Derivation and assessment of relativistic hyperfine-coupling tensors on the basis of orbital-optimized second-order Møller–Plesset perturbation theory and the second-order Douglas–Kroll–Hess transformation

The accurate calculation of hyperfine-coupling tensors requires a good description of the electronic spin density, especially close to and at the nucleus. Thus, dynamic correlation as well as relativistic effects have to be included in the quantum-chemical calculation of this quantity. In this paper, orbital-optimized second-order Møller–Plesset perturbation theory (MP2) is combined with the second-order Douglas–Kroll–Hess (DKH) transformation to yield an efficient and accurate ab initio method for the calculation of hyperfine couplings for larger molecules including heavy elements. Particular attention is paid to the derivation of the hyperfine-coupling tensor in the DKH framework. In the presence of a magnetic field, the DKH-transformation is not unique. Two different versions can be found in the literature. In this paper, a detailed derivation of one-electron contributions to the hyperfine-coupling tensor as they arise in linear-response theory is given for both DKH-transformations. It turns out that one of the two variants produces divergent hyperfine-coupling constants. The possibility to remove this divergence through a physically motivated finite-nucleus model taking into account the different extent of charge and magnetization distribution is discussed. Hyperfine-coupling values obtained at the orbital-optimized MP2 level with second-order DKH corrections for the non-divergent variant are presented. The influence of a Gaussian nucleus model is studied. The method is compared to four-component, high-accuracy calculations for a number of cations and atoms. Comparison to B3LYP and B2PLYP is made for a set of transition-metal complexes of moderate size.