Coordination-Driven Classification and Energetic Scaling of Boron Fullerenes and Borophene

We present a comprehensive first-principles investigation of boron fullerenes and two-dimensional boron sheets, unified under a coordination-based framework. By classifying over a dozen boron nanostructures, including B$_{12}$, B$_{40}$, B$_{38}$, B$_{65}$, B$_{80}$, and B$_{92}$, according to their local atomic environments (4-, 5-, and 6-fold coordination), we identify clear trends in structural stability, electronic properties, and magnetism. A universal energetic scaling relation $E_c(n) = -a/n^b + E_c^{\mathrm{sheet}}$, with $b = 0.4$ or $b = 0.9$ depending on the coordination family, captures the convergence of fullerene cohesive energies toward those of 2D boron phases. Notably, we establish one-to-one structural correspondences between select cages and experimentally accessible borophenes: B$_{40}$ mirrors the $\chi^3$-sheet, B$_{65}$ the $\beta_{12}$-sheet, B$_{80}$ the $\alpha$-sheet, and B$_{92}$ the $bt$-sheet. Our analysis reveals two distinct families of boron nanostructures based on the scaling exponent: one with $b = 0.9$, comprising structures with significant 6-fold coordination, and another with $b = 0.4$, which includes B$_{40}$, B$_{38}$, and two experimentally observed borophenes. These clusters also exhibit large HOMO--LUMO gaps (e.g., $E_g = 1.78$~eV for B$_{40}$, 1.14~eV for B$_{92}$), contrasting with the metallicity of their 2D counterparts and, in the case of B$_{65}$, spontaneous spin polarization ($M = 3\mu_B$). Our findings provide a predictive strategy for designing boron nanostructures by leveraging coordination fingerprints, and are further validated by the recent experimental synthesis of the B$_{80}$ cage. This work bridges zero- and two-dimensional boron chemistry, offering a roadmap for the future synthesis and application of boron-based materials.