magnetic anisotropy of Ni$_{5}$ chain
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In the above subsection we have chosen two magnetization directions A-\[1$\bar{1}$0\] and C-\[110\]. In this subsection, we study the magnetic anisotropy of the Ni$_{5}$ chain and show that C-\[110\] direction is along the easy magnetization axis. The magnetic anisotropy energies in S$_{1}$ and S$_{2}$ planes are defined as $\delta E_{1}=E^{C}-E^{A}$ and $\delta E_{2}=E^{C}-E^{B}$ respectively. $E^{A}$, $E^{B}$ and $E^{C}$ are the total energies when magnetizing the chain along A-\[1$\bar{1}$0\] ($\theta=0^{\circ}$), B-\[001\] ($\phi=0^{\circ}$)and C-\[110\] ($\theta=90^{\circ}$) directions [@Force1]. The magnetic anisotropy of one-dimensional structure is more prominent than compact structures such as surface and bulk. Fig.\[fig8\] shows the total energies, the spin and orbital moments change with two magnetization angles in above two planes. We find from Fig. \[fig8\](a,b) that there are weak magnetic anisotropy in S$_{1}$ plane and relative strong the magnetic anisotropy in S$_{2}$ plane. The magnetic anisotropy energies are 0.158(meV/atom) and 0.524(meV/atom) respectively in the two planes. The shape anisotropy determines the large difference of magnetic anisotropy energy in two plane because of one-dimension characteristic of mono-atomic chain. In the same plane such as the plane S$_{1}$, the magnetic anisotropy is determined by spin-orbit coupling. The easy magnetization directions in both S$_{1}$ and S$_{2}$ 