Data-Driven Learning of Total and Local Energies in Elemental Boron

The allotropes of boron continue to challenge structural elucidation and solid-state theory. Here we use machine learning combined with random structure searching (RSS) algorithms to systematically construct an interatomic potential for boron. Starting from ensembles of randomized atomic configurations, we use alternating single-point quantum-mechanical energy and force computations, Gaussian approximation potential (GAP) fitting, and GAP-driven RSS to iteratively generate a representation of the element’s potential-energy surface. Beyond the total energies of the very different boron allotropes, our model readily provides atom-resolved, local energies and thus deepened insight into the frustrated $\beta$-rhombohedral boron structure. Our results open the door for the efficient and automated generation of GAPs, and other machine-learning-based interatomic potentials, and suggest their usefulness as a tool for materials discovery.

Figure 1

The iterative construction of a machine-learning model for the potential-energy surface of boron. (a) An energy–volume plot for the training database of DFT reference computations. Starting with an initial seed of random structures (light green), we iteratively generate new seeds and relax them using the previous generation of the GAP, adding data points after 5 (teal) and up to 200 (purple) conjugate-gradient (CG) steps. The optimized $\alpha \text{−}{\mathrm{B}}_{12}$ structure is set as the energy zero. (b) The evolution of DFT energies in the progressively generated database, for (i) less constrained and (ii) more constrained initial seed structures; (iii) GAP-RSS snapshots after 5 CG steps; (iv) the same after up to 200 CG steps; see text for details. Boxes denote percentiles corresponding to one standard deviation (68%); bold horizontal lines denote the median, and whiskers the entire range of data. (c) RMSE energy errors for test sets of distorted crystal structures with small unit cells. After step $\mathbf{4}$, these crystal structures were added to the database (“$+\mathrm{C}$”).

Figure 2

The energy–volume plots for boron allotropes. The symbols denote the results of DFT computations, whereas lines connect GAP energies for the same structures. Two versions of the potential have been fitted to the complete reference database: one (dashed lines) using the smooth settings as in the iterative protocol, and one (thick lines) with the final, tighter settings.

Figure 3

The stability of randomized mixed-occupation models for $\beta \text{−}\mathrm{B}$ as a test for the transferability of the potential. The structural drawing shows the hexagonal unit-cell setup according to Ref. [7]; fully occupied Wyckoff sites are indicated by small, light gray atoms, whereas partially occupied sites are shown by larger, black atoms. On the latter sites, we randomly distribute 24 B atoms, which, together with 297 atoms on fully occupied sites, constitute the $\beta \text{−}{\mathrm{B}}_{321}$ model. The right-hand side shows energies for ten arbitrary, discrete structural models created in this fashion, computed using DFT and the final version of our GAP, respectively.

Figure 4

A local energy analysis using our GAP, comparing the simplistic $\beta \text{−}{\mathrm{B}}_{105}$ model without partial occupations (a) to the more disordered and favorable $\beta \text{−}{\mathrm{B}}_{106}$ model (b). The characteristic fragment at the center of the $\beta \text{−}\mathrm{B}$ unit cell is shown, with atoms color coded according to more favorable (blue) to less favorable (red) environments. Crystallographic sites are labeled as in the original literature; the analysis was performed for optimized structures. Visualization was done using ovito [46]. Note that while a vacancy has been created locally, the structural model in (b) contains one more atom per unit cell overall, due to the additional occupation of B16 sites (not shown).