  We introduce a fingerprint representation of molecules based on a Fourier
series of atomic radial distribution functions. This fingerprint is unique
(except for chirality), continuous, and differentiable with respect to atomic
coordinates and nuclear charges. It is invariant with respect to translation,
rotation, and nuclear permutation, and requires no pre-conceived knowledge
about chemical bonding, topology, or electronic orbitals. As such it meets many
important criteria for a good molecular representation, suggesting its
usefulness for machine learning models of molecular properties trained across
chemical compound space. To assess the performance of this new descriptor we
have trained machine learning models of molecular enthalpies of atomization for
training sets with up to 10k organic molecules, drawn at random from a
published set of 134k organic molecules. We validate the descriptor on all
remaining molecules of the 134k set. For a training set of 5k molecules the
fingerprint descriptor achieves a mean absolute error of 8.0 kcal/mol,
respectively. This is slightly worse than the performance attained using the
Coulomb matrix, another popular alternative, reaching 6.2 kcal/mol for the same
training and test sets.
