e3x.so3.irreps.solid_harmonics

e3x.so3.irreps.solid_harmonics(r, max_degree, cartesian_order=True)[source]

Real Cartesian (regular) solid harmonics \(R_\ell^m(\vec{r})\).

\[R_{\ell}^{m}(\vec{r}) = \lVert\vec{r}\rVert^\ell Y_{\ell}^{m}(\vec{r})\,,\]

where \(Y_{\ell}^{m}(\vec{r})\) are spherical harmonics using Racah’s normalization. See also spherical_harmonics for more details.

Parameters:

r (<class 'Float[Array, '... 3']'>) – Array of shape (..., 3) containing Cartesian vectors \(\vec{r}\).
max_degree (int) – Maximum degree \(L\) of the solid harmonics.
cartesian_order (bool, default: True) – If True, solid harmonics are returned in Cartesian order.

Return type:

<class 'Float[Array, '... (max_degree+1)**2']'>

Returns:

The values \(R_\ell^m(\vec{r})\) of all solid harmonics up to \(\ell\) = max_degree. Values are returned in an Array of shape (..., (max_degree+1)**2) ordered \([R_{0}^{0}\ R_{1}^{1}\ R_{1}^{-1}\ R_{1}^{0}\ R_{2}^{2}\ \cdots]\). If cartesian_order = False, values are ordered \([R_{0}^{0}\ R_{1}^{-1}\ R_{1}^{0}\ R_{1}^{1}\ R_{2}^{-2}\ \cdots]\) instead.

Raises:

ValueError – If r has an invalid shape (not a 3-vector), or
max_degree` is not positive or zero –