In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions ${\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }$. There are two kinds: the regular solid harmonics ${\displaystyle R_{\ell }^{m}(\mathbf {r} )}$, which are well-defined at the origin and the irregular solid harmonics ${\displaystyle I_{\ell }^{m}(\mathbf {r} )}$, 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:

$${\displaystyle R_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1))}\;r^{\ell }Y_{\ell }^{m}(\theta ,\varphi )}$$

$${\displaystyle I_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1))}\;{\frac {Y_{\ell }^{m}(\theta ,\varphi )}{r^{\ell +1))))$$

Derivation, relation to spherical harmonics

Introducing r, θ, and φ for the spherical polar coordinates of the 3-vector r, and assuming that ${\displaystyle \Phi }$ is a (smooth) function ${\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }$, we can write the Laplace equation in the following form

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

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

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

$${\displaystyle {\hat {l))^{2}Y_{\ell }^{m}\equiv \left[((\hat {l))_{x))^{2}+{\hat {l))_{y}^{2}+{\hat {l))_{z}^{2}\right]Y_{\ell }^{m}=\ell (\ell +1)Y_{\ell }^{m}.}$$

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,

$${\displaystyle {\frac {1}{r)){\frac {\partial ^{2)){\partial r^{2))}rF(r)={\frac {\ell (\ell +1)}{r^{2))}F(r)\Longrightarrow F(r)=Ar^{\ell }+Br^{-\ell -1}.}$$

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

$${\displaystyle R_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1))}\;r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),}$$

$${\displaystyle I_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1))}\;{\frac {Y_{\ell }^{m}(\theta ,\varphi )}{r^{\ell +1))}.}$$

Racah's normalization

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

$${\displaystyle \int _{0}^{\pi }\sin \theta \,d\theta \int _{0}^{2\pi }d\varphi \;R_{\ell }^{m}(\mathbf {r} )^{*}\;R_{\ell }^{m}(\mathbf {r} )={\frac {4\pi }{2\ell +1))r^{2\ell ))$$

Addition theorems

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

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

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

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

$${\displaystyle I_{\ell }^{m}(\mathbf {r} +\mathbf {a} )=\sum _{\lambda =0}^{\infty }{\binom {2\ell +2\lambda +1}{2\lambda ))^{1/2}\sum _{\mu =-\lambda }^{\lambda }R_{\lambda }^{\mu }(\mathbf {r} )I_{\ell +\lambda }^{m-\mu }(\mathbf {a} )\;\langle \lambda ,\mu ;\ell +\lambda ,m-\mu |\ell m\rangle }$$

${\displaystyle |r|\leq |a|\,}$

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

The addition theorems were proved in different manners by several authors.^[1][2]

Complex form

The regular solid harmonics are homogeneous, polynomial solutions to the Laplace equation ${\displaystyle \Delta R=0}$. Separating the indeterminate ${\displaystyle z}$ and writing ${\textstyle R=\sum _{a}p_{a}(x,y)z^{a))$, the Laplace equation is easily seen to be equivalent to the recursion formula

$${\displaystyle p_{a+2}={\frac {-\left(\partial _{x}^{2}+\partial _{y}^{2}\right)p_{a)){\left(a+2\right)\left(a+1\right)))}$$

${\displaystyle p_{0}(x,y)}$

${\displaystyle \ell }$

${\displaystyle p_{1}(x,y)}$

${\displaystyle \ell -1}$

${\displaystyle k}$

${\displaystyle \left\{(x^{2}+y^{2})^{m}(x\pm iy)^{k-2m}\mid 0\leq m\leq k/2\right\))$

${\displaystyle SO(2)}$

${\displaystyle \rho _{\alpha ))$

${\displaystyle \alpha \in [0,2\pi ]}$

${\displaystyle e^{\pm i(k-2m)\alpha ))$

${\displaystyle (x^{2}+y^{2})^{m}(x+iy)^{k-2m))$

If we combine the degree ${\displaystyle \ell }$ basis and the degree ${\displaystyle \ell -1}$ basis with the recursion formula, we obtain a basis of the space of harmonic, homogeneous polynomials (in three variables this time) of degree ${\displaystyle \ell }$ consisting of eigenvectors for ${\displaystyle SO(2)}$ (note that the recursion formula is compatible with the ${\displaystyle SO(2)}$-action because the Laplace operator is rotationally invariant). These are the complex solid harmonics:

$${\displaystyle {\begin{aligned}R_{\ell }^{\pm \ell }&=(x\pm iy)^{\ell }z^{0}\\R_{\ell }^{\pm (\ell -1)}&=(x\pm iy)^{\ell -1}z^{1}\\R_{\ell }^{\pm (\ell -2)}&=(x^{2}+y^{2})(x\pm iy)^{\ell -2}z^{0}+{\frac {-(\partial _{x}^{2}+\partial _{y}^{2})\left((x^{2}+y^{2})(x\pm iy)^{\ell -2}\right)}{1\cdot 2))z^{2}\\R_{\ell }^{\pm (\ell -3)}&=(x^{2}+y^{2})(x\pm iy)^{\ell -3}z^{1}+{\frac {-(\partial _{x}^{2}+\partial _{y}^{2})\left((x^{2}+y^{2})(x\pm iy)^{\ell -3}\right)}{2\cdot 3))z^{3}\\R_{\ell }^{\pm (\ell -4)}&=(x^{2}+y^{2})^{2}(x\pm iy)^{\ell -4}z^{0}+{\frac {-(\partial _{x}^{2}+\partial _{y}^{2})\left((x^{2}+y^{2})^{2}(x\pm iy)^{\ell -4}\right)}{1\cdot 2))z^{2}+{\frac {(\partial _{x}^{2}+\partial _{y}^{2})^{2}\left((x^{2}+y^{2})^{2}(x\pm iy)^{\ell -4}\right)}{1\cdot 2\cdot 3\cdot 4))z^{4}\\R_{\ell }^{\pm (\ell -5)}&=(x^{2}+y^{2})^{2}(x\pm iy)^{\ell -5}z^{1}+{\frac {-(\partial _{x}^{2}+\partial _{y}^{2})\left((x^{2}+y^{2})^{2}(x\pm iy)^{\ell -5}\right)}{2\cdot 3))z^{3}+{\frac {(\partial _{x}^{2}+\partial _{y}^{2})^{2}\left((x^{2}+y^{2})^{2}(x\pm iy)^{\ell -5}\right)}{2\cdot 3\cdot 4\cdot 5))z^{5}\\&\;\,\vdots \end{aligned))}$$

$${\displaystyle R_{\ell }^{\pm m}={\begin{cases}\sum _{k}(\partial _{x}^{2}+\partial _{y}^{2})^{k}\left((x^{2}+y^{2})^{(\ell -m)/2}(x\pm iy)^{m}\right){\frac {(-1)^{k}z^{2k)){(2k)!))&\ell -m{\text{ is even))\\\sum _{k}(\partial _{x}^{2}+\partial _{y}^{2})^{k}\left((x^{2}+y^{2})^{(\ell -1-m)/2}(x\pm iy)^{m}\right){\frac {(-1)^{k}z^{2k+1)){(2k+1)!))&\ell -m{\text{ is odd))\end{cases))}$$

${\displaystyle 0\leq m\leq \ell }$

Plugging in spherical coordinates ${\displaystyle x=r\cos(\theta )\sin(\varphi )}$, ${\displaystyle y=r\sin(\theta )\sin(\varphi )}$, ${\displaystyle z=r\cos(\varphi )}$ and using ${\displaystyle x^{2}+y^{2}=r^{2}\sin(\varphi )^{2}=r^{2}(1-\cos(\varphi )^{2})}$ one finds the usual relationship to spherical harmonics ${\displaystyle R_{\ell }^{m}=r^{\ell }e^{im\phi }P_{\ell }^{m}(\cos(\vartheta ))}$ with a polynomial ${\displaystyle P_{\ell }^{m))$, which is (up to normalization) the associated Legendre polynomial, and so ${\displaystyle R_{\ell }^{m}=r^{\ell }Y_{\ell }^{m}(\theta ,\varphi )}$ (again, up to the specific choice of normalization).

Real form

By a simple linear combination of solid harmonics of ±m these functions are transformed into real functions, i.e. functions ${\displaystyle \mathbb {R} ^{3}\to \mathbb {R} }$. The real regular solid harmonics, expressed in Cartesian coordinates, are real-valued homogeneous polynomials of order ${\displaystyle \ell }$ 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

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

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

${\displaystyle P_{\ell }(\cos \theta )}$

The following expression defines the real regular solid harmonics:

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

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

z-dependent part

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

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

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

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

(x,y)-dependent part

Consider next, recalling that x = r sin θ cos φ and y = r sin θ sin φ,

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

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

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

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

In total

$${\displaystyle C_{\ell }^{m}(x,y,z)=\left[{\frac {(2-\delta _{m0})(\ell -m)!}{(\ell +m)!))\right]^{1/2}\Pi _{\ell }^{m}(z)\;A_{m}(x,y),\qquad m=0,1,\ldots ,\ell }$$

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

List of lowest functions

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

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

The lowest functions ${\displaystyle A_{m}(x,y)\,}$ and ${\displaystyle B_{m}(x,y)\,}$ are:

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