of TM-GY. By treating the SOC Hamilton as the second perturbations, the occupied( $\psiup_o$) and unoccupied($\psiup_u$) states can interact via the matrix element $<\psiup_u|\hat{H}_{SO}^{sc}|\psiup_o>$, the associated energy lowering can be expressed in simplified form as:[@55; @56] $$\Delta E_{SOC} \propto \sum_{\psiup_u,\psiup_o} \frac{<\psiup_u|\hat{H}_{SO}^{sc}|\psiup_o>^2}{E_\emph{u}-E_\emph{o}} \label{eq8}$$ The interaction between occupied and unoccupied states determines the preferred spin orientations. The $d$ states with the same $|m|$ value interact through the operator $\hat{L}_z$ give nonzero matrix element, the different $|m|$ values interact through the ladder operator $\hat{L}_+$ and $\hat{L}_-$.[@55; @56]\

![Majority spin (red curves) and minority spin (blue curves) $m_l$-resolved PDOS for V (a), Cr (c), Mn (d), Fe (e) and Co (f) under FM state and Cr (b) under AFM state, respectively. The Fermi level is set to zero.[]{data-label="fig7"}](Fig7.eps "fig:"){width="3.5in"}\

For V chain adsorbed on GY, the system exhibit easy-axis anisotropy. The results as shown in Fig.\[fig7\](a) indicate that both HOMO and LUMO of V-chain-GY belong to the same spin channel (majority spin), and they are mainly derived from the $|m|$=1($d_{yz},d_{xz}$) and $|m|$=2($d_{x^2+y^2},d_{xy}$) states. Therefore, the energy lowering associated with the HOMO-LUMO interaction induced by SCO depends on the matrix element: $$<d_{yz},d_{xz}|\hat{H}_{SO}^{sc}|d_{yz},d_{xz}> ,  \   \
<d_{x^2+y^2},d_{xy}|\hat{H}_{SO}^{sc}|d_{x^2+y^2},d_{xy}>$$ Using the Eq.\[eq7\], we can easily get the nonzero 