Atomic Structure Optimization with Machine-Learning Enabled Interpolation between Chemical Elements

We introduce a computational method for global optimization of structure and ordering in atomic systems. The method relies on interpolation between chemical elements, which is incorporated in a machine-learning structural fingerprint. The method is based on Bayesian optimization with Gaussian processes and is applied to the global optimization of Au-Cu bulk systems, Cu-Ni surfaces with CO adsorption, and Cu-Ni clusters. The method consistently identifies low-energy structures, which are likely to be the global minima of the energy. For the investigated systems with 23–66 atoms, the number of required energy and force calculations is in the range 3–75.

Figure 1

A single local optimization run of element fractions in AuCu bulk with 64 atoms in a unit cell. The training set consists of four DFT calculations. The structures at different steps are shown above with the color saturation denoting the corresponding element fractions (black, Au; white, Cu). The black lines indicate the computational supercell boundaries. Middle: the evolution of the element fractions (where Au corresponds to 1 and Cu to 0) is shown for all atoms along the optimization. Lower: the energy predicted by the surrogate model is shown. The optimization is seen to proceed to the ground state without getting trapped in metastable states.

Figure 2

Success curves for optimizing the ordering of ${\mathrm{Au}}_x{\mathrm{Cu}}_y$ in a bulk fcc lattice using ice-beacon. For each system, ten individual optimization runs are performed. The number of DFT evaluations in the training set is shown on the $x$ axis, while the fraction of successful runs, where the correct structure is identified, is shown on the $y$ axis. The indicated uncertainties are calculated with bootstrapping.

Figure 3

Initial and optimal CuNi surface structures with 16 Cu atoms out of a total of 64. (a) Typical initial structure before optimization of the clean surface. The colors are linearly scaled between those of Ni and Cu corresponding to the element fractions of the atoms. (b) Optimal structure for the clean surface system as found with ice-beacon with simultaneous optimization of coordinates and fractions. The structure is found after three DFT calculations in all of four runs. (c) Optimal structure after introducing a CO molecule at the surface. We see that the CO molecule pulls a Ni atom to the surface due to the stronger chemisorption. During the optimizations, the bottommost layer is fixed to be Ni in fixed positions. Colors: Cu, brown; Ni, light gray; C, dark gray; O, red.

Figure 4

(a) Example of a random CuNi initial structure of a 23-atom cluster generated in a cubic box (black lines) before optimization in ice-beacon. The colors are linearly scaled between those of Ni and Cu, corresponding to the element fractions of the atoms. (b)–(d) The identified global minimum structures for ${\mathrm{Cu}}_{20}{\mathrm{Ni}}_3$, ${\mathrm{Cu}}_{18}{\mathrm{Ni}}_5$, and ${\mathrm{Cu}}_{12}{\mathrm{Ni}}_{11}$, respectively, as determined with ice-beacon. (e) Success curves for optimizing the clusters with beacon and ice-beacon up to 75 DFT calculations based on 20 runs each. ice-beacon requires considerably less DFT calculations than beacon, and in the case of 11 nickel atoms, where the number of possible orderings exceeds ${10}^6$, only ice-beacon finds the optimal structure.