A generalized and adaptable tensor-contraction-based cluster expansion formalism for multicomponent solids

TL;DR

The paper introduces the Tensor Cluster Expansion (TCE), a generalization of the cluster expansion that expresses correlation functions as mixed tensor contractions using precomputed topology tensors, enabling handling of exotic and low-symmetry lattices without enumerating clusters. This formulation yields an energy model $\,\mathcal{H}_\text{eff}(\mathbf{X})$ that is linear in a contracted feature vector and supports universal, incremental energy updates suitable for MC/MD simulations on massively parallel hardware. The authors implement the approach in the open-source tce-lib and validate it through three benchmarks: a timing study showing near-constant energy-difference computation with system size, a TaW heat-of-mixing calculation from DFT data, and Cowley SRO parameters for the CoNiCrFeMn high-entropy alloy using a MEAM potential. In both TaW and CoNiCrFeMn tests, the derived CE models reproduce ground-truth energies and ordering metrics with high fidelity, demonstrating the method's accuracy and scalability for multicomponent solids. Limitations include fixed lattice mappings and long-range interactions, with proposed extensions to variable-lattice CE and electrostatic terms to broaden applicability.

Abstract

Density functional theory (DFT)-based simulations of materials have first-principles accuracy, but are very computationally expensive. For simulating various properties of multi-component alloys, the cluster expansion (CE) technique has served as the standard workaround to improve computational efficiency. However, the standard CE technique is difficult to extend to exotic and/or low-symmetry lattices, often implemented via iteration over particular cluster types, which must be enumerated per lattice structure. In this work, we introduce the tensor cluster expansion (TCE), implemented in the open-source code tce-lib, which maps correlation functions to mixed tensor contractions, eliminating the need to iterate over cluster types and additionally making the calculation of correlation functions well-suited for massively parallel architectures like GPUs. We show that local interaction energies are an immediate consequence of the TCE formalism, yielding nearly $\mathcal{O}(1)$ energy difference calculations. We then use this formalism to fit CE models for the TaW and CoNiCrFeMn systems, and use these models to respectively compute the enthalpy of mixing curve and Cowley short-range order parameters, showing excellent agreement with ground truth data.

Figures (7)

• Figure 1: Naive vs. shortcut computation time benchmarked over various system sizes with respective power law fits. The time complexities presented in the legends are determined via power-law fits (denoted by dashed lines) to the presented data points.
• Figure 2: Distribution of SROs in training data generated from genetic algorithm (top left), and 32 example configurations (top right) out of the 500 total, starting from a set of ordered seeds (bottom). All atomic configurations are rendered using OVITO.
• Figure 3: TaW cross validation residual errors. Each plot represents a different $90\%/10\%$ train-test split. One outlier (approximately $-40meV/atom$) is excluded in the top left split.
• Figure 4: TaW enthalpy of mixing calculated via our CE and the DFT-MC routine.
• Figure 5: Distribution of SROs in training data for each alloying type pair generated from surrogate models (top) and 32 example configurations (bottom) out of the 500 total.