In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions
There are two kinds: the regular solid harmonics
, which are well-defined at the origin and the irregular solid harmonics
Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:
Introducing r, θ, and φ for the spherical polar coordinates of the 3-vector r, and assuming that
where L2 is the square of the nondimensional angular momentum operator,
Substitution of Φ(r) = F(r) Ymℓ into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution,
The particular solutions of the total Laplace equation are regular solid harmonics:
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
(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,
The similar expansion for irregular solid harmonics gives an infinite series,
The quantity between pointed brackets is again a Clebsch-Gordan coefficient,
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
, the Laplace equation is easily seen to be equivalent to the recursion formula
One particular basis of the space of homogeneous polynomials (in two variables) of degree
Note that it is the (unique up to normalization) basis of eigenvectors of the rotation group
basis with the recursion formula, we obtain a basis of the space of harmonic, homogeneous polynomials (in three variables this time) of degree
one finds the usual relationship to spherical harmonics
The real regular solid harmonics, expressed in Cartesian coordinates, are real-valued homogeneous polynomials of order
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.
The following expression defines the real regular solid harmonics:
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
Since z = r cos θ it follows that this derivative, times an appropriate power of r, is a simple polynomial in z,
We list explicitly the lowest functions up to and including ℓ = 5.