The values of initial moments of Ni atoms are all the same, however the values after the self-consistent calculations are probably not the same for all Ni atoms. The orientations of magnetic moment may manually change to simulate the different magnetizing directions. In a single self-consistent calculation, the directions of magnetic moments change very small for the ferromagnetic Ni chains although we have used the unconstrained Density Functional calculations (see the table.(\[table2\])).

The chains become short after the relaxations and the average nearest-nearby distance of Ni$_{5}$ chain is about 2.75 $\AA$ shorter than 2.88 $\AA$ that of initial structure \[the insert figure of Fig.\[fig5\](b) and Fig.\[fig1\]\]. We choose two magnetizing directions: A-\[1$\bar{1}$0\] direction parallel to the chain and C-\[110\] direction perpendicular to the chain. The size-dependence magnetic moments are shown in Fig. \[fig5\] (a). From this figure we can see that the spin moments of a single absorbed Ni atom are small, 0.147$\mu_{_{B}}$ for the magnetization along C-\[110\] direction and 0.127$\mu_{_{B}}$ for A-\[1$\bar{1}$0\] direction. The spin moments pre atom of the Ni$_{2}$ chain increase to 0.455 $\mu_{_{B}}$ for A-\[1$\bar{1}$0\] direction and 0.464 $\mu_{_{B}}$ for C-\[110\] directions. The spin moments of Ni$_{3}$ have slightly decrease for C-\[110\] direction but large decrease 