Density-matrix embedding based multi-reference perturbation theory approach to single-ion magnets

Multi-configurational wave-function theory (MC-WFT) that combines the complete active space self-consistent field (CASSCF) approach with subsequent state interaction treatment of spin-orbit coupling, abbreviated as CASSCF-SO, plays important roles in the microscopic understanding of single-ion magnets (SIMs) with different central transition metal or lanthanide ions and various coordination environments, but its application to SIMs with complex structures is severely limited due to its highly demanding computational cost. Density-matrix embedding theory (DMET) provides a systematic and mathematically rigorous framework to combine low-level mean-field approaches like Hartree-Fock and high-level MC-WFT methods like CASSCF-SO, which is particularly promising for SIMs. As a continuation of our previous work on DMET + CASSCF for 3d SIMs [Ai et al., J. Phys. Chem. Lett. 13, 10627 (2022)], we extend the methodology by considering dynamic correlation on top of CASSCF using the second-order n-electron valence perturbation theory (NEVPT2) in the DMET framework, abbreviated as DMET + NEVPT2, and benchmark the accuracy of this approach to molecular magnetic anisotropy in a set of typical transition metal complexes. We found that DMET + NEVPT2 can give results very close to all-electron treatment, and can be systematically improved for higher accuracy by expanding the region treated as the central cluster, while the computational cost is dramatically reduced due to the reduction of the number of orbitals by DMET construction. Our findings suggest that the dynamic correlation treated at the NEVPT2 level, which is important for magnetic anisotropy in typical SIMs, can be well described in the DMET framework, which can facilitate high-accuracy ab initio spin-phonon relaxation study and high-throughput computations.