Entropy-driven electron density and effective model Hamiltonian for boron systems

The unique electron deficiency of boron makes it challenging to determine the stable structures, leading to a wide variety of forms. In this work, we introduce a statistical model based on grand canonical ensemble theory that incorporates the octet rule to determine electron density in boron systems. This parameter-free model, referred to as the bonding free-energy (BFE) model, aligns well with first-principles calculations and accurately predicts total energies. For borane clusters, the model successfully predicts isomer energies, hydrogen diffusion pathways, and optimal charge quantity for closo-boranes. In all-boron clusters, the absence of B-H bond constraints enables increased electron delocalization and flexibility. The BFE model systematically explains the geometric structures and chemical bonding in boron clusters, revealing variations in electron density that clarify their structural diversity. For borophene, the BFE model predicts that hexagonal vacancy distributions are influenced by bonding entropy, with uniform electron density enhancing stability. Notably, our model predicts borophenes with a vacancy concentration of $\frac{1}{6}$ to exhibit increased stability with long-range periodicity. Therefore, the BFE model serves as a practical criterion for structure prediction, providing essential insights into the stability and physical properties of boron-based systems.