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 $E_c(n)=-a\mathrm{/}{n}^b+E_c^{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{e}\mathrm{t}}$, 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.