Added doc explanation on rotation equivariant features

Author: ecignoni

docs/src/explanations/index.rst

@@ -13,3 +13,4 @@ all about.
 
     concepts
     soap
+    rotation_adapted

docs/src/explanations/rotation_adapted.rst

@@ -0,0 +1,88 @@
+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 I first rotate the structure and compute the descriptor, I 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 structure as :math:`A`, the function computing the descriptor as :math:`f(\cdot)`, and the rotation operator as :math:`\hat{R}`, rotation equivariance can be expressed as:
+
+.. math::
+   :name: eq:equivariance
+
+   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 :math:`\rho_i(\mathbf{r})`.
+We have:
+
+.. math::
+
+    \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
+
+and noting that :math:`Y_l^m(\hat{R} \hat{\mathbf{r}}) = \hat{R} Y_l^m(\hat{\mathbf{r}})` and :math:`\hat{R}r = r`, equation :ref:`(1) <eq:equivariance>` is satisfied and we conclude that the expansion coefficients :math:`c_{nlm}^{i}` are rotation equivariant.
+Indeed, each :math:`c_{nlm}^{i}` transforms under rotation as the spherical harmonics :math:`Y_l^m(\hat{\mathbf{r}})`.
+
+Using the Dirac notation, the coefficient :math:`c_{nlm}^{i}` can be expressed as :math:`\braket{nlm\vert\rho_i}`.
+Equivalently, and to stress the fact that this coefficient describes something that transforms under rotation as a spherical harmonics :math:`\ket{lm}`, it is sometimes written as :math:`\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:
+
+.. math::
+    :name: eq:cg_coupling
+
+    \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 :math:`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 :math:`c_{nlm}^{i}` and :math:`Y_l^m`,
+:ref:`(2) <eq:cg_coupling>` 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 :ref:`(2) <eq:cg_coupling>`:
+
+.. math::
+    :name: eq:cg_decoupling
+
+    \ket{l_1 m_1} \ket{l_2 m_2} = \sum_{l m} C_{m_1 m_2 m}^{l_1 l_2 l m} \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: :math:`\lambda`-SOAP
+-----------------------------
+
+A straightforward application of :ref:`(2) <eq:cg_coupling>` is the construction of :math:`\lambda`-SOAP features.
+Indeed, :math:`\lambda`-SOAP was created in order to have a rotation and inversion equivariant version of the 3-body density correlations.
+The :math:`\lambda` represents the degree of a spherical harmonics, :math:`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 :ref:`(2) <eq:cg_coupling>`.
+The :math:`\lambda`-SOAP descriptor is then:
+
+.. math::
+
+    \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 real spherical harmonics coefficients.