From Local Motifs to Long-Range Order: Adaptive Diffusion of the Complex Au–S Surface Reconstruction

Surface reconstructions present a challenging structure-prediction problem because locally favorable atomic motifs must be accommodated within the periodicity of the underlying crystal, often producing large and complex surface unit cells. Here, we extend adaptive diffusion to jointly explore continuous atomic coordinates and discrete surface cell transformations, allowing favorable atomic arrangements and periodicities to be discovered together. We demonstrate the approach for the complex sulfur-induced reconstruction of Au(111), whose atomic structure has remained debated despite extensive experimental investigation. The search identifies a particularly stable rectangular Au 4 S 4 motif and a preference for a transformed surface cell. The exploration also reveals how favorable motifs can be transferred between different periodic environments. Guided by this observation, we construct and relax an arrangement of ten Au 4 S 4 motifs in the experimentally observed large quasi-rectangular unit cell that reproduces the reported sulfur coverage, and is consistent with the local ordering observed by scanning tunneling microscopy.