In physics and mathematics , the solid harmonics are solutions of the Laplace equation in spherical polar coordinates , assumed to be (smooth) functions R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } . There are two kinds: the regular solid harmonics R ℓ m ( r ) {\displaystyle R_{\ell }^{m}(\mathbf {r} )} , which are well-defined at the origin and the irregular solid harmonics I ℓ m ( r ) {\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:
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 )} 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}}}} Derivation, relation to spherical harmonics [ edit ] 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
l 2 is the square of the nondimensional
angular momentum operator ,
l ^ = − i ( r × ∇ ) . {\displaystyle \mathbf {\hat {l}} =-i\,(\mathbf {r} \times \mathbf {\nabla } ).} It is known that spherical harmonics Y m ℓ are eigenfunctions of l 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}.} 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,
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}.} 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
harmonic homogeneous polynomials , i.e. homogeneous polynomials which are solutions to
Laplace's equation .
Racah's normalization [ edit ] Racah '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.
Addition theorems [ edit ] 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
Clebsch–Gordan coefficient 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}.} 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
Clebsch-Gordan coefficient ,
⟨ 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}.} The addition theorems were proved in different manners by several authors.[1] [2]
Complex form [ edit ] 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
eigenvectors 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}} .
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 } .
Plugging in spherical coordinates 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 associated Legendre polynomial , 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).
Real form [ edit ] By a simple linear combination of solid harmonics of ±m 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 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 [ edit ] 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
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:
( 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
m = 0:
C ℓ 0 ≡ R ℓ 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.
z -dependent part[ edit ] Upon writing u = cos θ 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
z = r cos θ 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}.} (x ,y )-dependent part [ edit ] Consider next, recalling that x = r sin θ cos φ and y = r sin θ sin φ ,
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}}.} In total [ edit ]
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 .} List of lowest functions [ edit ] We list explicitly the lowest functions up to and including ℓ = 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).}
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}}} The lowest functions A m ( x , y ) {\displaystyle A_{m}(x,y)\,} and B m ( x , y ) {\displaystyle B_{m}(x,y)\,} are:
m A m B m 0 1 {\displaystyle 1\,} 0 {\displaystyle 0\,} 1 x {\displaystyle x\,} y {\displaystyle y\,} 2 x 2 − y 2 {\displaystyle x^{2}-y^{2}\,} 2 x y {\displaystyle 2xy\,} 3 x 3 − 3 x y 2 {\displaystyle x^{3}-3xy^{2}\,} 3 x 2 y − y 3 {\displaystyle 3x^{2}y-y^{3}\,} 4 x 4 − 6 x 2 y 2 + y 4 {\displaystyle x^{4}-6x^{2}y^{2}+y^{4}\,} 4 x 3 y − 4 x y 3 {\displaystyle 4x^{3}y-4xy^{3}\,} 5 x 5 − 10 x 3 y 2 + 5 x y 4 {\displaystyle x^{5}-10x^{3}y^{2}+5xy^{4}\,} 5 x 4 y − 10 x 2 y 3 + y 5 {\displaystyle 5x^{4}y-10x^{2}y^{3}+y^{5}\,}
References [ edit ] ^ 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) 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 .