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Machine learning (ML) force fields are one of the most common applications ofML methods in the field of physical and chemical science. In the optimal case,they are able to reach accuracy close to the first principles methods withsignificantly lowered computational cost. However, often the training of the MLmethods rely on full atomic structures alongside their potential energies, andapplying the force information needs special modifications to standardalgorithms. Here we apply distance-based ML methods to predict force norms andestimate the directions of the force vectors of the thiolate-protected goldnanoclusters. The method relies only on local structural information withoutenergy evaluations. We apply the atomic ML forces on the structure optimizationof the gold-thiolate rings, $\text{Au}_{25}(\text{SCH}_{3})_{18}$ nanoclusterand two known structural isomers of the $\text{Au}_{38}(\text{SCH}_{3})_{24}$nanocluster. The results demonstrate that the method is well-suited for thestructural optimizations of the gold-thiolate systems, where the atomic bondinghas a covalent nature in the ligand shell and at the metal-ligand interface.The methodology could be seen as an early attempt to introduce equivariantlearning to distance-based ML methods.

Orientation Adaptive Minimal Learning Machine: Application to Thiolate-Protected Gold Nanoclusters and Gold-Thiolate Rings