Beyond pair entropy: orientational many-body correlations in supercooled glass-forming liquids from a four-point...

The conventional pair entropy, derived from the isotropic radial distribution function g(r), systematically underestimates structural ordering in supercooled glass-forming liquids. We argue that this failure arises because S₂, constructed from the isotropic g(r), is inherently insensitive to the many-body orientational correlations that become accessible only through a four-point conditional distribution evaluated in a local particle-centered frame. To recover this information, we introduce a three-dimensional four-point structural entropy S_3D, constructed from the four-point conditional distribution function g(r,θ,ϕ) evaluated in a local particle-centered reference frame, and derive its exact decomposition into a radial contribution S₂ and a weighted orientational entropy S_Ω. Applying this framework to the canonical KA binary mixture, we find that S_Ω accounts for a substantial fraction of S_3D across the full temperature range studied, reflecting genuine icosahedral and dodecahedral angular ordering present at all temperatures-as independently established by Zhang and Kob-rather than a numerical artifact. These results demonstrate that g(r,θ,ϕ) encodes angular structural information entirely invisible to the conventional g(r) and to the thermodynamic excess entropy S_ex, and that S_3D provides a tractable thermodynamic measure of packing-driven orientational ordering across the supercooled regime.