In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates. There are two kinds: the regular solid harmonics, which are well-defined at the origin and the irregular solid harmonics, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics

Derivation, relation to spherical harmonics

Introducing , , and for the spherical polar coordinates of the 3-vector where is the square of the angular momentum operator

It is known that spherical harmonics are eigenfunctions of

Substitution of 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 and irregular 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

Racah (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

where the Clebsch–Gordan coefficient

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.

Complex form

. Note that it is the (unique up to normalization) basis of eigenvectors.

.

Plugging in spherical coordinates, which is (up to normalization) the associated Legendre polynomial (again, up to the specific choice of normalization).

Real form

By a simple linear combination of solid harmonics of 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

is a Legendre polynomial of order . The dependent phase is known as the Condon–Shortley phase.

and for Since the transformation is by a unitary matrix the normalization of the real and the complex solid harmonics is the same.

z-dependent part

Upon writing the -th derivative of the Legendre polynomial can be written as the following expansion in Since it follows that this derivative, times an appropriate power of , is a simple polynomial in

(x,y)-dependent part

Consider next, recalling that and

In total

List of lowest functions

We list explicitly the lowest functions up to and including

are:

References