Solid harmonics

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

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 $${\displaystyle \nabla ^{2}\Phi (\mathbf {r} )=\left({\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r-{\frac {{\hat {l}}^{2}}{r^{2}}}\right)\Phi (\mathbf {r} )=0,\qquad \mathbf {r} \neq \mathbf {0} ,}$$ where l² is the square of the nondimensional angular momentum operator, $${\displaystyle \mathbf {\hat {l}} =-i\,(\mathbf {r} \times \mathbf {\nabla } ).}$$

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: $${\displaystyle R_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),}$$ and irregular solid harmonics: $${\displaystyle I_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;{\frac {Y_{\ell }^{m}(\theta ,\varphi )}{r^{\ell +1}}}.}$$ The regular solid harmonics correspond to harmonic homogeneous polynomials, i.e. homogeneous polynomials which are solutions to Laplace's equation.

Racah's normalization (also known as Schmidt's semi-normalization) is applied to both functions $${\displaystyle \int _{0}^{\pi }\sin \theta \,d\theta \int _{0}^{2\pi }d\varphi \;R_{\ell }^{m}(\mathbf {r} )^{*}\;R_{\ell }^{m}(\mathbf {r} )={\frac {4\pi }{2\ell +1}}r^{2\ell }}$$ (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, $${\displaystyle R_{\ell }^{m}(\mathbf {r} +\mathbf {a} )=\sum _{\lambda =0}^{\ell }{\binom {2\ell }{2\lambda }}^{1/2}\sum _{\mu =-\lambda }^{\lambda }R_{\lambda }^{\mu }(\mathbf {r} )R_{\ell -\lambda }^{m-\mu }(\mathbf {a} )\;\langle \lambda ,\mu ;\ell -\lambda ,m-\mu |\ell m\rangle ,}$$ where the Clebsch–Gordan coefficient is given by $${\displaystyle \langle \lambda ,\mu ;\ell -\lambda ,m-\mu |\ell m\rangle ={\binom {\ell +m}{\lambda +\mu }}^{1/2}{\binom {\ell -m}{\lambda -\mu }}^{1/2}{\binom {2\ell }{2\lambda }}^{-1/2}.}$$

The similar expansion for irregular solid harmonics gives an infinite series, $${\displaystyle I_{\ell }^{m}(\mathbf {r} +\mathbf {a} )=\sum _{\lambda =0}^{\infty }{\binom {2\ell +2\lambda +1}{2\lambda }}^{1/2}\sum _{\mu =-\lambda }^{\lambda }R_{\lambda }^{\mu }(\mathbf {r} )I_{\ell +\lambda }^{m-\mu }(\mathbf {a} )\;\langle \lambda ,\mu ;\ell +\lambda ,m-\mu |\ell m\rangle }$$ with ${\displaystyle |r|\leq |a|\,}$. The quantity between pointed brackets is again a Clebsch-Gordan coefficient, $${\displaystyle \langle \lambda ,\mu ;\ell +\lambda ,m-\mu |\ell m\rangle =(-1)^{\lambda +\mu }{\binom {\ell +\lambda -m+\mu }{\lambda +\mu }}^{1/2}{\binom {\ell +\lambda +m-\mu }{\lambda -\mu }}^{1/2}{\binom {2\ell +2\lambda +1}{2\lambda }}^{-1/2}.}$$

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

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 $${\displaystyle p_{a+2}={\frac {-\left(\partial _{x}^{2}+\partial _{y}^{2}\right)p_{a}}{\left(a+2\right)\left(a+1\right)}}}$$ so that any choice of polynomials ${\displaystyle p_{0}(x,y)}$ of degree ${\displaystyle \ell }$ and ${\displaystyle p_{1}(x,y)}$ of degree ${\displaystyle \ell -1}$ gives a solution to the equation. One particular basis of the space of homogeneous polynomials (in two variables) of degree ${\displaystyle k}$ is ${\displaystyle \left\{(x^{2}+y^{2})^{m}(x\pm iy)^{k-2m}\mid 0\leq m\leq k/2\right\}}$. Note that it is the (unique up to normalization) basis of eigenvectors of the rotation group ${\displaystyle SO(2)}$: The rotation ${\displaystyle \rho _{\alpha }}$ of the plane by ${\displaystyle \alpha \in [0,2\pi ]}$ acts as multiplication by ${\displaystyle e^{\pm i(k-2m)\alpha }}$ on the basis vector ${\displaystyle (x^{2}+y^{2})^{m}(x+iy)^{k-2m}}$.

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: $${\displaystyle {\begin{aligned}R_{\ell }^{\pm \ell }&=(x\pm iy)^{\ell }z^{0}\\R_{\ell }^{\pm (\ell -1)}&=(x\pm iy)^{\ell -1}z^{1}\\R_{\ell }^{\pm (\ell -2)}&=(x^{2}+y^{2})(x\pm iy)^{\ell -2}z^{0}+{\frac {-(\partial _{x}^{2}+\partial _{y}^{2})\left((x^{2}+y^{2})(x\pm iy)^{\ell -2}\right)}{1\cdot 2}}z^{2}\\R_{\ell }^{\pm (\ell -3)}&=(x^{2}+y^{2})(x\pm iy)^{\ell -3}z^{1}+{\frac {-(\partial _{x}^{2}+\partial _{y}^{2})\left((x^{2}+y^{2})(x\pm iy)^{\ell -3}\right)}{2\cdot 3}}z^{3}\\R_{\ell }^{\pm (\ell -4)}&=(x^{2}+y^{2})^{2}(x\pm iy)^{\ell -4}z^{0}+{\frac {-(\partial _{x}^{2}+\partial _{y}^{2})\left((x^{2}+y^{2})^{2}(x\pm iy)^{\ell -4}\right)}{1\cdot 2}}z^{2}+{\frac {(\partial _{x}^{2}+\partial _{y}^{2})^{2}\left((x^{2}+y^{2})^{2}(x\pm iy)^{\ell -4}\right)}{1\cdot 2\cdot 3\cdot 4}}z^{4}\\R_{\ell }^{\pm (\ell -5)}&=(x^{2}+y^{2})^{2}(x\pm iy)^{\ell -5}z^{1}+{\frac {-(\partial _{x}^{2}+\partial _{y}^{2})\left((x^{2}+y^{2})^{2}(x\pm iy)^{\ell -5}\right)}{2\cdot 3}}z^{3}+{\frac {(\partial _{x}^{2}+\partial _{y}^{2})^{2}\left((x^{2}+y^{2})^{2}(x\pm iy)^{\ell -5}\right)}{2\cdot 3\cdot 4\cdot 5}}z^{5}\\&\;\,\vdots \end{aligned}}}$$ and in general $${\displaystyle R_{\ell }^{\pm m}={\begin{cases}\sum _{k}(\partial _{x}^{2}+\partial _{y}^{2})^{k}\left((x^{2}+y^{2})^{(\ell -m)/2}(x\pm iy)^{m}\right){\frac {(-1)^{k}z^{2k}}{(2k)!}}&\ell -m{\text{ is even}}\\\sum _{k}(\partial _{x}^{2}+\partial _{y}^{2})^{k}\left((x^{2}+y^{2})^{(\ell -1-m)/2}(x\pm iy)^{m}\right){\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}&\ell -m{\text{ is odd}}\end{cases}}}$$ for ${\displaystyle 0\leq m\leq \ell }$.

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).

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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.

We write in agreement with the earlier definition $${\displaystyle R_{\ell }^{m}(r,\theta ,\varphi )=(-1)^{(m+|m|)/2}\;r^{\ell }\;\Theta _{\ell }^{|m|}(\cos \theta )e^{im\varphi },\qquad -\ell \leq m\leq \ell ,}$$ with $${\displaystyle \Theta _{\ell }^{m}(\cos \theta )\equiv \left[{\frac {(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\,\sin ^{m}\theta \,{\frac {d^{m}P_{\ell }(\cos \theta )}{d\cos ^{m}\theta }},\qquad m\geq 0,}$$ where ${\displaystyle P_{\ell }(\cos \theta )}$ 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: $${\displaystyle {\begin{pmatrix}C_{\ell }^{m}\\S_{\ell }^{m}\end{pmatrix}}\equiv {\sqrt {2}}\;r^{\ell }\;\Theta _{\ell }^{m}{\begin{pmatrix}\cos m\varphi \\\sin m\varphi \end{pmatrix}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}(-1)^{m}&\quad 1\\-(-1)^{m}i&\quad i\end{pmatrix}}{\begin{pmatrix}R_{\ell }^{m}\\R_{\ell }^{-m}\end{pmatrix}},\qquad m>0.}$$ and for m = 0: $${\displaystyle C_{\ell }^{0}\equiv R_{\ell }^{0}.}$$ Since the transformation is by a unitary matrix the normalization of the real and the complex solid harmonics is the same.

Upon writing u = cos θ the m-th derivative of the Legendre polynomial can be written as the following expansion in u $${\displaystyle {\frac {d^{m}P_{\ell }(u)}{du^{m}}}=\sum _{k=0}^{\left\lfloor (\ell -m)/2\right\rfloor }\gamma _{\ell k}^{(m)}\;u^{\ell -2k-m}}$$ with $${\displaystyle \gamma _{\ell k}^{(m)}=(-1)^{k}2^{-\ell }{\binom {\ell }{k}}{\binom {2\ell -2k}{\ell }}{\frac {(\ell -2k)!}{(\ell -2k-m)!}}.}$$ Since z = r cos θ it follows that this derivative, times an appropriate power of r, is a simple polynomial in z, $${\displaystyle \Pi _{\ell }^{m}(z)\equiv r^{\ell -m}{\frac {d^{m}P_{\ell }(u)}{du^{m}}}=\sum _{k=0}^{\left\lfloor (\ell -m)/2\right\rfloor }\gamma _{\ell k}^{(m)}\;r^{2k}\;z^{\ell -2k-m}.}$$

Consider next, recalling that x = r sin θ cos φ and y = r sin θ sin φ, $${\displaystyle r^{m}\sin ^{m}\theta \cos m\varphi ={\frac {1}{2}}\left[(r\sin \theta e^{i\varphi })^{m}+(r\sin \theta e^{-i\varphi })^{m}\right]={\frac {1}{2}}\left[(x+iy)^{m}+(x-iy)^{m}\right]}$$ Likewise $${\displaystyle r^{m}\sin ^{m}\theta \sin m\varphi ={\frac {1}{2i}}\left[(r\sin \theta e^{i\varphi })^{m}-(r\sin \theta e^{-i\varphi })^{m}\right]={\frac {1}{2i}}\left[(x+iy)^{m}-(x-iy)^{m}\right].}$$ Further $${\displaystyle A_{m}(x,y)\equiv {\frac {1}{2}}\left[(x+iy)^{m}+(x-iy)^{m}\right]=\sum _{p=0}^{m}{\binom {m}{p}}x^{p}y^{m-p}\cos(m-p){\frac {\pi }{2}}}$$ and $${\displaystyle B_{m}(x,y)\equiv {\frac {1}{2i}}\left[(x+iy)^{m}-(x-iy)^{m}\right]=\sum _{p=0}^{m}{\binom {m}{p}}x^{p}y^{m-p}\sin(m-p){\frac {\pi }{2}}.}$$

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

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	${\displaystyle x^{5}-10x^{3}y^{2}+5xy^{4}\,}$	${\displaystyle 5x^{4}y-10x^{2}y^{3}+y^{5}\,}$

• Steinborn, E. O.; Ruedenberg, K. (1973). "Rotation and Translation of Regular and Irregular Solid Spherical Harmonics". In Lowdin, Per-Olov (ed.). Advances in quantum chemistry. Vol. 7. Academic Press. pp. 1–82. ISBN 9780080582320.
• Thompson, William J. (2004). Angular momentum: an illustrated guide to rotational symmetries for physical systems. Weinheim: Wiley-VCH. pp. 143–148. ISBN 9783527617838.