Fig. 3: Rank deficiency in high-component ionic systems.
From: Approaches for handling high-dimensional cluster expansions of ionic systems
a Illustration of the cluster α with single site basis functions labelled by color. Since the zeroth label is independent of occupation and is always 1, the colors are the same (orange). However, since the rest of the basis functions (\(\gamma _{\alpha _j > 0,n_i}\left( {\sigma _i} \right)\)) are dependent on occupation, they are differently colored. b Rank of all orbits B for the fully random set of structures and the (c) rank deficiency of all orbits B for the set of structures in this study. The size of the cluster generating orbit B is indicated for points (1), pairs (2), triplets (3), quadruplets (4), and quintuplets (5).
