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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. 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The DKH approach has meanwhile been superseded by approaches based on the matrix formulation of the Dirac-Kohn-Sham method [4,5].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].

### References

[1] *Scalar relativistic calculations of hyperfine coupling tensors using the Douglas-Kroll-Hess method* I. Malkin, O. L. Malkina, V. G. Malkin, M. Kaupp *Chem. Phys. Lett.* **2004**, *396*, 268-276.

[2] *Scalar relativistic calculations of hyperfine coupling tensors using the Douglas-Kroll-Hess method with a finite-size nucleus model* E. Malkin, I. Malkin, O. L. Malkina, V. G. Malkin, M. Kaupp *Phys. Chem. Chem. Phys. ***2006**, * 8*, 4079-4085.

[3] *Relativistic two-component calculations of electronic g-tensors that include spin polarization* I. Malkin, O. L. Malkina, V. G. Malkin, M. Kaupp *J. Chem. Phys.* **2005**, *123*, 244103/1-16.

[4] *Resolution of identity Dirac-Kohn-Sham method using the large component only. Calculations of g-tensor and hyperfine tensor* S. Komorovský, M. Repiský, O. L. Malkina, V. G. Malkin, I. Malkin, M. Kaupp J. *Chem. Phys.* **2006**, *124*, 084108-1–8.

[5] *A fully relativistic method for calculation of nuclear magnetic shielding tensors with a restricted magnetically balanced basis in the framework of the modified matrix Dirac-Kohn-Sham equation* S. Komorovský, M. Repiský, O. L. Malkina, V. G. Malkin, I. Malkin Ondík, M. Kaupp *J. Chem. Phys.* **2008**, *128*, 104101/1-15.

[6] *Assessment of Higher-Order Spin-Orbit Effects on Electronic g-Tensors of d1 Transition-Metal Complexes by Relativistic Two- and Four-Component Methods* P. Hrobárik, M. Repiský, S. Komorovský, V. Hrobáriková, M. Kaupp *Theor. Chem. Acc.* **2011**, *129*, 715-725.

[7] *Relativistic Four-Component DFT Calculations of 1H NMR Chemical Shifts in Transition-Metal Hydride Complexes: Unusual High-Field Shifts Beyond the Buckingham-Stephens Model* P. Hrobárik, V. Hrobáriková, F. Meier, M. Repiský, S. Komorovský, M. Kaupp *J. Phys. Chem. A* **2011**, *115*, 5654-5659.