## Smooth overlap of atomic positions (SOAP) SOAP represents the atomic density around an atom $j$ as a sum of Gaussians centered on each surrounding atom of species $a$. The sum can be cast into a smooth, local probability amplitude by employing a cut-off. SOAP is permutation invariant, rotation invariant, translation invariant, and can detect atomic environment. And it is smooth & continuous, so more suitable for ML compared to raw coordinates. >[!Note] >In the form, each atom is described as a smoothed summation of Gaussians, multiplied by a spherical harmonic function to describe its angular dependence. And finally, all atomic contributions are added together and form a character vector (last step is the so-called 'Smooth overlap') >[!Info] >This doc may help to understand this concept: https://singroup.github.io/dscribe/0.3.x/tutorials/soap.html#:~:text=Smooth%20Overlap%20of%20Atomic%20Positions%20(SOAP)%20is%20a%20descriptor%20that,harmonics%20and%20radial%20basis%20functions.