such as the strong magnetic anisotropy, the enhanced spin and orbital moments, some new phenomena have been found, such as the temperature and time dependent magnetization [@Elmers1; @Shen1] which make the realistic applications of these novel experimental devices face new serious problem such as the instability of magnetic structure.

It is valuable to investigate theoretically the magnetic stability of surface-supported novel structures [@Dreysse1]. Many theoretical and computational methods have been used to studied the magnetism of one-dimensional structure on surface such as the KKR [@Bellini1; @Wildberger1; @Lazarovits1; @Eisenbach1], and TB-LMTO [@Komelj1] and PAW methods [@Spisak1; @Spisak2] based on Density Functional Theory, and the numerically tight-binding self-consistent calculation [@Dovila1] and analytically tight-binding calculation [@Druzinic1]. Some common arguments have been found such as the enhancement of surface magnetism, the magnetic anisotropy are close related to the orbital moments and the spin-orbit coupling interaction (SOC).

The orbital polarization [@Brooks1; @Eriksson1; @Solovyev1] can remedy the small orbital moments in the calculations of density functional theory to approach the experimental values, and especially reproduce the correct phase transition in Ce companying with the volume collapses [@Eriksson1]. The orbital polarization plays the opposite role to crystal field which quenches the orbital moment to small value in 