  Machine learning frameworks based on correlations of interatomic positions
begin with a discretized description of the density of other atoms in the
neighbourhood of each atom in the system. Symmetry considerations support the
use of spherical harmonics to expand the angular dependence of this density,
but there is as yet no clear rationale to choose one radial basis over another.
Here we investigate the basis that results from the solution of the Laplacian
eigenvalue problem within a sphere around the atom of interest. We show that
this generates the smoothest possible basis of a given size within the sphere,
and that a tensor product of Laplacian eigenstates also provides the smoothest
possible basis for expanding any higher-order correlation of the atomic density
within the appropriate hypersphere. We consider several unsupervised metrics of
the quality of a basis for a given dataset, and show that the Laplacian
eigenstate basis has a performance that is much better than some widely used
basis sets and is competitive with data-driven bases that numerically optimize
each metric. In supervised machine learning tests, we find that the optimal
function smoothness of the Laplacian eigenstates leads to comparable or better
performance than can be obtained from a data-driven basis of a similar size
that has been optimized to describe the atom-density correlation for the
specific dataset. We conclude that the smoothness of the basis functions is a
key and hitherto largely overlooked aspect of successful atomic density
representations.
