1 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Calculation of EPR g tensors for spatially non-degenerate high-spin radicals with density functional theory S. Patchkovskii and T. Ziegler Department of Chemistry, University of Calgary, 2500 University Dr. NW, Calgary, Alberta, T2N 1N4 Canada I am on the Web:

2 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Conclusions and outlookConclusions and outlook Accurate techniques for ab initio prediction of EPR g-tensors of small spatially non- degenerate doublet radicals has been available for some time [1]. Recently, with the introduction of density functional formulations by Schreckenbach and Ziegler [2], and by van Lenthe et al [3], calculations on larger systems, including transitions metal complexes, also became possible. These techniques have been applied to small main group radicals, as well as to transition metal complexes. In favourable cases, the results for changes in g tensor components are approaching the accuracy of a typical powder-spectra experiment. However, these techniques are currently limited to spatially non-degenerate radicals with the effective spin S=½. Although the GUHF technique of Jayatilaka [5] is, in principle, not limited to such Kramers-type systems, it is not justified for radicals with non-negligible zero-field splitting tensors D, and may be difficult to extend to correlated approaches. At the same time, high-spin radicals are ubiquitous in transition metal chemistry, and are found in many enzymatic systems of current research interest [6]. Experimental analysis of EPR parameters for such radicals in terms of structural features may become involved, and can be facilitated by reasonably accurate accurate theoretical techniques. In this work, we show that the previously developed DFT formulation of the g-tensors [2] can be easily extended to arbitrary spatially non-degenerate radicals. The first results obtained with this technique are encouraging, particularly for main group radicals. Introduction

3 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Effective spin-Hamiltonian The simplest form of the effective spin-Hamiltonian, with hyperfine terms omitted, is: For small deviations from the free-electron g e value, energy levels are given by: In the high-field limit, the D tensor can be ignored (q=0). For consistency, all spin-spin coupling terms contributing to D will have to be omitted in the microscopic Hamiltonian as well. The resulting spin-Hamiltonian can be diagonalized exactly, giving: Magnetic field2.0023… Bohr magnetonEffective spinZero-field splitting Deviation from free electron Energy level, Field direction, In the free-electron limit (p=0) two of the eigenfunctions are given by simple products: ;

4 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Microscopic Hamiltonian The corresponding microscopic DFT Hamiltonian, is given by (see [2,7] for the exact forms of the individual operators): Scalar field- free operators Spin-Zeeman (free electron) terms Diamagnetic (gauge) terms Paramagnetic (spin-current) terms In the absence of the magnetic field, solutions of Kohn-Sham equations are given by: Since the spin-Zeeman operator commutes with the scalar field-free Hamiltonian,  i 0 are still eigenfunctions of the full Hamiltonian, provided that p=0: ; The corresponding single-determinantal non-interacting reference KS wavefunction: is then the direct equivalent of of the effective spin-Hamiltonian treatment.

5 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton EPR g-tensor “Unfreezing” the spin-orbit terms in the microscopic and effective Hamiltonians (p  1), and comparing the corresponding energy expressions for k =, we obtain: Examining this expression for different orientations of the magnetic field (  ), and substituting n  -n  for the effective spin, we obtain for individual components of the g tensor: which is analogous to the spin-doublet expression (n  -n  =1) considered previously [2]. The expression for the energy derivative on the right-hand side is unchanged compared to the doublet case, and is, in fact, evaluated by the same computer program. The critical assumption, made in deriving the expression, is that the non-interacting reference wavefunction is not changed by either the magnetic field, or spin-orbit coupling, in the zeroth order. This is equivalent to the requirement of a spatially non-degenerate electronic ground state, which remains in force for the present formulation as well.

6 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Methods

7 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Results: Main group diatomics Ge 2 +, 4  Calculated  g  (BP86) Experimental  g , parts per thousand (ppt) GaAs +, 4  NI, 3  S 2, 3  SeO, 3  SiAl, 4  Si 2 +, 4  SiB, 4  AlC, 4  C 2 +, 4  Ge 2 +, 4  B 2, 3  NH, 3  SO, 3  O 2, 3  PH, 3  BC, 4  Gas-phase value BP86VWN Points 16 Average error, ppt absolute error, ppt RMS error, ppt The gas-phase experimental values (circled) for  g  were computed from microwave spectral parameters using Curl equation [13]. The remaining experimental values are noble gas matrices. In all cases, the parallel component  g || is close to the free electron value, both in theory and in experiment

8 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Main group-transition metal diatomics BP86VWN Points 15 Average error, ppt absolute error, ppt 8.0 RMS error, ppt 9.8 Gas-phase value Calculated  g  (BP86) Experimental  g , parts per thousand (ppt) NbO, 4  YAl +, 4  YB +, 4  CrN, 4  CrF, 6  VO, 4  MnI, 7  MnBr, 7  MnS, 6  MnH, 7  MnO, 6  MnCl, 7  MoN, 4  CrH, 6  MnF, 7 

9 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Transition metal-transition metal diatomics BP86VWN Points 13 Average error, ppt absolute error, ppt 3738 RMS error, ppt 43 Calculated  g  (BP86) Experimental  g , parts per thousand (ppt) WCu, 6  HfV, 4  ZrNb, 4  Mn 2 +, 12  CrAu, 6  TiV, 4  V 2 +, 4  ZrV, 4  MnAg, 7  CrAg, 6  TiNb, 4  WAg, 6  WAu, 6 

10 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton 3 B u trans-(CNSSS) ±2º 21º g2g2 g3g3 experiment calculated  g 1, ppt  g 2, ppt  g 3, ppt Exp [14] VWN BP (CNSSS) 2 2+ provides a rare example of a thermally accessible excited triplet state, for which an accurate measurement of complete EPR g and D tensors is available [14]. Both the calculated magnitudes of the principal components, and their orientations, are in a good agreement with the experimental values.

11 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Summary  An existing second-order perturbation treatment of EPR g-tensors [2] was extended to arbitrary spatially non-degenerate radicals.  The approach is generally successful in predicting magnitudes and orientations of g- tensor principal components of first and second row main group radicals. Somewhat larger errors for heavier radicals may result from higher-order spin-orbit coupling terms.  Description of g-tensors in transition metal diatomic molecules is unsatisfactory. Errors in the theoretical values appear to be largely unsystematic, which likely indicates and inadequate description of the metal-metal chemical bonds in these molecules.  Transition metal radicals with no metal-metal chemical bonds show an intermediate behavior, with large, but often systematic errors in the calculated g-tensors. Outlook  Computing contributions in second order of spin-orbit coupling  Computing g-tensors for radicals with spatially degenerate ground states

12 EPR g-tensors of high-spin radicals with DFT DFT 2000, Menton Acknowledgements This work has been supported by the National Sciences and Engineering Research Council of Canada (NSERC), as well as by the donors of the Petroleum Research Fund, administered by the American Chemical Society (ACS-PRF No AC3). Dr. Georg Schreckenbach is gratefully acknowledged for making the original GIAO-DFT implementation of the EPR g tensors available to the authors. References