Applying Statistical Mechanics to Improve Computational Sampling Algorithms and Interatomic Potentials

In this dissertation the application of statistical mechanics is presented to improve classical simulated annealing and machine learning-based interatomic potentials. Classical simulated annealing is known to be among the most robust global optimization methods. Therefore, many variations of this method have been developed over the last few decades. This dissertation introduces simulated annealing with adaptive cooling and shows its efficiency with respect to the classical simulated annealing. Adaptive cooling simulated annealing makes use of the on-the-fly evaluation of the sta- tistical mechanical properties to adaptively adjust the cooling rate. In this case, the cooling rate is adaptively adjusted based on the instantaneous evaluations of the heat capacities, with the possible future extension to the density of states. Results are presented for Lennard-Jones clusters optimized by adaptive cooling sim- ulated annealing and the classical simulated annealing. The adaptive cooling approach proved to be more efficient than the classical simulated annealing. Statistical mechanics was also used to improve the quality and transferability of machine learning- based interatomic potentials. Machine learning (ML)-based interatomic potentials are currently garnering a lot of attention as they strive to achieve the accuracy of electronic structure methods at the computational cost of empirical potentials. Given their generic functional forms, the transferability of these potentials is highly dependent on the quality of the training set, the generation of which is a highly labor-intensive activity. Good training sets should at once contain a very diverse set of configurations while avoiding redundancies that incur cost without providing benefits. We formalize these requirements in a local entropy maximization framework and propose an automated sampling scheme to sample from this objective function. We show that this approach generates much more diverse training sets than unbiased sampling and is competitive with hand-crafted training sets[1]

Transferable prediction of formation energy across lattices of increasing size

In this study, we show the transferability of graph convolutional neural network (GCNN) predictions of the formation energy of the nickel-platinum (NiPt) solid solution alloy across atomic structures of increasing sizes. The original dataset was generated with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) using the second nearest-neighbor modified embedded-atom method (2NN MEAM) empirical interatomic potential. Geometry optimization was performed on the initially randomly generated face centered cubic (FCC) crystal structures and the formation energy has been calculated at each step of the geometry optimization, with configurations spanning the whole compositional range. Using data from various steps of geometry optimization, we first trained the GCNN on a lattice of 256 atoms, which accounts well for the short-range interactions. Using this data, we predicted the formation energy for lattices of 864 atoms and 2,048 atoms, which resulted in lower-than-expected accuracy due to the long-range interactions present in these larger lattices. We accounted for the long-range interactions by including a small amount of training data representative for those two larger sizes, whereupon the predictions of the GCNN scaled linearly with the size of the lattice. Therefore, our strategy ensured scalability while reducing significantly the computational cost of training on larger lattice sizes

Ab initio approaches to high-entropy alloys: a comparison of CPA, SQS, and supercell methods

We present a comparative study of different modeling approaches to the electronic properties of the $\textrm{Hf}_{0.05}\textrm{Nb}_{0.05}\textrm{Ta}_{0.8}\textrm{Ti}_{0.05}\textrm{Zr}_{0.05}$ high entropy alloy. Common to our modeling is the methodology to compute the one-particle Green's function in the framework of density functional theory. We demonstrate that the special quasi-random structures modeling and the supercell, i.e. the locally self-consistent multiple-scatering methods provide very similar results for the ground state properties such as the spectral function (density of states) and the equilibrium lattice parameter. To reconcile the multiple-scattering single-site coherent potential approximation with the real space supercell methods, we included the effect of screening of the net charges of the alloy components. Based on the analysis of the total energy and spectral functions computed within the density functional theory, we found no signature for the long-range or local magnetic moments formation in the $\textrm{Hf}_{0.05}\textrm{Nb}_{0.05}\textrm{Ta}_{0.8}\textrm{Ti}_{0.05}\textrm{Zr}_{0.05}$ high entropy alloy, instead we find possible superconductivity below $\sim 9$K