  We introduce and explore an approach for constructing force fields for small
molecules, which combines intuitive low body order empirical force field terms
with the concepts of data driven statistical fits of recent machine learned
potentials. We bring these two key ideas together to bridge the gap between
established empirical force fields that have a high degree of transferability
on the one hand, and the machine learned potentials that are systematically
improvable and can converge to very high accuracy, on the other. Our framework
extends the atomic Permutationally Invariant Polynomials (aPIP) developed for
elemental materials in [Mach. Learn.: Sci. Technol. 2019 1 015004] to molecular
systems. The body order decomposition allows us to keep the dimensionality of
each term low, while the use of an iterative fitting scheme as well as
regularisation procedures improve the extrapolation outside the training set.
We investigate aPIP force fields with up to generalised 4-body terms, and
examine the performance on a set of small organic molecules. We achieve a high
level of accuracy when fitting individual molecules, comparable to those of the
many-body machine learned force fields. Fitted to a combined training set of
short linear alkanes, the accuracy of the aPIP force field still significantly
exceeds what can be expected from classical empirical force fields, while
retaining reasonable transferability to both configurations far from the
training set and to new molecules.
