Rotation-Adapted Features

Equivariance

Descriptors like SOAP are translation, rotation, and permutation invariant. Indeed, such invariances are extremely useful if one wants to learn an invariant target (e.g., the energy). Being already encoded in the descriptor, the learning algorithm does not have to learn such a physical requirement.

The situation is different if the target is not invariant. For example, one may want to learn a dipole. The dipole rotates with a rotation of the molecule, and as such, invariant descriptors do not have the required symmetries for this task.

Instead, one would need a rotation equivariant descriptor. Rotation equivariance means that, if we first rotate the system and compute the descriptor, we obtain the same result as first computing the descriptor and then applying the rotation, i.e., the descriptor behaves correctly upon rotation operations. Denoting a system as \(A\), the function computing the descriptor as \(f(\cdot)\), and the rotation operator as \(\hat{R}\), rotation equivariance can be expressed as:

(1)\[f(\hat{R} A) = \hat{R} f(A)\]

Of course, invariance is a special case of equivariance.

Rotation Equivariance of the Spherical Expansion

The spherical expansion is a rotation equivariant descriptor. Let’s consider the expansion coefficients of \(\rho_i(\mathbf{r})\). We have:

\[\begin{split}\hat{R} \rho_i(\mathbf{r}) &= \sum_{nlm} c_{nlm}^{i} R_n(r) \hat{R} Y_l^m(\hat{\mathbf{r}}) \nonumber \\ &= \sum_{nlmm'} c_{nlm}^{i} R_n(r) D_{m,m'}^{l}(\hat{R}) Y_l^{m'}(\hat{\mathbf{r}}) \nonumber \\ &= \sum_{nlm} \left( \sum_{m'} D_{m',m}^l(\hat{R}) c_{nlm'}^{i}\right) B_{nlm}(\mathbf{r}) \nonumber\end{split}\]

and noting that \(Y_l^m(\hat{R} \hat{\mathbf{r}}) = \hat{R} Y_l^m(\hat{\mathbf{r}})\) and \(\hat{R}r = r\), equation (1) is satisfied and we conclude that the expansion coefficients \(c_{nlm}^{i}\) are rotation equivariant. Indeed, each \(c_{nlm}^{i}\) transforms under rotation as the spherical harmonics \(Y_l^m(\hat{\mathbf{r}})\).

Using the Dirac notation, the coefficient \(c_{nlm}^{i}\) can be expressed as \(\braket{nlm\vert\rho_i}\). Equivalently, and to stress the fact that this coefficient describes something that transforms under rotation as a spherical harmonics \(Y_l^m(\hat{\mathbf{r}})\), it is sometimes written as \(\braket{n\vert\rho_i;lm}\), i.e., the atomic density is “tagged” with a label that tells how it transforms under rotations.

Completeness Relations of Spherical Harmonics

Spherical harmonics can be combined together using rules coming from standard theory of angular momentum:

(2)\[\ket{lm} \propto \ket{l_1 l_2 l m} = \sum_{m_1 m_2} C_{m_1 m_2 m}^{l_1 l_2 l} \ket{l_1 m_1} \ket{l_2 m_2}\]

where \(C_{m_1 m_2 m}^{l_1 l_2 l}\) is a Clebsch-Gordan (CG) coefficient.

Thanks to the one-to-one correspondence (under rotation) between \(c_{nlm}^{i}\) and \(Y_l^m\), (2) means that one can take products of two spherical expansion coefficients (which amounts to considering density correlations), and combine them with CG coefficients to get new coefficients that transform as a single spherical harmonics. This process is known as coupling, from the uncoupled basis of angular momentum (formed by the product of rotation eigenstates) to a coupled basis (a single rotation eigenstate).

One can also write the inverse of (2):

(3)\[\ket{l_1 m_1} \ket{l_2 m_2} = \sum_{l m} C_{m_1 m_2 m}^{l_1 l_2 l} \ket{l_1 l_2 l m}\]

that express the product of two rotation eigenstates in terms of one. This process is known as decoupling.

Example: \(\lambda\)-SOAP

A straightforward application of (2) is the construction of \(\lambda\)-SOAP features. Indeed, \(\lambda\)-SOAP was created in order to have a rotation and inversion equivariant version of the 3-body density correlations. The \(\lambda\) represents the degree of a spherical harmonics, \(Y_{\lambda}^{\mu}(\hat{\mathbf{r}})\), and it indicates that this descriptor can transform under rotations as a spherical harmonics, i.e., it is rotation equivariant.

It is then obtained by considering two expansion coefficients of the atomic density, and combining them with a CG iteration to a coupled basis, as in (2). The \(\lambda\)-SOAP descriptor is then:

\[\braket{n_1 l_1 n_2 l_2\vert\overline{\rho_i^{\otimes 2}, \sigma, \lambda \mu}} = \frac{\delta_{\sigma, (-1)^{l_1 + l_2 + \lambda}}}{\sqrt{2 \lambda + 1}} \sum_{m} C_{m (\mu-m) \mu}^{l_1 l_2 \lambda} c_{n_1 l_1 m}^{i} c_{n_2 l_2 (\mu - m)}^{i}\]

where we have assumed complex spherical harmonics coefficients.