Beyond the Binary Hull: Higher-Order Thermodynamic Stabilization in Inorganic Ternary Compounds
Stable ternary compounds lie below every competing phase on a conventional convex hull, but this does not reveal whether their stability is inherited from already favorable binaries or created by bringing three elements into one crystal. We define the emergent ternary stabilization energy, delta E3, as the energy by which a stable ternary lies below the complete unary-binary hull at the same composition. Applied to 19,209 Materials Project compounds calculated within the GGA framework, delta E3 has a median of 63.3 meV/atom and spans from nearly binary-degenerate phases to compounds stabilized by several hundred meV/atom. The distribution separates recognizable chemical limits: intermetallics have the smallest median stabilization, single-anion compounds occupy a broad intermediate regime, and polyanion compounds receive the largest gains. Within single anion families, large delta E3 is associated with cation electronegativity contrast, bandgap opening, and reconstruction of coordination, bond geometry, and local neighbour chemistry relative to the actual binary decomposition products. Experimentally linked compounds obtain a median 4.92% of their formation energy from ternary stabilization, compared with 3.53% for theoretical compounds, with the clearest separation in sulphides and selenides. delta E3 therefore provides a compact measure of the thermodynamic value of compositional complexity: low values identify phases whose stability is largely reproducible by binaries, while high values identify compounds that gain substantially from ternary specific electronic and structural organization.