Ni$_{n}$ where $n$ less than 6 and large box size $23 \AA$ along the direction parallel the chain so that the ends of chains are far away from the boundaries of the box cell. The energetically favorable configuration of one-dimensional free-standing chains has zigzag shapes and not straight line. The Ni$_{n}$ atomic chains in the trough of Au(110)-(1$\times$2) surface are straight. As we optimize the structures of free-standing chains, all atoms are only allowed modified their position along the direction of chains and the optimized free-standing Ni chains still keep a straight line shape. We try to compare the results of surface-support straight chains with those of free-standing straight chains. The quantization axis is still along the C-\[110\] direction in this section.

\[Ni\_chain\_15\] Magnetism of Ni$_{n}$ (n=1-5)
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All Ni atoms in the cell have their initial magnetic moments in our calculations. We calculate the total energies, the spin and orbital magnetic moments of the supported Ni chains when they are magnetized along A-\[1$\bar{1}$0\] direction parallel to the chains or C-\[110\] direction perpendicular to the chains respectively. The magnetization of an atomic chain in this paper means that the magnetic moments for every Ni atom align along the same direction. 