On representing chemical environments

We review some recently published methods to represent atomic neighbourhood\nenvironments, and analyse their relative merits in terms of their faithfulness\nand suitability for fitting potential energy surfaces. The crucial properties\nthat such representations (sometimes called descriptors) must have are\ndifferentiability with respect to moving the atoms, and invariance to the basic\nsymmetries of physics: rotation, reflection, translation, and permutation of\natoms of the same species. We demonstrate that certain widely used descriptors\nthat initially look quite different are specific cases of a general approach,\nin which a finite set of basis functions with increasing angular wave numbers\nare used to expand the atomic neighbourhood density function. Using the example\nsystem of small clusters, we quantitatively show that this expansion needs to\nbe carried to higher and higher wave numbers as the number of neighbours\nincreases in order to obtain a faithful representation, and that variants of\nthe descriptors converge at very different rates. We also propose an altogether\nnew approach, called Smooth Overlap of Atomic Positions (SOAP), that sidesteps\nthese difficulties by directly defining the similarity between any two\nneighbourhood environments, and show that it is still closely connected to the\ninvariant descriptors. We test the performance of the various representations\nby fitting models to the potential energy surface of small silicon clusters and\nthe bulk crystal.\n