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    Predicting stable planar boron clusters: Bonding rules, stability criteria, and global structure search accelerated by a machine learning force field

    Lu Qiu1,2,3,*,†, Cheng Qian4,*, Yuewen Mu5, Yuhan Liu1, Xiuyun Zhang6, Sung Youb Kim2,3,‡, and Feng Ding4,§

    • *These authors contributed equally to this work.
    • †Contact author: luqiu6@cityu.edu.hk
    • ‡Contact author: sykim@unist.ac.kr
    • §Contact author: dingf@szlab.ac.cn

    Phys. Rev. B 114, 254105 – Published 5 November, 2026

    DOI: https://doi.org/10.1103/x3bz-g9zz

    Abstract

    Boron's electron deficiency favors multicenter bonding and enables planar or quasiplanar boron nanostructures with unusually stable pristine edges and corners. Yet a compact bonding picture that explains these stability trends remains elusive. Here we introduce a bonding model for triangular-lattice-based planar boron clusters, defined by four bonding rules that use only 3c-2e and 2c-2e bonding and σ bond resonance. The model allows intuitive construction of σ bonding configurations without extensive quantum calculations, revealing the origin of the high stability of planar boron clusters and rationalizing their effective self-termination at pristine edges and specific corners. From this model we extract three stability criteria: energetically favored clusters tend to have (i) obtuse-angle convex and/or concave corners, (ii) flat edges with an even number of atoms, and (iii) an increasing number of hexagonal holes as the cluster grows. Applying these criteria, we predict B49, B70, and B84 as particularly stable planar clusters beyond the well-known B36, reproducing density-functional theory trends in formation energies. Independent validation using a machine learning force field–accelerated global structure search across both planar and three-dimensional manifolds confirms these predictions, while energy gaps, in this size range, remain small enough that planar forms stay competitive and are further stabilized on substrates. This resulting framework provides fundamental insights into boron chemistry and a practical route to rational design and scalable prediction of boron-based nanostructures.

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