to find from this figure that, when magnetizing the Ni chains parallel to the chain, the orbital moments are larger than that perpendicular to the chains. The spin moments for every Ni atom of the ferromagnetic Ni chain almost don’t change their directions after the Kohn-Sham self-consistent calculations for all three magnetization directions \[table. (\[table2\])\]. The spin moment of the middle atom is smaller than that of other atoms and thus the short Ni chain isn’t in prefect ferromagnetic order. The larger spin-moments for atoms at two ends are because of the smaller coordination number compared with the middle atoms.

The interaction between supported Ni$_{n}$ chain and surface can be measured with the interaction energy $E_{I}=E^{tot}-E^{s}-E^{f}$ where $E^{tot}$ is the total energy, $E^{s}$ the energy of slab with relaxed surface and $E^{f}$ the energy of free-standing chain. We generally calculate the interaction energy per atom, which removes the effects of simple summation. The interaction energies are generally negative. Fig. \[fig5\](b) shows that the interaction energies of the Ni chains with the surface increase with the chain lengths. In our classical molecular dynamics simulation, we have found the oscillation of interaction energy for longer chains induced by lattice misfit [@Fan1]. These 