Spherical Harmonics

Spherical harmonics provide a concrete realization of the irreducible representations of \(\mathrm{SO}(3)\) as scalar functions on the unit sphere. This chapter fixes the relation between the abstract function, its spherical components, and the coefficients of a field expanded in those functions.

Following the notation used in Wikipedia, this chapter uses \(Y_\ell^m\) for complex spherical harmonics and reserves \(Y_{\ell m}\) for real spherical harmonics.

Complex Spherical Harmonics

Let

\[ S^2 = \left\{ \widehat{\boldsymbol r}\in\mathbb R^3: \lvert\widehat{\boldsymbol r}\rvert=1 \right\}. \]

Here \(\widehat{\boldsymbol r}\) is a point on the unit sphere, or equivalently a direction in three-dimensional space. In the braket notation below, \(|\widehat{\boldsymbol r}\rangle\) is the direction-basis ket labeled by that point, so \(\langle\widehat{\boldsymbol r}|\ell m\rangle\) simply means the value of the angular state \(|\ell m\rangle\) in that direction.

For each \(\ell=0,1,2,\ldots\), the complex spherical harmonics are \(2\ell+1\) scalar functions

\[ \left\{ Y_\ell^m \right\}_{m=-\ell}^{\ell} \]

on \(S^2\). Under rotations, the functions at fixed \(\ell\) mix among themselves according to the irrep \(V_\ell\) of \(\mathrm{SO}(3)\).

Denote the corresponding abstract angular basis by \(\{|\ell m\rangle\}_{m=-\ell}^{\ell}\). The spherical harmonics are its coordinate functions:

\[ Y_\ell^m(\widehat{\boldsymbol r}) \equiv \langle \widehat{\boldsymbol r} | \ell m \rangle . \]

The basis is orthonormal:

\[ \langle\ell m|\ell' m'\rangle = \int_{S^2} \mathrm d\Omega\, \left[ Y_\ell^m(\widehat{\boldsymbol r}) \right]^* Y_{\ell'}^{m'}(\widehat{\boldsymbol r}) = \delta_{\ell\ell'}\delta_{mm'}. \]

One common convention, used in NIST DLMF §14.30, normalizes each spherical harmonic to unit \(L^2\) norm and includes the Condon–Shortley phase in \(P_\ell^m\):

\[ \boxed{ Y_\ell^m(\theta,\phi) = \sqrt{ \frac{2\ell+1}{4\pi} \frac{(\ell-m)!}{(\ell+m)!} }\, P_\ell^m(\cos\theta)e^{\mathrm i m\phi} }. \]

In this convention,

\[ \left[ Y_\ell^m \right]^* = (-1)^mY_\ell^{-m}. \]

The formula fixes one useful reference basis. Other phase choices or mixtures among the functions at fixed \(\ell\) change component values, Wigner matrices, and CG coefficients.

Spherical Coefficients and Rotations

For a function \(f\) on \(S^2\), define its spherical-harmonic coefficients by

\[ c_{\ell m} = \langle\ell m|f\rangle = \int_{S^2} \mathrm d\Omega\, \left[ Y_\ell^m(\widehat{\boldsymbol r}) \right]^* f(\widehat{\boldsymbol r}) . \]

The reason to introduce these coefficients here is that they have a simple rotation law. The Wigner matrix in this basis is

\[ \mathcal D^\ell_{m'm}(R) = \langle\ell m'| \rho_\ell(R) |\ell m\rangle. \]

Therefore the basis functions obey

\[ Y_\ell^m \left( R^{-1}\widehat{\boldsymbol r} \right) = \sum_{m'=-\ell}^{\ell} Y_\ell^{m'}(\widehat{\boldsymbol r}) \mathcal D^\ell_{m'm}(R) . \]

If the function is actively rotated,

\[ f'(\widehat{\boldsymbol r}) = f(R^{-1}\widehat{\boldsymbol r}), \]

then its coefficients transform as

\[ c_{\ell m'}' = \sum_{m=-\ell}^{\ell} \mathcal D^\ell_{m'm}(R)c_{\ell m} . \]

Thus the \(2\ell+1\) coefficients at fixed \(\ell\) form one \(V_\ell\) block. Rotations mix \(m\) within that block but do not mix different values of \(\ell\).

Parity

Spherical harmonics have natural parity

\[ Y_\ell^m(-\widehat{\boldsymbol r}) = (-1)^\ell Y_\ell^m(\widehat{\boldsymbol r}) . \]

For an ordinary scalar-valued function on the sphere, a degree-\(\ell\) coefficient block therefore has \(\mathrm{O}(3)\) type

\[ \left(\ell,(-1)^\ell\right). \]

An intrinsic pseudoscalar sign or another value-space parity would multiply this spatial parity and must be stated separately.

Real Spherical Harmonics

Real spherical harmonics are obtained by mixing the complex \(+m\) and \(-m\) functions at fixed \(\ell\) into a real orthonormal basis. For the complex convention above, one common real reference is

\[ Y_{\ell m} = \begin{cases} \sqrt{2}(-1)^m \operatorname{Im}Y_\ell^{|m|}, &m<0,\\ Y_\ell^0, &m=0,\\ \sqrt{2}(-1)^m \operatorname{Re}Y_\ell^m, &m>0. \end{cases} \]

In a real orthonormal basis:

a real scalar field has real coefficients;
Wigner matrices are real orthogonal.

Convention Requirements

A concrete spherical representation is not fixed until the following are specified together:

active or passive rotation convention;
complex or real component basis;
component ordering and phases;
normalization of the spherical harmonics;
matching Wigner \(\mathcal D\) and Clebsch–Gordan conventions;
whether an array contains basis-function values, expansion coefficients, projections onto a dual basis, or nonlinear quantities derived from them.

ELFES change-of-basis code requires exact versioned choices for these items. Those implementation choices and the e3nn reference basis do not alter the theory above.