crystal due to the splitting of ground state energy levels. However there are still many contradictions among the already known theoretical results. The density functional calculations with OP (orbital polarization) + SOC (spin-orbit coupling) have given the correct orbital moments but too large MAE (Magnetic Anisotropy Energy) for tetragonal and cubic Ni. If the OP term switches off the theoretical value of the magnetic anisotropy energy is close its experimental values [@Hjortstarn1]. The magnetic anisotropy energy of CoPt calculated using the so-called c-RPA [@Solovyev2] are weakly dependent on the new introduced OP interaction. The magnetic anisotropy energies are surprisingly consistent with experiments even only under LSDA approximation without the OP effects. The orbital moments of transition metals are well consistent with experiments. This method has been combined with PAW method to calculate quasi-particle spectrum of materials [@Fuchs1].

Based on Brook’s theory, the magnetic anisotropy is induced by the spin-orbital interaction [@Brooks2]. Although the orbital moments are underestimated, it’s still valuable to study the magnetic anisotropy properties in the absence of (OP) orbital polarization term. Theoretical works based on Density Functional Theory have also discovered others interesting properties of surface supported magnetic chains. Such as in Ref. [@Spisak2], the authors have 