Machine-Learned Potential Energy Surfaces for Free Sodium Clusters with Density Functional Accuracy: Applications to Melting

Gaussian Process Regression-based Gaussian Approximation Potential has been used to develop machine-learned interatomic potentials having density-functional accuracy for free sodium clusters. The training data was generated from a large sample of over 100,000 data points computed for clusters in the size range of N = 40 - 200, using the density-functional method as implemented in the VASP package. Two models have been developed, model M1 using data for N=55 only, and model M2 using additional data from larger clusters. The models are intended for computing thermodynamic properties using molecular dynamics. Hence, particular attention has been paid to improve the fitting of the forces. Interestingly, it turns out that the best fit can be obtained by carefully selecting a smaller number of data points viz. 1,900 and 1,300 configurations, respectively, for the two models M1 and M2. Although it was possible to obtain a good fit using the data of Na55 only, additional data points from larger clusters were needed to get better accuracies in energies and forces for larger sizes. Surprisingly, the model M1 could be significantly improved by adding about 50 data points per cluster from the larger sizes. Both models have been deployed to compute the heat capacities of Na55 and Na147 and to obtain about 40 isomers for larger clusters of sizes N = 147, 200, 201, and 252. There is an excellent agreement between the computed and experimentally measured melting temperatures. The geometries of these isomers when further optimized by DFT, the mean absolute error in the energies between DFT results and those of our models is about 7 meV/atom or less. The errors in the interatomic bond lengths are estimated to be below 2% in almost all the cases.