  Atomic cluster expansion (ACE) methods provide a systematic way to describe
particle local environments of arbitrary body order. For practical applications
it is often required that the basis of cluster functions be symmetrized with
respect to rotations and permutations. Existing methodologies yield sets of
symmetrized functions that are over-complete. These methodologies thus require
an additional numerical procedure, such as singular value decomposition (SVD),
to eliminate redundant functions. In this work, it is shown that analytical
linear relationships for subsets of cluster functions may be derived using
recursion and permutation properties of generalized Wigner symbols. From these
relationships, subsets (blocks) of cluster functions can be selected such that,
within each block, functions are guaranteed to be linearly independent. It is
conjectured that this block-wise independent set of permutation-adapted
rotation and permutation invariant (PA-RPI) functions forms a complete,
independent basis for ACE. Along with the first analytical proofs of block-wise
linear dependence of ACE cluster functions and other theoretical arguments,
numerical results are offered to demonstrate this. The utility of the method is
demonstrated in the development of an ACE interatomic potential for tantalum.
Using the new basis functions in combination with Bayesian compressive sensing
sparse regression, some high degree descriptors are observed to persist and
help achieve high-accuracy models.
