Exploring the chemical space of transition-metal cluster thermodynamics via automated first-principles calculations and machine learning.

Atomic clusters serve as the embryos of materials, yet their enormous and compositionally complex chemical space has long hindered systematic exploration of thermodynamic stability. Here, we show that a unified first-principles and machine-learning framework enables large-scale mapping of transition metal cluster thermodynamics. By integrating a progressive sampling strategy with a composition-based deep-learning model, we alleviate the intrinsic sampling bottleneck associated with exponentially expanding chemical spaces. Based on ≈ 93,000 global-minimum structures obtained via automated first-principles calculations, our model-the cluster-transformer-encoder network-enables reliable predictions of atomization energies (≈ 40 meV per atom accuracy) for 9.13 million metal cluster compositions, covering 30 d-block metals and 4 chemically relevant ligand elements (C, N, O, and S). The resulting thermodynamic trends show strong correlations between cluster atomization energies and bulk cohesive energies, and identify heteronuclear stabilization and the stability of noble-metal-doped oxide clusters. This work establishes a scalable, composition-driven strategy for efficient first-pass exploration of chemically complex small-cluster spaces, providing insights into collective stability trends of metal systems.