  The applications of machine learning techniques to chemistry and materials
science become more numerous by the day. The main challenge is to devise
representations of atomic systems that are at the same time complete and
concise, so as to reduce the number of reference calculations that are needed
to predict the properties of different types of materials reliably. This has
led to a proliferation of alternative ways to convert an atomic structure into
an input for a machine-learning model. We introduce an abstract definition of
chemical environments that is based on a smoothed atomic density, using a
bra-ket notation to emphasize basis set independence and to highlight the
connections with some popular choices of representations for describing atomic
systems. The correlations between the spatial distribution of atoms and their
chemical identities are computed as inner products between these feature kets,
which can be given an explicit representation in terms of the expansion of the
atom density on orthogonal basis functions, that is equivalent to the smooth
overlap of atomic positions (SOAP) power spectrum, but also in real space,
corresponding to $n$-body correlations of the atom density. This formalism lays
the foundations for a more systematic tuning of the behavior of the
representations, by introducing operators that represent the correlations
between structure, composition, and the target properties. It provides a
unifying picture of recent developments in the field and indicates a way
forward towards more effective and computationally affordable machine-learning
schemes for molecules and materials.
