Fast proper orthogonal descriptors for many-body interatomic potentials

The development of differentiable invariant descriptors for accurate representations of atomic environments plays a central role in the success of interatomic potentials for chemistry and materials science. We introduce a method to generate fast proper orthogonal descriptors for the construction of many-body interatomic potentials, and we discuss its relation to existing empirical and machine-learning interatomic potentials. A traditional way of implementing the proper orthogonal descriptors has a computational complexity that scales exponentially with the body order in terms of the number of neighbors. We present an algorithm to compute the proper orthogonal descriptors with a computational complexity that scales linearly with the number of neighbors irrespective of the body order. We show that our method can enable a more efficient implementation for a number of existing potentials, and we provide a scalable systematic framework to construct new many-body potentials. The new potentials are demonstrated on a data set of density functional theory calculations for tantalum and compared with other interatomic potentials.