where E [001] and E [100] are the energies with the magnetization direction parallel and perpendicular to the MgO(001) surface, respectively. A positive E MAE indicates easy-plane anisotropy, favoring in-plane magnetization, while a negative E MAE corresponds to easy-axis anisotropy, where magnetization aligns along the z-axis. We achieved MAE convergence using a 2×2×1 k-point mesh for periodic models and a 1×1×1 k-point mesh for cluster models (see Table ). given as the energy difference between the ground M J = ±9/2 state and the first excited