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1 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Prediction of EPR g-Tensors of Transition Metal Compounds with Density Functional Theory: First Applications to some Axial d 1 MEX 4 Systems S. Patchkovskii and T. Ziegler Department of Chemistry, University of Calgary, 2500 University Dr. NW, Calgary, Alberta, T2N 1N4 Canada

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2 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Introduction Electron Paramagnetic Resonance (EPR) is an important tool in experimental studies of systems containing unpaired electrons [1]. The traditional application areas for EPR include studies of transition metal complexes, stable organic radicals, transient reaction intermediates, as well as solid state and surface defects. In many cases, the extreme sensitivity of EPR allows experimental access to electronic structure and molecular environment parameters which would be impossible to measure otherwise. Extraction of this information from experimental spectra is however not always straightforward, and can be greatly facilitated by quantum-chemical calculations. The fundamental physical laws that determine the g-tensor of EPR are well understood [2]. Even so, traditional ab initio approaches to g-tensor calculations require large basis sets and sofisticated treatment of the electron correlation, making such calculations very expensive [3]. Not surprisingly, g-tensors of transition metal complexes largely remain beyond the reach of the existing classical ab initio machinery. Density Functional Theory (DFT), on the other hand, allows for an inexpensive treatment of the electron correlation and has been remarkably successful in studies of other properties of transition metal compounds [4], making it the tool of choice for such calculations. In this work, we apply the recently developed DFT implementation of the EPR g- tensors [5] to a series of axial pentacoordinated d 1 complexes with spatially non-degenerate ground states. To our knowledge, this is the first systematic application of a rigorous first- principles theoretical approach to EPR g-tensors of transition metal compounds.

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3 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Theory Quasi-relativistic DFT formulation of the EPR g-tensors used in this work distinguishes between several contributions to the g-tensor [5,6] : free-electron g value ( ) diamagnetic termparamagnetic term kinetic energy correction mass-velocity correction Darwin term Relativistic corrections The paramagnetic term dominates deviation of g from the free-electron value for complexes considered here, and can be in turn separated into several contributions: frozen core contributions occupied-occupied coupling terms occupied-virtual coupling terms The occ-vir term is usually the most qualitatively important contribution.

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4 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 The contribution is given by (atomic units): the effective potential -spin current due to unit magnetic field along s=x,y, or z -spin current The form of the occupied-virtual paramagnetic contribution the the EPR g-tensor is analogous to the expression for the paramagnetic part of the NMR shielding tensor for a nucleus N, given by: The similarity between the two quantities is extremely useful both in the evaluation and in analysis of g tensor, and is unique to our DFT implementation.

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5 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 The spin-current density for a spin arising due to the coupling between occupied and virtual MOs caused by the external magnetic field B 0 is given by: magnetic coupling coefficient unperturbed occupied MOunperturbed virtual MO field strength in the direction s (=x,y,z) The principal contribution to the coupling coefficient u is in turn given by: unperturbed orbital energiesunperturbed MO coefficients atomic orbitals (AOs) applies to each AO

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6 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Qualitative analysis of the g-tensor contributions can be considerably simplified if and MOs and corresponding orbital energies are constrained to be identical (the spin- restricted approach). In this case, and coupling coefficients u become numerically the same, so that most contributions to the g tensor cancel. The only surviving contributions involve coupling with the singly occupied MO (SOMO) and are given by: -SOMO (singly occupied) -SOMO (singly unoccupied) virtual MOs occupied -MOs Operator appearing in this expression is, apart from a constant, the spin-reduced form of the spin-orbit (SO) term in the first-order Pauli Hamiltonian. Matrix elements of this operator are primarily determined by one-centre contributions from the atom-like regions surrounding the nuclei.

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7 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Methods

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8 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 EPR g-tensor in axial MEX 4 systems Due to the high C 4v symmetry of the MEX 4 complexes, only two independent g-tensor components are possible in this system [1]. The isotropic value is then given simply by: z yx parallel component, field along the C 4 (Z) axis doubly degenerate perpendicular component, field perpendicular to the C 4 axis For the analysis of the contributions to the g tensor components, it is expedient to introduce deviations g from the free-electron g value, defined by: For the transition metal complexes studied here, experimental g values are typically accurate to 0.001, and can therefore be conveniently measured in parts per thousands (ppt).

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9 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Principal MO contributions to g 2*e 1 (d xz,d yz )b 2 (d xy )2*e 1 SOMO -spin charge-transfer contribution is usually small and positive. It increases in relative importance for heavy ligands coordinating 3d or 4d metal. This term dominates in TcNBr 4 1- -spin ligand field-type contribution is typically large and negative, and dominates g for most MEX 4 complexes. It becomes larger in the 3d 4d 5d series

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10 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Principal MO contributions to g || Negative -spin ligand field-type contribution dominates g || in most cases. It increases for heavier metals and decreases for heavier ligands. b 2 (d xy ) b 1 ( *) b1b1 b 1 ( ) SOMO Positive -spin contribution increases both for heavier metals and heavier ligands. It dominates in complexes of 3d and 4d metals with heavy ligands.

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11 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Molecular geometry and g-tensor Calculated g-tensors are relatively insensitive to the choice of the molecular geometries. For complexes where experimental geometries are known, the differences in the g-tensor components calculated at the optimized VWN and experimental geometries usually do not exceed 10 ppt, and have no impact on the qualitative trends. Interestingly, g ||, g , and g iso show qualitatively different dependence on the structural parameters in MEX 4 complexes: g shows no dependence on the distance between the metal and singly bonded ligand (R M-X ), while g || is insensitive to the position of the doubly bound ligand (R M=E ). At the same time, both components are highly sensitive to the bond angle EMX. The reasons for these trends are immediately apparent from the composition of the MOs giving dominant contributions to each of the components. ReOF 4

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12 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Relativity and g-tensor Within the present quasi-relativistic approach, it is possible to distinguish several sources of relativistic contributions to g, namely: Direct relativistic contributions: kinetic energy correction, mass-velocity correction, and Darwin term. Taken together, the three direct relativistic terms contribute less than 1 ppt to the g- tensor, and can be ignored. Relativistic bond contraction: can change bond lengths by a few percent for 5d metals and heavy ligands, leading to about 10 ppt change in the g-tensor for such systems. Does not affect qualitative trends, but is important for accurate predictions Relativistic changes in Kohn-Sham orbitals: small changes (under 10 ppt) for 3d and 4d complexes. Up to 60 ppt for complexes of W and Re (or ¼ of the total g for ReOBr 4 ). The parallel component g || is highly sensitive, while the perpendicular component g is almost unaffected.

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13 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Approximate functionals and g-tensor Table 1. Average errors (in ppt) for the g-tensor components computed using VWN and BP86 approximate density functionals. “ave” is the average signed error, “abs” is the average absolute error, and “rms” is the root mean square deviation. N val gives the number of comparison for each component. We examined performance of several approximate density functionals, including X , VWN, and VWN-Stoll local functionals, as well as BP86 and BLYP gradient-corrected functionals, for prediction of the g-tensor components. Calculated g tensors show only a marginal dependence on the specific functional. GGAs, which are tend to be more accurate than local functionals for other molecular properties, produce essentially identical results for the g tensor (see Table 1). As can be seen from Table 1, calculated g-tensor components are always above the experimental values, with much higher deviations observed for the parallel component g || compared to g . It is therefore instructive to examine the patterns in the g values for the individual complexes.

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14 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 The errors in the g component are strongly correlated with the metal’s transition row: values for 3d complexes are slightly underestimated (by 3 ppt); predictions for the 4d complexes are somewhat too high (by 15 ppt); while the 5d complexes show large positive deviations (93 ppt on average) in the calculated g . Systematic errors: g component Experiment Calculated (VWN) ReOBr 4 ReOCl 4 TcNBr 4 1-

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15 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 The errors in the g || component are also strongly correlated with the metal’s transition row: values for 3d and 4d complexes are somewhat too high (by respectively 26 and 52 ppt); while the 5d complexes show large positive deviations (112 ppt on average) in the calculated g ||. Systematic errors: g || component Experiment Calculated (VWN) ReOF 4 TcNBr 4 1-

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16 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Empirical corrections to calculated g || and g Given the strong correlation between the residual errors calculated g-tensor components and the metal’s transition row, it is possible to introduce empirical additive corrections for the g-tensor components, such that: The requisite corrections are shown in Table 2. As can be seen from Table 3 below, this reduces the residual errors by more than a factor of 2 in all cases. Table 3. Average errors (in ppt) for the g-tensor components computed using VWN with and without the empirical correction given above. The average signed error (“ave”) for g iso does not vanish since the c(g iso ) correction was computed from c(g || ) and c(g ) rather than fitted independently.

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17 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Some bigger systems Systems where theory can contribute to the analysis of the experimental EPR spectra are typically much larger than the MEX 4 complexes we considered before. It is interesting to see well we can describe one of the “real world” series - MoO(SPh) 4 1-, MoO(SePh) 4 1-, WO(SPh) 4 1-, and WO(SPh) 4 1- : MoO(SePh) 4 1-, C 4v MoO(SePh) 4 1-, C 4 So, we can describe qualitative trends due to gross changes in molecular geometry (C 4v vs C 4 values for MoO(SePh) 4 1- ) and periodic trends correctly in this series. We are still far from approaching the experimental accuracy, however.

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18 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 Conclusions and Outlook We presented the first extensive application of density functional theory to EPR g-tensors of transition metal complexes. The approach covers all physically significant sources of contributions to the g-tensor, and allows for g values both above and below the free- electron value naturally and without introducing any artificial assumptions. Calculated g-tensors are relatively insensitive to the molecular geometry, so that theoretical LDA VWN geometries are satisfactory. Relativistic effects on molecular geometry and Kohn-Sham orbitals are important for 5d complexes, where they contribute up to one fourth of the total g-shift, but can be ignored for lighter complexes. Calculated tensor components are insensitive to the choice of the approximate density functional, with local (VWN) and gradient-corrected (BP86) functionals giving essentially identical results. The g-tensor components are overestimated by all approximate functionals. Systematic errors in calculated g-tensors can be traced back to the overestimation of covalent bonding by popular approximate functionals, leading to subtle deficiencies in the shapes and relative energies of the -bonding MOs. Calculations of EPR g tensors can thus provide a stringent test on the local behavior of an approximate functional. Future extensions of this work may include applications to large systems, e.g. radical intermediates in catalytic processes and metal-containing enzyme reaction centers. Extension of the method to systems with spatially degenerate ground states and more than one unpaired electron is also desirable.

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19 Prediction of EPR g-Tensors with DFT, CSC’82, 1999 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 GIAO- DFT implementation of the EPR g tensors available to the authors. References

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