Solid harmonics – Wikipedia

Posted on September 15, 2017 by lordneo

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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} }

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${displaystyle mathbb {R} ^{3}to mathbb {C} }$ . There are two kinds: the regular solid harmonics

{displaystyle R_{ell }^{m}(mathbf {r} )}

$R_{ell }^{m}({mathbf {r}})$ , which are well-defined at the origin and the irregular solid harmonics

{displaystyle I_{ell }^{m}(mathbf {r} )}

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

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

Table of Contents

Derivation, relation to spherical harmonics[edit]

Introducing $r$ , $θ$ , and $φ$ for the spherical polar coordinates of the 3-vector $r$ , and assuming that

{displaystyle Phi }

$Phi$ is a (smooth) function

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{displaystyle mathbb {R} ^{3}to mathbb {C} }

${displaystyle mathbb {R} ^{3}to mathbb {C} }$ , 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 $l 2$ 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 $l 2$ :

{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 {4pi }{2ell +1}}};r^{ell }Y_{ell }^{m}(theta ,varphi ),}

and irregular solid harmonics:

{displaystyle I_{ell }^{m}(mathbf {r} )equiv {sqrt {frac {4pi }{2ell +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

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

(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,

{displaystyle R_{ell }^{m}(mathbf {r} +mathbf {a} )=sum _{lambda =0}^{ell }{binom {2ell }{2lambda }}^{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 mrangle ,}

where the Clebsch–Gordan coefficient is given by

{displaystyle langle lambda ,mu ;ell -lambda ,m-mu |ell mrangle ={binom {ell +m}{lambda +mu }}^{1/2}{binom {ell -m}{lambda -mu }}^{1/2}{binom {2ell }{2lambda }}^{-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 {2ell +2lambda +1}{2lambda }}^{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 mrangle }

with

{displaystyle |r|leq |a|,}

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

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

{displaystyle Delta R=0}

${displaystyle Delta R=0}$ . Separating the indeterminate

{displaystyle z}

$z$ and writing

{textstyle R=sum _{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

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

so that any choice of polynomials

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

${displaystyle p_{0}(x,y)}$ of degree

{displaystyle ell }

$ell$ and

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

${displaystyle p_{1}(x,y)}$ of degree

{displaystyle ell -1}

${displaystyle ell -1}$ gives a solution to the equation. One particular basis of the space of homogeneous polynomials (in two variables) of degree

{displaystyle k}

$k$ is

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

${displaystyle left{(x^{2}+y^{2})^{m}(xpm iy)^{k-2m}mid 0leq mleq k/2right}}$ . Note that it is the (unique up to normalization) basis of eigenvectors of the rotation group

{displaystyle SO(2)}

$SO(2)$ : The rotation

{displaystyle rho _{alpha }}

${displaystyle rho _{alpha }}$ of the plane by

{displaystyle alpha in [0,2pi ]}

${displaystyle alpha in [0,2pi ]}$ acts as multiplication by

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

${displaystyle e^{pm i(k-2m)alpha }}$ on the basis vector

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

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

If we combine the degree

{displaystyle ell }

$ell$ basis and the degree

{displaystyle ell -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

{displaystyle ell }

$ell$ consisting of eigenvectors for

{displaystyle SO(2)}

$SO(2)$ (note that the recursion formula is compatible with the

{displaystyle SO(2)}

$SO(2)$ -action because the Laplace operator is rotationally invariant). These are the complex solid harmonics:

{displaystyle {begin{aligned}R_{ell }^{pm ell }&=(xpm iy)^{ell }z^{0}\R_{ell }^{pm (ell -1)}&=(xpm iy)^{ell -1}z^{1}\R_{ell }^{pm (ell -2)}&=(x^{2}+y^{2})(xpm iy)^{ell -2}z^{0}+{frac {-(partial _{x}^{2}+partial _{y}^{2})left((x^{2}+y^{2})(xpm iy)^{ell -2}right)}{1cdot 2}}z^{2}\R_{ell }^{pm (ell -3)}&=(x^{2}+y^{2})(xpm iy)^{ell -3}z^{1}+{frac {-(partial _{x}^{2}+partial _{y}^{2})left((x^{2}+y^{2})(xpm iy)^{ell -3}right)}{2cdot 3}}z^{3}\R_{ell }^{pm (ell -4)}&=(x^{2}+y^{2})^{2}(xpm iy)^{ell -4}z^{0}+{frac {-(partial _{x}^{2}+partial _{y}^{2})left((x^{2}+y^{2})^{2}(xpm iy)^{ell -4}right)}{1cdot 2}}z^{2}+{frac {(partial _{x}^{2}+partial _{y}^{2})^{2}left((x^{2}+y^{2})^{2}(xpm iy)^{ell -4}right)}{1cdot 2cdot 3cdot 4}}z^{4}\R_{ell }^{pm (ell -5)}&=(x^{2}+y^{2})^{2}(xpm iy)^{ell -5}z^{1}+{frac {-(partial _{x}^{2}+partial _{y}^{2})left((x^{2}+y^{2})^{2}(xpm iy)^{ell -5}right)}{2cdot 3}}z^{3}+{frac {(partial _{x}^{2}+partial _{y}^{2})^{2}left((x^{2}+y^{2})^{2}(xpm iy)^{ell -5}right)}{2cdot 3cdot 4cdot 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}(xpm 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}(xpm iy)^{m}right){frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}&ell -m{text{ is odd}}end{cases}}}

for

{displaystyle 0leq mleq ell }

${displaystyle 0leq mleq ell }$ .

Plugging in spherical coordinates

{displaystyle x=rcos(theta )sin(varphi )}

${displaystyle x=rcos(theta )sin(varphi )}$ ,

{displaystyle y=rsin(theta )sin(varphi )}

${displaystyle y=rsin(theta )sin(varphi )}$ ,

{displaystyle z=rcos(varphi )}

${displaystyle z=rcos(varphi )}$ and using

{displaystyle x^{2}+y^{2}=r^{2}sin(varphi )^{2}=r^{2}(1-cos(varphi )^{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

{displaystyle R_{ell }^{m}=r^{ell }e^{imphi }P_{ell }^{m}(cos(vartheta ))}

${displaystyle R_{ell }^{m}=r^{ell }e^{imphi }P_{ell }^{m}(cos(vartheta ))}$ with a polynomial

{displaystyle P_{ell }^{m}}

$P_ell^m$ , which is (up to normalization) the associated Legendre polynomial, and so

{displaystyle R_{ell }^{m}=r^{ell }Y_{ell }^{m}(theta ,varphi )}

${displaystyle R_{ell }^{m}=r^{ell }Y_{ell }^{m}(theta ,varphi )}$ (again, up to the specific choice of normalization).

Real form[edit]

By a simple linear combination of solid harmonics of $\pm m$ these functions are transformed into real functions, i.e. functions

{displaystyle mathbb {R} ^{3}to mathbb {R} }

${displaystyle mathbb {R} ^{3}to mathbb {R} }$ . The real regular solid harmonics, expressed in Cartesian coordinates, are real-valued homogeneous polynomials of order

{displaystyle ell }

$ell$ 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

{displaystyle R_{ell }^{m}(r,theta ,varphi )=(-1)^{(m+|m|)/2};r^{ell };Theta _{ell }^{|m|}(cos theta )e^{imvarphi },qquad -ell leq mleq 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 )}{dcos ^{m}theta }},qquad mgeq 0,}

where

{displaystyle P_{ell }(cos theta )}

$P_{ell }(cos theta )$ 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 mvarphi \sin mvarphi end{pmatrix}}={frac {1}{sqrt {2}}}{begin{pmatrix}(-1)^{m}&quad 1\-(-1)^{m}i&quad iend{pmatrix}}{begin{pmatrix}R_{ell }^{m}\R_{ell }^{-m}end{pmatrix}},qquad m>0.}

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