Bonding-Property Relationships in HfNbTaTiMoX (X = V, Fe, Zr, Re) Refractory High-Entropy Alloys.*

We present a comparative first-principles study of four refractory high-entropy alloys (RHEAs) of the form HfNbTaTiMoX(X = V, Fe, Zr, Re) using density functional theory (DFT). Atomic disorder was modeled with a 108-atom special quasi-random structure (SQS) based on a 3 × 3 × 3 supercell to capture statistical randomness. We evaluated formation energies, mechanical and thermodynamic properties, electronic structures, and charge transfer characteristics, revealing a complex interplay among them. To quantify bonding strength and internal cohesion, we employed the total bond order density (TBOD), a quantum-mechanical metric that correlates with mechanical performance. Among the alloys, Re-HEA exhibits the highest TBOD and stiffness, while Zr-HEA shows the lowest TBOD and greatest ductility. Density of states analysis indicates that Re and Fe significantly contribute near the Fermi level through strong d–dorbital hybridization. Partial charge analysis further supports the enhanced electronic stability of Re-HEA and highlights charge imbalance and weaker bonding in Zr-HEA, consistent with all other calculations. These findings demonstrate that atomic size and d-state occupancy govern the physical properties of RHEAs. More broadly, they underscore the role of elemental substitution in tuning the balance between strength and ductility, establishing Re as a promising additive for structural applications under demanding conditions.