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## Development of one- two- and four-component relativistic all-electron approaches (Douglas-Kroll-Hess transformation and matrix Dirac-Kohn-Sham approaches)

The perturbation theoretical
treatment of spin-orbit coupling meets its limits once the spin-orbit
effects become too large relative to the electronic energies (energy
differences between different nonrelativistic states). This is the
case when going to very heavy elements, or when having particularly
small energy differences in transition metal complexes. Then a
**variational treatment of spin-orbit coupling** becomes
necessary. Moreover, the calculation of the NMR or EPR parameters of
heavy nuclei (chemical shifts, spin-spin coupling constants, hyperfine
coupling constants, nuclear quadrupole coupling constants) requires a
**relativistic all-electron method**.

In collaboration with the group of V. G. Malkin and O. L. Malkina (Bratislava), we have thus recently implemented one-, two- and four-component relativistic all-electron density functional approaches into the ReSpect program [1]. The initial work has been based on the Douglas-Kroll Hamiltonian (Douglas-Kroll-Hess method, DKH). This requires the correct transformation of the property operators from the Dirac picture to the DKH picture to include the so-called “picture change” effects. After initial work of the Bratislava group on one-component, analytical DKH calculations of nuclear quadrupole coupling constants, the method has meanwhile been extended significantly. Initial work on scalar relativistic calculations of hyperfine couplings is the first of this kind [1] and was demonstrated to give rather accurate results on heavy-atom hyperfine couplings. The method has been extended to incorporate a finite-size nucleus model, which further improved the accuracy [2]. The long-standing problem of including spin polarization into two-component g-tensor calculations has been solved by a non-collinear spin-density approach [3].

The DKH approach has meanwhile been superseded by approaches based on the matrix formulation of the Dirac-Kohn-Sham method [4,5]. Initial work was in the two-component framework of an elimination of the small component using restricted kinetic balance [4], and tests on g-tensors and hyperfine tensors have been provided. Meanwhile, an extension to a four-component framework and second-order magnetic properties (at the moment nuclear shieldings) has been achieved by applying magnetic balance conditions [5]. The great advantage of these DKS-based approaches over DKH is that no picture-change problem arises in property calculations. Excellent work by M. Repiský and S. Komorovský has made the implementation so efficient, that relatively large systems can now be tackled at the relativistic 4-component level. First applications have been to g-tensors of transition-metal complexes [6], and in particular to the spectacular high-field NMR chemical shifts of transition-metal hydride complexes [7].

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