Spherical Harmonics

Spherical harmonics are basis functions on the sphere used to represent angular structure.

They are commonly written as:

$$Y_{\ell }^m(\theta ,\varphi )$$

where $\ell$ is the degree and $m$ is the order. Higher $\ell$ captures higher-frequency angular structure.

For a unit direction $\hat{r}\in S^2$, a spherical harmonic maps direction to an angular basis value:

$$Y_{\ell }^m:S^2\to \mathbb{C},-\ell \le m\le \ell$$

The degree $\ell$ has $2\ell +1$ components. In equivariant neural networks, these components transform together under rotation:

$$Y_{\ell }(R\hat{r})=D^{\ell }(R)Y_{\ell }(\hat{r})$$

where $R\in SO(3)$ and $D^{\ell }(R)$ is the Wigner-D matrix for degree $\ell$.

Radial and Angular Split

For two points $x_i,x_j\in {\mathbb{R}}^3$, define:

$$r_{ij}=x_j-x_i,d_{ij}=\Vert r_{ij}{\Vert }_2,{\hat{r}}_{ij}=\frac{r_{ij}}{\Vert r_{ij}{\Vert }_2+ϵ}$$

Many 3D models separate distance and direction:

$$\varphi (r_{ij})=R_n(d_{ij})Y_{\ell }^m({\hat{r}}_{ij})$$

where $R_n$ is a radial basis and $Y_{\ell }^m$ is an angular basis. Distances are rotation-invariant; directions carry rotation-equivariant information.

Why It Matters

• They encode directional information in rotation-aware neural networks.
• They provide a practical bridge between 3D geometry and irreducible representations.
• In molecular models, they can describe relative directions between atoms or residues.
• They help decide whether a model needs only scalar distance features or higher-order angular features.

Degree Choice

Maximum Degree	Captures	Cost/Risk
${\ell }_{\max }=0$	scalar radial information only	cannot represent direction-dependent interactions
${\ell }_{\max }=1$	vector-like directional structure	moderate cost, often enough for vector fields
${\ell }_{\max }\ge 2$	higher-order angular patterns	higher tensor-product and memory cost

For docking, conformers, and protein-ligand interactions, angular features can matter when orientation, directionality, or local geometry is part of the claim. If the task only predicts an invariant scalar from a fixed graph, distances and scalar features may be sufficient.

Checks

• Are angular features actually needed, or are distances sufficient?
• What maximum degree is used, and what compute cost does it add?
• Are directions normalized and numerically stable?
• Does the model combine radial and angular information cleanly?

Related

• SO(3)
• Tensor field network
• Equivariance
• Structure