In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions mathbb{R}^3 to mathbb{C}. There are two kinds: the regular solid harmonics R^m_ell(mathbf{r}), which are well-defined at the origin and the irregular solid harmonics I^m_{ell}(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^m_{ell}(mathbf{r}) equiv sqrt{frac{4pi}{2ell+1}}; r^ell Y^m_{ell}(theta,varphi) I^m_{ell}(mathbf{r}) equiv sqrt{frac{4pi}{2ell+1}} ; frac{ Y^m_{ell}(theta,varphi)}{r^{ell+1}}

Derivation, relation to spherical harmonics

Introducing {{mvar|r}}, {{mvar|θ}}, and {{mvar|φ}} for the spherical polar coordinates of the 3-vector {{math|r}}, and assuming that Phi is a (smooth) function mathbb{R}^3 to mathbb{C}, we can write the Laplace equation in the following form where {{math|l2}} is the square of the nondimensional angular momentum operator, It is known that spherical harmonics {{math|Y{{su|p=m|b=â„“|lh=1em}}}} are eigenfunctions of {{math|l2}}:hat l^2 Y^m_{ell}equiv left[ {hat l_x}^2 +hat l^2_y+hat l^2_zright]Y^m_{ell} = ell(ell+1) Y^m_{ell}.Substitution of {{math|1=Φ(r) = F(r) Y{{su|p=m|b=â„“|lh=1em}}}} into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution,frac{1}{r}frac{partial^2}{partial r^2}r F(r) = frac{ell(ell+1)}{r^2} F(r)Longrightarrow F(r) = A r^ell + B r^{-ell-1}.The particular solutions of the total Laplace equation are regular solid harmonics:R^m_{ell}(mathbf{r}) equiv sqrt{frac{4pi}{2ell+1}}; r^ell Y^m_{ell}(theta,varphi), and irregular solid harmonics:I^m_{ell}(mathbf{r}) equiv sqrt{frac{4pi}{2ell+1}} ; frac{ Y^m_{ell}(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

Racah’s normalization (also known as Schmidt’s semi-normalization) is applied to both functions int_{0}^{pi}sintheta, dtheta int_0^{2pi} dvarphi; R^m_{ell}(mathbf{r})^*; R^m_{ell}(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

The translation of the regular solid harmonic gives a finite expansion, langle lambda, mu; ell-lambda, m-mu| ell m rangle,where the Clebsch–Gordan coefficient is given bylangle lambda, mu; ell-lambda, m-mu| ell m rangle

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, langle lambda, mu; ell+lambda, m-mu| ell m ranglewith |r| le |a|,. The quantity between pointed brackets is again a Clebsch-Gordan coefficient,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{2ell+2lambda+1}{2lambda}^{-1/2}.The addition theorems were proved in different manners by several authors. 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)

Complex form

The regular solid harmonics are homogeneous, polynomial solutions to the Laplace equation Delta R=0. Separating the indeterminate z and writing R = sum_a p_a(x,y) z^a, the Laplace equation is easily seen to be equivalent to the recursion formulap_{a+2} = frac{-left(partial_x^2 + partial_y^2right) p_a}{left(a+2right) left(a+1right)}so that any choice of polynomials p_0(x,y) of degree ell and p_1(x,y) of degree ell-1 gives a solution to the equation. One particular basis of the space of homogeneous polynomials (in two variables) of degree k is 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 SO(2): The rotation rho_alpha of the plane by alphain[0,2pi] acts as multiplication by e^{pm i(k-2m)alpha} on the basis vector (x^2+y^2)^m (x+iy)^{k-2m}.If we combine the degree ell basis and the degree 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 ell consisting of eigenvectors for SO(2) (note that the recursion formula is compatible with the SO(2)-action because the Laplace operator is rotationally invariant). These are the complex solid harmonics:begin{align}R_ell^{pmell} &= (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)}{1cdot 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)}{2cdot 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)}{1cdot 2} z^2 + frac{ (partial_x^2+partial_y^2)^2 left( (x^2+y^2)^2(x pm iy)^{ell-4}right)}{1cdot 2 cdot 3cdot 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)}{2cdot 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 3cdot 4cdot 5}z^5 &;,vdotsend{align}and in generalR_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 0leq mleq ell.Plugging in spherical coordinates x = rcos(theta)sin(varphi), y = rsin(theta)sin(varphi), z = rcos(varphi) and using x^2+y^2=r^2 sin(varphi)^2 = r^2(1-cos(varphi)^2) one finds the usual relationship to spherical harmonics R_ell^m = r^ell e^{imphi} P_ell^m(cos(vartheta)) with a polynomial P_ell^m, which is (up to normalization) the associated Legendre polynomial, and so R_ell^m = r^ell Y_ell^m(theta,varphi) (again, up to the specific choice of normalization).

Real form

{{Unreferenced section|date=October 2010}}By a simple linear combination of solid harmonics of {{math|±m}} these functions are transformed into real functions, i.e. functions mathbb{R}^3 to mathbb{R}. The real regular solid harmonics, expressed in Cartesian coordinates, are real-valued homogeneous polynomials of order 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 R_ell^m(r,theta,varphi) = (-1)^{(m+|m|)/2}; r^ell ;Theta_{ell}^{|m|} (costheta) withTheta_{ell}^m (costheta) equiv left[frac{(ell-m)!}{(ell+m)!}right]^{1/2} ,sin^mtheta, frac{d^m P_ell(costheta)}{dcos^mtheta}, qquad mge 0,where P_ell(costheta) is a Legendre polynomial of order {{mvar|ℓ}}.The {{mvar|m}} dependent phase is known as the Condon–Shortley phase.The following expression defines the real regular solid harmonics:begin{pmatrix}C_ell^{m} S_ell^{m}end{pmatrix}equiv sqrt{2} ; r^ell ; Theta^{m}_ellbegin{pmatrix}cos mvarphi sin mvarphiend{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 {{math|1=m = 0}}: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“>

z-dependent part

Upon writing {{math|1=u = cos θ}} the {{mvar|m}}-th derivative of the Legendre polynomial can be written as the following expansion in {{mvar|u}}frac{d^m P_ell(u)}{du^m} =sum_{k=0}^{left lfloor (ell-m)/2right rfloor} gamma^{(m)}_{ell k}; u^{ell-2k-m}withgamma^{(m)}_{ell k} = (-1)^k 2^{-ell} binom{ell}{k}binom{2ell-2k}{ell} frac{(ell-2k)!}{(ell-2k-m)!}.Since {{math|1=z = r cos θ}} it follows that this derivative, times an appropriate power of {{mvar|r}}, is a simple polynomial in {{mvar|z}},Pi^m_ell(z)equivr^{ell-m} frac{d^m P_ell(u)}{du^m} =sum_{k=0}^{left lfloor (ell-m)/2right rfloor} gamma^{(m)}_{ell k}; r^{2k}; z^{ell-2k-m}.x,y)-dependent part“>

(x,y)-dependent part

Consider next, recalling that {{math|1=x = r sin θ cos φ}} and {{math|1=y = r sin θ sin φ}},r^m sin^mtheta cos mvarphi = frac{1}{2} left[ (r sintheta e^{ivarphi})^m + (r sintheta e^{-ivarphi})^m right] =frac{1}{2} left[ (x+iy)^m + (x-iy)^m right]Likewiser^m sin^mtheta sin mvarphi = frac{1}{2i} left[ (r sintheta e^{ivarphi})^m - (r sintheta e^{-ivarphi})^m right] =frac{1}{2i} left[ (x+iy)^m - (x-iy)^m right].FurtherA_m(x,y) equivfrac{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}andB_m(x,y) equivfrac{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

C^m_ell(x,y,z) = left[frac{(2-delta_{m0}) (ell-m)!}{(ell+m)!}right]^{1/2} Pi^m_{ell}(z);A_m(x,y),qquad m=0,1, ldots,ellS^m_ell(x,y,z) = left[frac{2 (ell-m)!}{(ell+m)!}right]^{1/2} Pi^m_{ell}(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 {{math|1=â„“ = 5}}.Here bar{Pi}^m_ell(z) equiv left[tfrac{(2-delta_{m0}) (ell-m)!}{(ell+m)!}right]^{1/2} Pi^m_{ell}(z) . The lowest functions A_m(x,y), and B_m(x,y), are:{| class=“wikitable”! m! Am! Bm| 0| 1,| 0,| 1| x,| y,| 2| x^2-y^2,| 2xy,| 3| x^3-3xy^2,| 3x^2y -y^3, | 4| x^4 - 6x^2 y^2 +y^4,| 4x^3y-4xy^3,| 5| x^5-10x^3y^2+ 5xy^4, | 5x^4y -10x^2y^3+y^5,

References

{{Reflist}}

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