Quantum-embedded equation-of-motion coupled-cluster approach to single-atom magnets on surfaces

We investigate electronic states and magnetic properties of transition-metal atoms on surfaces using projection-based density embedding that combines equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) theory with density functional theory (DFT). As a case study, we explore Co adsorbed on MgO(001), an ideal model for single-atom magnet design, known for its record magnetic anisotropy among transition-metal adatoms. Periodic DFT-based calculations of the magnetic anisotropy energy, i.e., the energy required to rotate the magnetization from parallel to perpendicular relative to the surface normal, predict in-plane magnetic anisotropy, contradicting the experimentally observed easy-axis anisotropy. This failure stems from the inability of the approximate density functionals to describe the multiconfigurational, non-aufbau spin states of Co/MgO(001). In contrast, embedded EOM-CCSD calculations on Co/Mg₉O₉ finite models of the adsorption complex capture the system's unquenched orbital angular momentum (L ≈ 3) and strong spin-orbit coupling, leading to easy-axis anisotropy and a spin-inversion energy barrier that agrees with experiment within spectroscopic accuracy. When treating both the oxygen adsorption site and the Co magnetic center at the EOM-CCSD level of theory, embedded calculations accurately reproduce the state ordering, spin-orbit coupling, and susceptibility curve of all-atom EOM-CCSD calculations. These results demonstrate that embedded EOM-CCSD provides a reliable description of the electronic states and magnetic properties of magnetic adsorbates on surfaces, offering a robust framework for future investigations of surface-bound magnetic systems.