From 2350c4b24244b57a4f0d68b91daf450201c4ddf8 Mon Sep 17 00:00:00 2001 From: Edoardo Cignoni Date: Tue, 23 May 2023 15:23:40 +0200 Subject: [PATCH] Added doc explanation on rotation equivariant features --- docs/src/explanations/index.rst | 1 + docs/src/explanations/rotation_adapted.rst | 88 ++++++++++++++++++++++ 2 files changed, 89 insertions(+) create mode 100644 docs/src/explanations/rotation_adapted.rst diff --git a/docs/src/explanations/index.rst b/docs/src/explanations/index.rst index 8643a7235..70e37a67f 100644 --- a/docs/src/explanations/index.rst +++ b/docs/src/explanations/index.rst @@ -13,3 +13,4 @@ all about. concepts soap + rotation_adapted diff --git a/docs/src/explanations/rotation_adapted.rst b/docs/src/explanations/rotation_adapted.rst new file mode 100644 index 000000000..b7a4b2ab3 --- /dev/null +++ b/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) ` 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) ` 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) `: + +.. 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) ` 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) `. +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.