surface is generally smaller than that of the crystal. Thus we optimized the slab with vacuum layer together. Based on the optimized lattice constant, the size of the super-cell is 11.523$\AA$$\times$8.148$\AA$$\times$15.52$\AA$. The vacuum layer is about twice thicker than thickness of the slab. For the optimized Au(111) surface, the size of super-cell is about 11.5$\AA$$\times$9.97$\AA$$\times$20$\AA$ including 64 gold atoms. In above optimizations, we only change the lattice constant. The initial structures are constructed by placing the 3d transition metal atoms on the hollow sites of the Au(111) and Au(110)-(1$\times$2) surface in the trough (Fig. \[fig1\]). The initial structures are optimized for all atoms except for the bottom atoms by having used the conjugate-gradient Methods. The K-points meshes of 2$\times$2$\times$1 or $\Gamma$ point are used to sampling the first Brillouin-zone in the corresponding calculations of the electronic structure for Au(110)-(1$\times$2) and Au(111) surfaces respectively. In this work we use smaller k-points mesh to more efficiently optimize surface structures.

We use both collinear and non-collinear density functional calculations but mainly present the non-collinear results. In non-collinear DFT calculation, we use the optimized structures from collinear DFT calculation. Based on the relaxed structures, we calculate more exactly the electronic structures using the RMM-DIIS 