DFT calculations

The spin polarized density functional theory calculations were performed using the DMol3 (ref. ³²) module in the Materials Studio software (Bio Accelrys). Exchange-correlation functional was based on Perdew-Burke-Ernzerh (PBE) within the generalized gradient approximation. The core electrons were treated with DFT semi-core pseudopotentials. The optimized lattice constant of Au is 4.180 Å, in good agreement with the experimental value of 4.078 Å. The Au(111) facet was modelled by an unreconstructed 4 × 4 surface slab, which contains four atomic layers with a vacuum region of ∼20 Å. The self-consistent-field (SCF) convergence criterion was set to be <10⁻⁵ Hartree and the force convergence was set to be lower than 0.002 Hartree per Å for all the optimizations of the adsorptions. In particular, for the optimization of the adsorption of Re atoms and W atoms on Au(111) facets, the bottom two layers of Au atoms were fixed at their bulk positions. A 5 × 5 × 1 k-point mesh was used to sample the Brillouin zone of the supercells. The adsorption energy of a X (X=Re or W) atom on the surfaces of M (M=Au(111) or WS₂(001)) substrate is defined as E_ads^{X, M}=E(M)+E(X)−E(M−X).