  A central problem in multicomponent lattice systems is to systematically
quantify multi-point ordering. Ordering in such systems is often described in
terms of pair correlations, even though this is not sufficient when three-point
and higher-order interactions are significant. Current models and parameters
for multi-point ordering are often only applicable for very specific cases
and/or require approximating a subset of correlated occupational variables on a
lattice as being uncorrelated. In this work, a multi-point cluster order
parameter is introduced to quantify arbitrary multi-point ordering motifs in
substitutional systems through direct calculations of normalized cluster
probabilities. These parameters can describe multi-point chemical ordering in
crystal systems with multiple sublattices, multiple components, and systems
with reduced symmetry. These are defined within and applied to quantify
four-point chemical ordering motifs in platinum/palladium alloy nanoparticles
that are practical interest to the synthesis of catalytic nanocages. Impacts of
chemical ordering on alloy nanocage stability are discussed. It is demonstrated
that approximating multi-point order parameters from combinations of low-order
correlations is not sufficient in cases where multi-point energetics are
significant. Conclusions about the formation mechanisms of nanocages may change
significantly when using common low-order approximations.
