magnetization axis which is stable direction of magnetic moments. We need the ’strongest’ magnetic field to change direction of magnetic moments away from the direction of the easy magnetization axis. However for the materials with more complex magnetic structure such as the canted and spiral magnetic structures, the relaxations of directions of magnetic moments are intrinsic, the use of unconstrained DFT is essentially important [@Hobbs1; @Hobbs2].

\[mag\_3d\] The magnetism of 3d transition metal atoms absorbed on Au(110)-(1$\times$2) and Au(111) surface
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The missing row Au(110) surface reconstructs from Au(110) surface by having missed closed-packed Au chains every others, and forms one-dimensional trough along the close-packed A-\[1$\bar{1}$0\] direction. We construct a crystal slab with 4 atomic layers including 28 gold atoms. At the first step we optimized the lattice constant of gold crystal which is about 4.2$\AA$. In the procession of optimization, the RMM-DIIS algorithm [@Kresse1] is applied to the total-energy minimum with plane-wave energy cutoff 229.9eV and the 6x6x6 Monkhorst-Pack K points mesh. Now the size of super-cell is 11.8$\AA$$\times$8.4$\AA$$\times$5.94$\AA$.

We add a vacuum layer by extending the super-cell along C-\[110\] direction. The size along C-\[110\] direction of super-cell including vacuum layer is equal to 16$\AA$. The lattice constant of 