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## NMR/EPR properties under perturbation theoretical inclusion of spin-orbit coupling

Many parameters of magnetic resonance depend critically on spin-orbit effects. This has often hampered the theoretical and computational access up to now [1]. Our group, in collaboration with the groups of V. G. Malkin and O. L. Malkina (Bratislava), J. Vaara (Helsinki), and B. Schimmelpfennig (now in Karlsruhe), has developed over the years a machinery for the efficient and accurate perturbation theoretical treatment of such spin-orbit effects [2-7,9,11]. While this was done initially within the framework of the deMon-NMR/EPR code, during the past ten years the work has concentrated on the new, more flexible ReSpect program package (and more recently a fully relativistic version thereof). A distinct advantage of these implementations is the use of efficient and accurate ab initio approximations to the full microscopic one- and two-electron spin-orbit operators. These are (i) the all-electron „atomic meanfield approximation“ (AMFI), and (ii) spin-orbit pseudopotentials. This allows us to treat, with moderate computational effort, large systems, without employing the commonly used empirical parameters [2-7,9,11]. However, semi-empirical SO-operators and even the full one- and two-electron Breit-Pauli SO Hamiltonian are also available in the MAG-ReSpect program.

The first implementations have focused on (i) **spin-orbit corrections to NMR chemical shifts** in triple perturbation theory [2,3], and on (ii) calculations of **electronic g-tensors **in double perturbation theory [4,5]. The methodology was then extended to the computation of the important **spin-orbit corrections to hyperfine coupling constants** in second-order perturbation theory [6,7]. These correction terms improve agreement with experiment considerably already for 3d-metal complexes, and they afford the experimentally accessible antisymmetric contributions to the hyperfine matrix.

A further important magnetic resonance parameter that is dominated by spin-orbit effects is **zero-field splitting (ZFS)**. This parameter is of central importance in the field of molecular magnetism, as it determines the tunneling splitting of single-molecule magnets. In this field, the ZFS has often been computed by a simple perturbation theoretical approach due to Pederson and Khanna [8]. We have implemented that approach (using AMFI SO operators and SO-ECPs), as well as a two-component non-collinear DFT approach and compared the two in detail for various species [9]. Meanwhile a full coupled-perturbed Kohn-Sham implementation is available, including the spin-spin contributions.

Last but not least, all of these spin-orbit-dependent MR parameters have to be combined to compute **NMR chemical shifts of open-shell systems**, as g-tensors, hyperfine tensors (with SO corrections), orbital shifts, and ZFS enter the perturbation expressions for this parameter. The first implementation for systems with arbitrary spin multiplicity has recently been reported [10] (see also ref. [11]). First applications have been made, for example, to the elucidation of enantioselective catalysis mechanisms [12], and to magnetic materials [13].

### References

[1] *Calculation of NMR and EPR Parameters. Theory and Applications* (Eds. M. Kaupp, M. Bühl, V. G. Malkin) Wiley-VCH, Weinheim 2004.

[2] *Spin-Orbit Corrections to NMR Shielding Constants from Density Functional Theory. How Important are the Two-Electron Terms?* O. L. Malkina, B. Schimmelpfennig, M. Kaupp, B. A. Hess, P. Chandra, U. Wahlgren, V. G. Malkin *Chem. Phys. Lett.* **1998**, *296*, 93-104.

[3] *Study of Relativistic Effects on Nuclear Shieldings Using Density-Functional Theory and Spin-Orbit Pseudopotentials* J. Vaara, O. L. Malkina, H. Stoll, V. G. Malkin, M. Kaupp *J. Chem. Phys.* **2001**, *114*, 61-71.

[4] *Density-Functional Calculations of Electronic g-Tensors Using Spin-Orbit Pseudopotentials and/or Mean-Field All-Electron Spin-Orbit Operators* O. L. Malkina, J. Vaara, B. Schimmelpfennig, M. L. Munzarová, V. G. Malkin, M. Kaupp *J. Am. Chem. Soc.* **2000**, *122*, 9206-9218.

[5] *Calculation of Electronic g-Tensors for Transition Metal Complexes Using Hybrid Density Functionals and Atomic Meanfield Spin-Orbit Operators* M. Kaupp, R. Reviakine, O. L. Malkina, A. Arbuznikov, B. Schimmelpfennig, V. G. Malkin *J. Comput. Chem.* **2002**, *23*, 794-803.

[6] *Relativistic Spin-Orbit Effects on Hyperfine Coupling Tensors by Density Functional Theory* A. V. Arbuznikov, J. Vaara, M. Kaupp *J. Chem. Phys. ***2004**, 120, 2127-2139.

[7] *Spin-Orbit Effects on Hyperfine Coupling Tensors in Transition Metal Complexes Using Hybrid Density Functionals and Accurate Spin-Orbit Operators* C. Remenyi, A. V. Arbuznikov, R. Reviakine, J. Vaara, M. Kaupp *J. Phys. Chem. A* **2004**, *108*, 5026-5033.

[8] M. R. Pederson, S.N. Khanna *Phys. Rev. B* **1999**, *60*, 9566.

[9] C*alculation of Zero-Field Splitting Parameters. Comparison of a Two-Component Non-Collinear Density Functional Method and a One-Component Perturbational Approach* R. Reviakine, A. V. Arbuznikov, J.-C. Tremblay, C. Remenyi, O. L. Malkina, V. G. Malkin, M. Kaupp *J. Chem. Phys. ***2006**, *125*, 054110/1-12.

[10] *Density Functional Calculations of NMR Chemical Shift Tensors for Paramagnetic Systems with Arbitrary Spin Multiplicity. Validation on 3d-Metallocenes *P. Hrobárik, R. Reviakine, A. V. Arbuznikov, O. L. Malkina, V. G. Malkin, F. H. Köhler, M. Kaupp *J. Chem. Phys.* **2007**, *126*, 024107/1-19.

[11] T. Pennanen, J. Vaara *Phys. Rev. Lett.* **2008**, *100*, 133002.

[12] *Jacobsen’s Catalyst for Hydrolytic Kinetic Resolution: Structure Elucidation of Paramagnetic Co(III) Salen Complexes in Solution via Combined NMR and Quantum Chemical Studies* S. Kemper, P. Hrobarik, M. Kaupp. N. E. Schloerer *J. Am. Chem. Soc.* **2009**, *131*, 4172-4173. *Erratum: J. Am. Chem. Soc.* **2009**, *131*, 6641-6641.

[13] *Combining NMR spectroscopy and quantum chemistry as tools to quantify spin density distributions in molecular magnetic compounds *M. Kaupp, F. H. Köhler *Coord. Chem. Rev.* **2009**, *253*, 2376-2386.

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## Chair

**Prof. Dr. M. Kaupp**

Theoretical Chemistry

Quantum Chemistry

http://www.quantenchemie.tu-berlin.de/menue/ueber_uns/team/prof_dr_martin_kaupp/parameter/en/