Theoretical Methods: Non-collinear Density Functional Theory and PAW methods
========================================================================================

We have calculated magnetism of Ni$_{n}$(n=1-5) chains supported on the Au(110)-(1$\times$2) surface based on Density Functional Theory [@Hohenberg1; @Kohn1] and the Methods of Projection of Augmentation Wave (PAW) [@Blochl1] with the plane-wave base set and GGA Perdew-Burke-Ernzerhof’s exchange-correlation potential [@Perdew1]. The PAW methods used in VASP program [@Kresse1; @Kresse3; @Hobbs1] is as accurate as frozen-core all-electron methods and hopefully improved to include all-electron relaxation. Our calculations include the non-collinear effects and the spin-orbit coupling which is proved important to heavy metals such as gold. In non-collinear formula, a magnetic moment as a vector can point to any direction in space. The orbital polarization (OP) interaction isn’t included in our calculations.

The basic theory used in this paper is Hobbs’s fully unconstrained non-collinear DFT formulas based on plane wave basis sets and all-electron PAW methods [@Hobbs1; @Hobbs2]. Instead of distributing local quantization axes for every atoms, which is used in programs based on atom sphere approximation (ASA) or others analogous methods [@Sandratskii1], Hobbs’s theory adopts only one global quantization axis and the vectors of magnetic electronic density are varying smoothly and continuously, which is more suited for the calculations of itinerant 