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Solid harmonics
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<h2 id="derivation-relation-to-spherical-harmonics">Derivation, relation to spherical harmonics</h2> <p>Introducing r, θ, and φ for the spherical polar coordinates of the 3-vector r, and assuming that Φ {\displaystyle \Phi } is a (smooth) function R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } , we can write the Laplace equation in the following form ∇ 2 Φ ( r ) = ( 1 r ∂ 2 ∂ r 2 r − L ^ 2 r 2 ) Φ ( r ) = 0 , r ≠ 0 , {\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 <i>L</i>2 is the square of the nondimensional <a href="/facts/Angular_momentum_operator/d55sXn5u">angular momentum operator</a>, L ^ = − i ( r × ∇ ) . {\displaystyle \mathbf {\hat {L}} =-i\,(\mathbf {r} \times \mathbf {\nabla } ).} </p><p>It is <a href="/facts/Angular_momentum/7TuSVFxq">known</a> that <a href="/facts/Spherical_harmonics/bn98Gv2L">spherical harmonics</a> <i>Y</i><i>m</i><i>ℓ</i> are eigenfunctions of <i>L</i>2: L ^ 2 Y ℓ m ≡ [ L ^ x 2 + L ^ y 2 + L ^ z 2 ] Y ℓ m = ℓ ( ℓ + 1 ) Y ℓ m . {\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}.} </p><p>Substitution of Φ(r) = <i>F</i>(<i>r</i>) <i>Y</i><i>m</i><i>ℓ</i> into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution, </p><p> 1 r ∂ 2 ∂ r 2 r F ( r ) = ℓ ( ℓ + 1 ) r 2 F ( r ) ⟹ F ( r ) = A r ℓ + B r − ℓ − 1 . {\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}.} </p><p>The particular solutions of the total Laplace equation are regular solid harmonics: R ℓ m ( r ) ≡ 4 π 2 ℓ + 1 r ℓ Y ℓ m ( θ , φ ) , {\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: I ℓ m ( r ) ≡ 4 π 2 ℓ + 1 Y ℓ m ( θ , φ ) r ℓ + 1 . {\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 <a href="/facts/Harmonic_functions/H6pg4dMT">harmonic</a> <a href="/facts/Homogeneous_polynomials/WUHL7Kct">homogeneous polynomials</a>, i.e. homogeneous polynomials which are solutions to <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a>. </p> <h3>Racah's normalization</h3> <p><a href="/facts/Giulio_Racah/6YIN3WhM">Racah</a>'s normalization (also known as Schmidt's semi-normalization) is applied to both functions ∫ 0 π sin θ d θ ∫ 0 2 π d φ R ℓ m ( r ) ∗ R ℓ m ( r ) = 4 π 2 ℓ + 1 r 2 ℓ {\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. </p> <h2 id="addition-theorems">Addition theorems</h2> <p>The translation of the regular solid harmonic gives a finite expansion, R ℓ m ( r + a ) = ∑ λ = 0 ℓ ( 2 ℓ 2 λ ) 1 / 2 ∑ μ = − λ λ R λ μ ( r ) R ℓ − λ m − μ ( a ) ⟨ λ , μ ; ℓ − λ , m − μ | ℓ m ⟩ , {\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 <a href="/facts/Clebsch%25E2%2580%2593Gordan_coefficient/r063mruI">Clebsch–Gordan coefficient</a> is given by ⟨ λ , μ ; ℓ − λ , m − μ | ℓ m ⟩ = ( ℓ + m λ + μ ) 1 / 2 ( ℓ − m λ − μ ) 1 / 2 ( 2 ℓ 2 λ ) − 1 / 2 . {\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}.} </p><p>The similar expansion for irregular solid harmonics gives an infinite series, I ℓ m ( r + a ) = ∑ λ = 0 ∞ ( 2 ℓ + 2 λ + 1 2 λ ) 1 / 2 ∑ μ = − λ λ R λ μ ( r ) I ℓ + λ m − μ ( a ) ⟨ λ , μ ; ℓ + λ , m − μ | ℓ m ⟩ {\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 | r | ≤ | a | {\displaystyle |r|\leq |a|\,} . The quantity between pointed brackets is again a <a href="/facts/Clebsch-Gordan_coefficient/r063mruI">Clebsch-Gordan coefficient</a>, ⟨ λ , μ ; ℓ + λ , m − μ | ℓ m ⟩ = ( − 1 ) λ + μ ( ℓ + λ − m + μ λ + μ ) 1 / 2 ( ℓ + λ + m − μ λ − μ ) 1 / 2 ( 2 ℓ + 2 λ + 1 2 λ ) − 1 / 2 . {\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}.} </p><p>The addition theorems were proved in different manners by several authors.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="complex-form">Complex form</h2> <p>The regular solid harmonics are homogeneous, polynomial solutions to the Laplace equation Δ R = 0 {\displaystyle \Delta R=0} . Separating the indeterminate z {\displaystyle z} and writing R = ∑ a p a ( x , y ) z a {\textstyle R=\sum _{a}p_{a}(x,y)z^{a}} , the Laplace equation is easily seen to be equivalent to the recursion formula p a + 2 = − ( ∂ x 2 + ∂ y 2 ) p a ( a + 2 ) ( a + 1 ) {\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 p 0 ( x , y ) {\displaystyle p_{0}(x,y)} of degree ℓ {\displaystyle \ell } and p 1 ( x , y ) {\displaystyle p_{1}(x,y)} of degree ℓ − 1 {\displaystyle \ell -1} gives a solution to the equation. One particular basis of the space of homogeneous polynomials (in two variables) of degree k {\displaystyle k} is { ( x 2 + y 2 ) m ( x ± i y ) k − 2 m ∣ 0 ≤ m ≤ k / 2 } {\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 <a href="/facts/Eigenvector/8TjEoT8u">eigenvectors</a> of the rotation group S O ( 2 ) {\displaystyle SO(2)} : The rotation ρ α {\displaystyle \rho _{\alpha }} of the plane by α ∈ [ 0 , 2 π ] {\displaystyle \alpha \in [0,2\pi ]} acts as multiplication by e ± i ( k − 2 m ) α {\displaystyle e^{\pm i(k-2m)\alpha }} on the basis vector ( x 2 + y 2 ) m ( x + i y ) k − 2 m {\displaystyle (x^{2}+y^{2})^{m}(x+iy)^{k-2m}} . </p><p>If we combine the degree ℓ {\displaystyle \ell } basis and the degree ℓ − 1 {\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 ℓ {\displaystyle \ell } consisting of eigenvectors for S O ( 2 ) {\displaystyle SO(2)} (note that the recursion formula is compatible with the S O ( 2 ) {\displaystyle SO(2)} -action because the Laplace operator is rotationally invariant). These are the complex solid harmonics: R ℓ ± ℓ = ( x ± i y ) ℓ z 0 R ℓ ± ( ℓ − 1 ) = ( x ± i y ) ℓ − 1 z 1 R ℓ ± ( ℓ − 2 ) = ( x 2 + y 2 ) ( x ± i y ) ℓ − 2 z 0 + − ( ∂ x 2 + ∂ y 2 ) ( ( x 2 + y 2 ) ( x ± i y ) ℓ − 2 ) 1 ⋅ 2 z 2 R ℓ ± ( ℓ − 3 ) = ( x 2 + y 2 ) ( x ± i y ) ℓ − 3 z 1 + − ( ∂ x 2 + ∂ y 2 ) ( ( x 2 + y 2 ) ( x ± i y ) ℓ − 3 ) 2 ⋅ 3 z 3 R ℓ ± ( ℓ − 4 ) = ( x 2 + y 2 ) 2 ( x ± i y ) ℓ − 4 z 0 + − ( ∂ x 2 + ∂ y 2 ) ( ( x 2 + y 2 ) 2 ( x ± i y ) ℓ − 4 ) 1 ⋅ 2 z 2 + ( ∂ x 2 + ∂ y 2 ) 2 ( ( x 2 + y 2 ) 2 ( x ± i y ) ℓ − 4 ) 1 ⋅ 2 ⋅ 3 ⋅ 4 z 4 R ℓ ± ( ℓ − 5 ) = ( x 2 + y 2 ) 2 ( x ± i y ) ℓ − 5 z 1 + − ( ∂ x 2 + ∂ y 2 ) ( ( x 2 + y 2 ) 2 ( x ± i y ) ℓ − 5 ) 2 ⋅ 3 z 3 + ( ∂ x 2 + ∂ y 2 ) 2 ( ( x 2 + y 2 ) 2 ( x ± i y ) ℓ − 5 ) 2 ⋅ 3 ⋅ 4 ⋅ 5 z 5 ⋮ {\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 R ℓ ± m = { ∑ k ( ∂ x 2 + ∂ y 2 ) k ( ( x 2 + y 2 ) ( ℓ − m ) / 2 ( x ± i y ) m ) ( − 1 ) k z 2 k ( 2 k ) ! ℓ − m is even ∑ k ( ∂ x 2 + ∂ y 2 ) k ( ( x 2 + y 2 ) ( ℓ − 1 − m ) / 2 ( x ± i y ) m ) ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! ℓ − m is odd {\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 0 ≤ m ≤ ℓ {\displaystyle 0\leq m\leq \ell } . </p><p>Plugging in <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a> x = r cos ( θ ) sin ( φ ) {\displaystyle x=r\cos(\theta )\sin(\varphi )} , y = r sin ( θ ) sin ( φ ) {\displaystyle y=r\sin(\theta )\sin(\varphi )} , z = r cos ( φ ) {\displaystyle z=r\cos(\varphi )} and using x 2 + y 2 = r 2 sin ( φ ) 2 = r 2 ( 1 − cos ( φ ) 2 ) {\displaystyle x^{2}+y^{2}=r^{2}\sin(\varphi )^{2}=r^{2}(1-\cos(\varphi )^{2})} one finds the usual relationship to spherical harmonics R ℓ m = r ℓ e i m ϕ P ℓ m ( cos ( ϑ ) ) {\displaystyle R_{\ell }^{m}=r^{\ell }e^{im\phi }P_{\ell }^{m}(\cos(\vartheta ))} with a polynomial P ℓ m {\displaystyle P_{\ell }^{m}} , which is (up to normalization) the <a href="/facts/Associated_Legendre_polynomial/cc74eGon">associated Legendre polynomial</a>, and so R ℓ m = r ℓ Y ℓ m ( θ , φ ) {\displaystyle R_{\ell }^{m}=r^{\ell }Y_{\ell }^{m}(\theta ,\varphi )} (again, up to the specific choice of normalization). </p> <h2 id="real-form">Real form</h2> <p>By a simple linear combination of solid harmonics of ±<i>m</i> these functions are transformed into real functions, i.e. functions R 3 → R {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } . The real regular solid harmonics, expressed in Cartesian coordinates, are real-valued homogeneous polynomials of order ℓ {\displaystyle \ell } in <i>x</i>, <i>y</i>, <i>z</i>. The explicit form of these polynomials is of some importance. They appear, for example, in the form of spherical <a href="/facts/Atomic_orbital/tKhgl6mL">atomic orbitals</a> and real <a href="/facts/Multipole_moments/Kao2WRBA">multipole moments</a>. The explicit Cartesian expression of the real regular harmonics will now be derived. </p> <h3>Linear combination</h3> <p>We write in agreement with the earlier definition R ℓ m ( r , θ , φ ) = ( − 1 ) ( m + | m | ) / 2 r ℓ Θ ℓ | m | ( cos θ ) e i m φ , − ℓ ≤ m ≤ ℓ , {\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 Θ ℓ m ( cos θ ) ≡ [ ( ℓ − m ) ! ( ℓ + m ) ! ] 1 / 2 sin m θ d m P ℓ ( cos θ ) d cos m θ , m ≥ 0 , {\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 P ℓ ( cos θ ) {\displaystyle P_{\ell }(\cos \theta )} is a <a href="/facts/Legendre_polynomial/nDvV4em0">Legendre polynomial</a> of order ℓ. The m dependent phase is known as the <a href="/facts/Spherical_harmonics/bn98Gv2L">Condon–Shortley phase</a>. </p><p>The following expression defines the real regular solid harmonics: ( C ℓ m S ℓ m ) ≡ 2 r ℓ Θ ℓ m ( cos m φ sin m φ ) = 1 2 ( ( − 1 ) m 1 − ( − 1 ) m i i ) ( R ℓ m R ℓ − m ) , m > 0. {\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 <i>m</i> = 0: C ℓ 0 ≡ R ℓ 0 . {\displaystyle C_{\ell }^{0}\equiv R_{\ell }^{0}.} Since the transformation is by a <a href="/facts/Unitary_matrix/8fsxv3KD">unitary matrix</a> the normalization of the real and the complex solid harmonics is the same. </p> <h3><i>z</i>-dependent part</h3> <p>Upon writing <i>u</i> = cos <i>θ</i> the m-th derivative of the Legendre polynomial can be written as the following expansion in u d m P ℓ ( u ) d u m = ∑ k = 0 ⌊ ( ℓ − m ) / 2 ⌋ γ ℓ k ( m ) u ℓ − 2 k − m {\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 γ ℓ k ( m ) = ( − 1 ) k 2 − ℓ ( ℓ k ) ( 2 ℓ − 2 k ℓ ) ( ℓ − 2 k ) ! ( ℓ − 2 k − m ) ! . {\displaystyle \gamma _{\ell k}^{(m)}=(-1)^{k}2^{-\ell }{\binom {\ell }{k}}{\binom {2\ell -2k}{\ell }}{\frac {(\ell -2k)!}{(\ell -2k-m)!}}.} Since <i>z</i> = <i>r</i> cos <i>θ</i> it follows that this derivative, times an appropriate power of r, is a simple polynomial in z, Π ℓ m ( z ) ≡ r ℓ − m d m P ℓ ( u ) d u m = ∑ k = 0 ⌊ ( ℓ − m ) / 2 ⌋ γ ℓ k ( m ) r 2 k z ℓ − 2 k − 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}.} </p> <h3>(<i>x</i>,<i>y</i>)-dependent part</h3> <p>Consider next, recalling that <i>x</i> = <i>r</i> sin <i>θ</i> cos <i>φ</i> and <i>y</i> = <i>r</i> sin <i>θ</i> sin <i>φ</i>, r m sin m θ cos m φ = 1 2 [ ( r sin θ e i φ ) m + ( r sin θ e − i φ ) m ] = 1 2 [ ( x + i y ) m + ( x − i y ) m ] {\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 r m sin m θ sin m φ = 1 2 i [ ( r sin θ e i φ ) m − ( r sin θ e − i φ ) m ] = 1 2 i [ ( x + i y ) m − ( x − i y ) m ] . {\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 A m ( x , y ) ≡ 1 2 [ ( x + i y ) m + ( x − i y ) m ] = ∑ p = 0 m ( m p ) x p y m − p cos ( m − p ) π 2 {\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 B m ( x , y ) ≡ 1 2 i [ ( x + i y ) m − ( x − i y ) m ] = ∑ p = 0 m ( m p ) x p y m − p sin ( m − p ) π 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}}.} </p> <h3>In total</h3> <p> C ℓ m ( x , y , z ) = [ ( 2 − δ m 0 ) ( ℓ − m ) ! ( ℓ + m ) ! ] 1 / 2 Π ℓ m ( z ) A m ( x , y ) , m = 0 , 1 , … , ℓ {\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 } S ℓ m ( x , y , z ) = [ 2 ( ℓ − m ) ! ( ℓ + m ) ! ] 1 / 2 Π ℓ m ( z ) B m ( x , y ) , m = 1 , 2 , … , ℓ . {\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 .} </p> <h3>List of lowest functions</h3> <p>We list explicitly the lowest functions up to and including <i>ℓ</i> = 5. Here Π ¯ ℓ m ( z ) ≡ [ ( 2 − δ m 0 ) ( ℓ − m ) ! ( ℓ + m ) ! ] 1 / 2 Π ℓ m ( z ) . {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)\equiv \left[{\tfrac {(2-\delta _{m0})(\ell -m)!}{(\ell +m)!}}\right]^{1/2}\Pi _{\ell }^{m}(z).} </p><p> Π ¯ 0 0 = 1 Π ¯ 3 1 = 1 4 6 ( 5 z 2 − r 2 ) Π ¯ 4 4 = 1 8 35 Π ¯ 1 0 = z Π ¯ 3 2 = 1 2 15 z Π ¯ 5 0 = 1 8 z ( 63 z 4 − 70 z 2 r 2 + 15 r 4 ) Π ¯ 1 1 = 1 Π ¯ 3 3 = 1 4 10 Π ¯ 5 1 = 1 8 15 ( 21 z 4 − 14 z 2 r 2 + r 4 ) Π ¯ 2 0 = 1 2 ( 3 z 2 − r 2 ) Π ¯ 4 0 = 1 8 ( 35 z 4 − 30 r 2 z 2 + 3 r 4 ) Π ¯ 5 2 = 1 4 105 ( 3 z 2 − r 2 ) z Π ¯ 2 1 = 3 z Π ¯ 4 1 = 10 4 z ( 7 z 2 − 3 r 2 ) Π ¯ 5 3 = 1 16 70 ( 9 z 2 − r 2 ) Π ¯ 2 2 = 1 2 3 Π ¯ 4 2 = 1 4 5 ( 7 z 2 − r 2 ) Π ¯ 5 4 = 3 8 35 z Π ¯ 3 0 = 1 2 z ( 5 z 2 − 3 r 2 ) Π ¯ 4 3 = 1 4 70 z Π ¯ 5 5 = 3 16 14 {\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}}} </p><p>The lowest functions A m ( x , y ) {\displaystyle A_{m}(x,y)\,} and B m ( x , y ) {\displaystyle B_{m}(x,y)\,} are: </p> <table><tbody><tr><th><i>m</i></th><th><i>A</i>m</th><th><i>B</i>m</th></tr><tr><td>0</td><td> 1 {\displaystyle 1\,} </td><td> 0 {\displaystyle 0\,} </td></tr><tr><td>1</td><td> x {\displaystyle x\,} </td><td> y {\displaystyle y\,} </td></tr><tr><td>2</td><td> x 2 − y 2 {\displaystyle x^{2}-y^{2}\,} </td><td> 2 x y {\displaystyle 2xy\,} </td></tr><tr><td>3</td><td> x 3 − 3 x y 2 {\displaystyle x^{3}-3xy^{2}\,} </td><td> 3 x 2 y − y 3 {\displaystyle 3x^{2}y-y^{3}\,} </td></tr><tr><td>4</td><td> x 4 − 6 x 2 y 2 + y 4 {\displaystyle x^{4}-6x^{2}y^{2}+y^{4}\,} </td><td> 4 x 3 y − 4 x y 3 {\displaystyle 4x^{3}y-4xy^{3}\,} </td></tr><tr><td>5</td><td> x 5 − 10 x 3 y 2 + 5 x y 4 {\displaystyle x^{5}-10x^{3}y^{2}+5xy^{4}\,} </td><td> 5 x 4 y − 10 x 2 y 3 + y 5 {\displaystyle 5x^{4}y-10x^{2}y^{3}+y^{5}\,} </td></tr></tbody></table> <ul><li>Steinborn, E. O.; Ruedenberg, K. (1973). "Rotation and Translation of Regular and Irregular Solid Spherical Harmonics". In Lowdin, Per-Olov (ed.). <i>Advances in quantum chemistry</i>. Vol. 7. Academic Press. pp. 1–82. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780080582320.</li> <li>Thompson, William J. (2004). <i>Angular momentum: an illustrated guide to rotational symmetries for physical systems</i>. Weinheim: Wiley-VCH. pp. 143–148. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783527617838.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>R. J. A. Tough and A. J. Stone, J. Phys. A: Math. Gen. Vol. 10, p. 1261 (1977) <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>M. J. Caola, J. Phys. A: Math. Gen. Vol. 11, p. L23 (1978) <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>