Zero-Field Splitting Parameters within Exact Two-Component Theory and Modern Density Functional Theory Using Seminumerical Integration

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin–dipole contribution and the response equations of the spin–orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations (meta-GGAs) and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn–Sham (CPKS) equations for the spin–orbit perturbation term. Second, the spin–orbit perturbation is formulated within the relativistic exact two-component (X2C) theory and the screened nuclear spin–orbit (SNSO) approximation. Accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. Additionally, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin–orbit mean field (SOMF) ansatz.