  Our recent study reveals that macroscopic structure in thermodynamically
equilibrium state and its temperature dependence for classical discrete system
can be well-characterized by a single specially-selected microscopic state
(which we call projection state: PS), whose structure can be known a priori
without any information about energy or temperature. Although PS can be
universally constructed for any number of components R, practical application
of PS to systems with R >= 3 is non-trivial compared with R = 2 (i.e., binary
system). This is because (i) essentially, multicomponent system should
inevitably requires linear transformation from conventional basis functions to
intuitively-interpreted cluster probability basis, i.e., multiple PS energies
are required to predict one chosen pair probability, leading to practically
accumulating numerical errors, and (ii) additionally, explicit formula for the
transformation from basis functions to pair probabilities should be required,
which has been explicitly provided up to ternary (R = 3) system so far. We here
derive modified formulation to directly determine probability for likeand
unlike-atom pair consisting of any chosen elements by using a single PS energy,
with providing explicit relationship between basis functions and pair
probabilities up to quinary (R = 5) systems. We demonstrate the validity of the
formulation by comparing temperature dependence of pair probabilities for
multicomponent systems from thermodynamic simulations.
