results indicate that short Ni chains are more energetically favorable on Au(110)-(1$\times$2) surface. This is consistent with the experiments [@Hitzke1] in which most of Ni chains are shorter than 6. Additionally, we calculate the density of states of the Ni$_{n}$ (n=1-5) chains. From the Fig. \[fig7\], we can see that the densities of states of 3d states of Ni$_{5}$ chain changes very small when the magnetization changes from C-\[110\] direction to A-\[1$\bar{1}$0\] direction and the spin magnetic moment does also change very small \[Fig. \[fig7\]\]. From the figure \[fig5\], the changes of orbital moments are more significant if the magnetizing directions change. Although the orbital moments are underestimated due to the absence of (OP) orbital polarization term, our results still show that magnetic anisotropy are closely related to the orbital degree of freedom. For compact magnetic islands on metal surface such as Co/Pt(111) [@Gambardella4], the orbital moments decrease with increasing island sizes. This is because only atoms at the edge of island have significant contributions to the total orbital moments. However, for one-dimensional island all atoms have significant contribution to the total orbital moments. The orbital moments per atom keep constant as the lengths of chains beyond 4.

\[Ni\_chain\_aniso\] The 