Global Optimization of Atomic Clusters via Physically-Constrained Tensor Train Decomposition

TL;DR

This work introduces a tensor-train (TT) based framework for the global optimization of atomic clusters, addressing the exponential growth of local minima via low-rank energy representations and two complementary optimizers, TTOpt and PROTES. By embedding physically-constrained encodings directly into discretization, the approach dramatically reduces the effective search dimensionality while preserving chemical realism. It demonstrates strong results on Lennard-Jones clusters up to 45 atoms and validates applicability to real systems by optimizing C20 structures with a machine-learned Moment Tensor Potential fitted to DFT data, yielding geometries consistent with quantum-accurate simulations. The framework offers a general, adaptable pathway for high-dimensional optimization in computational materials science, with a tunable balance between efficiency and reliability depending on problem size and landscape ruggedness.

Abstract

The global optimization of atomic clusters represents a fundamental challenge in computational chemistry and materials science due to the exponential growth of local minima with system size (i.e., the curse of dimensionality). We introduce a novel framework that overcomes this limitation by exploiting the low-rank structure of potential energy surfaces through Tensor Train (TT) decomposition. Our approach combines two complementary TT-based strategies: the algebraic TTOpt method, which utilizes maximum volume sampling, and the probabilistic PROTES method, which employs generative sampling. A key innovation is the development of physically-constrained encoding schemes that incorporate molecular constraints directly into the discretization process. We demonstrate the efficacy of our method by identifying global minima of Lennard-Jones clusters containing up to 45 atoms. Furthermore, we establish its practical applicability to real-world systems by optimizing 20-atom carbon clusters using a machine-learned Moment Tensor Potential, achieving geometries consistent with quantum-accurate simulations. This work establishes TT-decomposition as a powerful tool for molecular structure prediction and provides a general framework adaptable to a wide range of high-dimensional optimization problems in computational material science.

Figures (6)

• Figure 1: Schematic representation of the TT-decomposition. The procedure for calculating element $[n_1, n_2, \ldots, n_d]$ using a tensor in the TT-format is presented at the top, and the tensor diagram corresponding to the TT-decomposition is demonstrated at the bottom.
• Figure 2: Schematic representation of the optimization method TTOpt.
• Figure 3: Schematic representation of the optimization method PROTES.
• Figure 4: Validation of optimized LJ cluster structures through pairwise distance analysis. The ground truth distributions (green) are closely matched by our method's predictions (red dashed) for clusters containing 5, 13, 38, and 43 atoms. The accurate reproduction of distance histograms confirms the structural correctness of the identified minima.
• Figure 5: Structures of the identified $\text{C}_{20}$ atomic clusters: monocyclic cap, fullerene cage structure, and buckyball.