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