Exact-Two-Component\nComplete Active Space Method with\nVariational Treatment of Magnetic Field and Spin–Orbit Coupling:\nApplication to X‑ray Magnetic Circular Dichroism Spectroscopy

We introduce an exact-two-component complete active space\nself-consistent-field\n(X2C-CASSCF) method formulated under the restricted-magnetic-balance\ncondition. This framework allows for the nonperturbative treatment\nof static magnetic fields using gauge-including atomic orbitals (GIAOs).\nThe GIAO-X2C-CASSCF methodology effectively captures all microstates\nwithin the same 2<i>J</i> + 1-degenerate manifold and their\nsplitting in a static magnetic field, which are not accessible through\nsingle-reference-based methods. We also present mathematical recursive\nexpressions for evaluating one-electron relativistic integrals by\nusing GIAOs in the presence of a finite magnetic field. Benchmark\nstudies include oxygen and nitrogen K-edge X-ray magnetic circular\ndichroism spectroscopy (XMCD) for closed-shell organic compounds,\nas well as L-edge XMCD spectroscopy for the high-spin open-shell transition\nmetal ion Mn<sup>2+</sup> and the tetrahedral Mn(II)O<sub>4</sub><sup>6–</sup> complex.