vanish at the origin and the irregular solid harmonics $I^m_{\ell}(\mathbf{r})$, 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

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

where l² is the square of the nondimensional angular momentum operator,

$\mathbf{\hat l} = -i\, (\mathbf{r} \times \mathbf{\nabla}) .$

It is known that spherical harmonics Y^m_l are eigenfunctions of l²:

$\hat l^2 Y^m_{\ell}\equiv \left[ {\hat l_x}^2 +\hat l^2_y+\hat l^2_z\right]Y^m_{\ell} = \ell(\ell+1) Y^m_{\ell}.$

Substitution of Φ(r) = F(r) Y^m_l 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.

Addition theorems

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

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

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,

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

with $|r| \le |a|\,$. The quantity between pointed brackets is again a Clebsch-Gordan coefficient,

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

References

The addition theorems were proved in different manners by several authors. For example, see the two different proofs in:

• R. J. A. Tough and A. J. Stone, J. Phys. A: Math. Gen. Vol. 10, p. 1261 (1977)
• M. J. Caola, J. Phys. A: Math. Gen. Vol. 11, p. L23 (1978)

Real form

By a simple linear combination of solid harmonics of ±m these functions are transformed into real functions. The real regular solid harmonics, expressed in Cartesian coordinates, are homogeneous polynomials of order l in x, y, z. The explicit form of these polynomials is of some importance. They appear, for example, in the form of spherical atomic orbitals and real multipole moments. The explicit cartesian expression of the real regular harmonics will now be derived.

Linear combination

We write in agreement with the earlier definition

$R_\ell^m(r,\theta,\varphi) = (-1)^{(m+|m|)/2}\; r^\ell \;\Theta_{\ell}^{|m|} (\cos\theta) e^{im\varphi}, \qquad -\ell \le m \le \ell,$

with

$\Theta_{\ell}^m (\cos\theta) \equiv \left[\frac{(\ell-m)!}{(\ell+m)!}\right]^{1/2} \,\sin^m\theta\, \frac{d^m P_\ell(\cos\theta)}{d\cos^m\theta}, \qquad m\ge 0,$

where $P_\ell(\cos\theta)$ is a Legendre polynomial of order l. The m dependent phase is known as the Condon-Shortley phase.

The following expression defines the real regular solid harmonics:

$\begin{pmatrix} C_\ell^{m} \\ S_\ell^{m} \end{pmatrix} \equiv \sqrt{2} \; r^\ell \; \Theta^{m}_\ell \begin{pmatrix} \cos m\varphi\\ \sin m\varphi \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} (-1)^m & \quad 1 \\ -(-1)^m i & \quad i \end{pmatrix} \begin{pmatrix} R_\ell^{m} \\ R_\ell^{-m} \end{pmatrix}, \qquad m > 0.$

and for m = 0:

$C_\ell^{0} \equiv R_\ell^{0} .$

Since the transformation is by a unitary matrix the normalization of the real and the complex solid harmonics is the same.

z-dependent part

Upon writing u = cos θ the mth derivative of the Legendre polynomial can be written as the following expansion in u

$\frac{d^m P_\ell(u)}{du^m} = \sum_{k=0}^{\left \lfloor (\ell-m)/2\right \rfloor} \gamma^{(m)}_{\ell k}\; u^{\ell-2k-m}$

with

$\gamma^{(m)}_{\ell k} = (-1)^k 2^{-\ell} \binom{\ell}{k}\binom{2\ell-2k}{\ell} \frac{(\ell-2k)!}{(\ell-2k-m)!}.$

Since z = r cosθ it follows that this derivative, times an appropriate power of r, is a simple polynomial in z,

$\Pi^m_\ell(z)\equiv r^{\ell-m} \frac{d^m P_\ell(u)}{du^m} = \sum_{k=0}^{\left \lfloor (\ell-m)/2\right \rfloor} \gamma^{(m)}_{\ell k}\; r^{2k}\; z^{\ell-2k-m}.$

(x,y)-dependent part

Consider next, recalling that x = r sinθcosφ and y = r sinθsinφ,

$r^m \sin^m\theta \cos m\varphi = \frac{1}{2} \left[ (r \sin\theta e^{i\varphi})^m + (r \sin\theta e^{-i\varphi})^m \right] = \frac{1}{2} \left[ (x+iy)^m + (x-iy)^m \right]$

Likewise

$r^m \sin^m\theta \sin m\varphi = \frac{1}{2i} \left[ (r \sin\theta e^{i\varphi})^m - (r \sin\theta e^{-i\varphi})^m \right] = \frac{1}{2i} \left[ (x+iy)^m - (x-iy)^m \right].$

Further

$A_m(x,y) \equiv \frac{1}{2} \left[ (x+iy)^m + (x-iy)^m \right]= \sum_{p=0}^m \binom{m}{p} x^p y^{m-p} \cos (m-p) \frac{\pi}{2}$

and

$B_m(x,y) \equiv \frac{1}{2i} \left[ (x+iy)^m - (x-iy)^m \right]= \sum_{p=0}^m \binom{m}{p} x^p y^{m-p} \sin (m-p) \frac{\pi}{2}.$

In total

$C^m_\ell(x,y,z) = \left[\frac{(2-\delta_{m0}) (\ell-m)!}{(\ell+m)!}\right]^{1/2} \Pi^m_{\ell}(z)\;A_m(x,y),\qquad m=0,1, \ldots,\ell$

$S^m_\ell(x,y,z) = \left[\frac{2 (\ell-m)!}{(\ell+m)!}\right]^{1/2} \Pi^m_{\ell}(z)\;B_m(x,y) ,\qquad m=1,2,\ldots,\ell.$

List of lowest functions

We list explicitly the lowest functions up to and including l = 5 . Here $\bar{\Pi}^m_\ell(z) \equiv \left[\tfrac{(2-\delta_{m0}) (\ell-m)!}{(\ell+m)!}\right]^{1/2} \Pi^m_{\ell}(z) .$

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$\begin{align} \bar{\Pi}^0_0 & = 1 & \bar{\Pi}^1_3 & = \frac{1}{4}\sqrt{6}(5z^2-r^2) & \bar{\Pi}^4_4 & = \frac{1}{8}\sqrt{35} \\ \bar{\Pi}^0_1 & = z & \bar{\Pi}^2_3 & = \frac{1}{2}\sqrt{15}\; z & \bar{\Pi}^0_5 & = \frac{1}{8}z(63z^4-70z^2r^2+15r^4) \\ \bar{\Pi}^1_1 & = 1 & \bar{\Pi}^3_3 & = \frac{1}{4}\sqrt{10} & \bar{\Pi}^1_5 & = \frac{1}{8}\sqrt{15} (21z^4-14z^2r^2+r^4) \\ \bar{\Pi}^0_2 & = \frac{1}{2}(3z^2-r^2) & \bar{\Pi}^0_4 & = \frac{1}{8}(35 z^4-30 r^2 z^2 +3r^4 ) & \bar{\Pi}^2_5 & = \frac{1}{4}\sqrt{105}(3z^2-r^2)z \\ \bar{\Pi}^1_2 & = \sqrt{3}z & \bar{\Pi}^1_4 & = \frac{\sqrt{10}}{4} z(7z^2-3r^2) & \bar{\Pi}^3_5 & = \frac{1}{16}\sqrt{70} (9z^2-r^2) \\ \bar{\Pi}^2_2 & = \frac{1}{2}\sqrt{3} & \bar{\Pi}^2_4 & = \frac{1}{4}\sqrt{5}(7z^2-r^2) & \bar{\Pi}^4_5 & = \frac{3}{8}\sqrt{35} z \\ \bar{\Pi}^0_3 & = \frac{1}{2} z(5z^2-3r^2) & \bar{\Pi}^3_4 & = \frac{1}{4}\sqrt{70}\;z & \bar{\Pi}^5_5 & = \frac{3}{16}\sqrt{14} \\ \end{align}$

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The lowest functions $A_m(x,y)\,$ and $B_m(x,y)\,$ are:

m	Am	Bm
0	$1\,$	$0\,$
1	$x\,$	$y\,$
2	$x^2-y^2\,$	$2xy\,$
3	$x^3-3xy^2\,$	$3x^2y -y^3\,$
4	$x^4 - 6x^2 y^2 +y^4\,$	$4x^3y-4xy^3\,$
5	$x^5-10x^3y^2+ 5xy^4\,$	$5x^4y -10x^2y^3+y^5\,$