Spherical Harmonics

The spherical harmonics \(Y^l(x)\) are functions defined on the sphere \(S^2\). They form a basis of the space on function on the sphere:

\[\mathcal{F} = \{ S^2 \longrightarrow \mathbb{R} \}\]

On this space it is natural how the group \(O(3)\) acts, Given \(p_a, p_v\) two scalar representations:

\[[L(g) f](x) = p_v(g) f(p_a(g) R(g)^{-1} x), \quad \forall f \in \mathcal{F}, x \in S^2\]

\(L\) is representation of \(O(3)\). But \(L\) is not irreducible. It can be decomposed via a change of basis into a sum of irreps, In a handwavey notation we can write:

\[Y^T L(g) Y = 0 \oplus 1 \oplus 2 \oplus 3 \oplus \dots\]

where the change of basis are the spherical harmonics! This notation is handwavey because \(x\) is a continuous variable, and therefore the change of basis \(Y\) is not a matrix.

As a consequence, the spherical harmonics are equivariant,

\[Y^l(R(g) x) = D^l(g) Y^l(x)\]

r = s2_grid()

r is a grid on the sphere.

Each point on the sphere has 3 components. If we plot the value of each of the 3 component separately we obtain the following figure:

plot(r, radial_abs=False)

x, y and z are represented as 3 scalar fields on 3 different spheres. To obtain a nicer figure (that looks like the spherical harmonics shown on Wikipedia) we can deform the spheres into a shape that has its radius equal to the absolute value of the plotted quantity:

plot(r)

\(Y^1\) is the identity function. Now let’s compute \(Y^2\), for this we take the tensor product \(r \otimes r\) and extract the \(L=2\) part of it.

tp = o3.ElementwiseTensorProduct("1o", "1o", ['2e'], irrep_normalization='norm')
y2 = tp(r, r)
plot(y2)

Similarly, the next spherical harmonic function \(Y^3\) is the \(L=3\) part of \(r \otimes r \otimes r\):

tp = o3.ElementwiseTensorProduct("2e", "1o", ['3o'], irrep_normalization='norm')
y3 = tp(y2, r)
plot(y3)

The functions below are more efficient versions not using e3nn.o3.ElementwiseTensorProduct:

Details

e3nn.o3.spherical_harmonics(l: int | List[int] | str | Irreps, x: Tensor, normalize: bool, normalization: str = 'integral')[source]

Spherical harmonics

https://user-images.githubusercontent.com/333780/79220728-dbe82c00-7e54-11ea-82c7-b3acbd9b2246.gif

Polynomials defined on the 3d space \(Y^l: \mathbb{R}^3 \longrightarrow \mathbb{R}^{2l+1}\)
Usually restricted on the sphere (with normalize=True) \(Y^l: S^2 \longrightarrow \mathbb{R}^{2l+1}\)
who satisfies the following properties:

are polynomials of the cartesian coordinates x, y, z
is equivariant \(Y^l(R x) = D^l(R) Y^l(x)\)
are orthogonal \(\int_{S^2} Y^l_m(x) Y^j_n(x) dx = \text{cste} \; \delta_{lj} \delta_{mn}\)

The value of the constant depends on the choice of normalization.

It obeys the following property:

\[ \begin{align}\begin{aligned}Y^{l+1}_i(x) &= \text{cste}(l) \; & C_{ijk} Y^l_j(x) x_k\\\partial_k Y^{l+1}_i(x) &= \text{cste}(l) \; (l+1) & C_{ijk} Y^l_j(x)\end{aligned}\end{align} \]

Where \(C\) are the wigner_3j.

Note

This function match with this table of standard real spherical harmonics from Wikipedia when normalize=True, normalization='integral' and is called with the argument in the order y,z,x (instead of x,y,z).

Parameters:

l (int or list of int) – degree of the spherical harmonics.
x (torch.Tensor) – tensor \(x\) of shape (..., 3).
normalize (bool) – whether to normalize the x to unit vectors that lie on the sphere before projecting onto the spherical harmonics
normalization ({'integral', 'component', 'norm'}) – normalization of the output tensors — note that this option is independent of normalize, which controls the processing of the input, rather than the output. Valid options: * component: \(\|Y^l(x)\|^2 = 2l+1, x \in S^2\) * norm: \(\|Y^l(x)\| = 1, x \in S^2\), component / sqrt(2l+1) * integral: \(\int_{S^2} Y^l_m(x)^2 dx = 1\), component / sqrt(4pi)

Returns:

a tensor of shape (..., 2l+1)

\[Y^l(x)\]

Return type:

torch.Tensor

Examples

>>> spherical_harmonics(0, torch.randn(2, 3), False, normalization='component')
tensor([[1.],
        [1.]])