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Prediction of EPR g-Tensors of
Transition Metal Compounds with
Density Functional Theory: First
Applications to some Axial d1 MEX4
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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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 gtensors[5] to a series of axial pentacoordinated d1 complexes with spatially non-degenerate
ground states. To our knowledge, this is the first systematic application of a rigorous firstprinciples theoretical approach to EPR g-tensors of transition metal compounds.
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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 (2.0023)
diamagnetic term
paramagnetic term
Darwin term
mass-velocity correction
kinetic energy correction
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-virtual coupling terms
occupied-occupied coupling terms
The occ-vir term is usually the most qualitatively important contribution.
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Prediction of EPR g-Tensors with DFT, CSC’82, 1999
The
contribution is given by (atomic units):
 0.00731
-spin current
-spin current due to unit magnetic
field along s=x,y, or z
the effective potential
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:

   1 r N  r
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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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 B0 is given by:
magnetic coupling coefficient
field strength in the direction s (=x,y,z)
unperturbed occupied MO
unperturbed virtual MO
The principal contribution to the coupling coefficient u is in turn given by:
applies i L̂sν to each AO 
atomic orbitals (AOs)
unperturbed orbital energies
unperturbed MO coefficients
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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 spinrestricted 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)
occupied -MOs
virtual  MOs
-SOMO (singly unoccupied)
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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Prediction of EPR g-Tensors with DFT, CSC’82, 1999
Methods
Theoretical approach: Density functional theory (DFT)
Program:
Amsterdam Density Functional (ADF) v. 2.3.3[7]
Implementation of the EPR g tensors due to Schreckenbach and
Ziegler[5,6]
Basis set:
Uncontracted triple- Slater on the ns, np, nd, (n+1)s, and
(n+1)p valence shells of the metal atom; ns and np on main
group elements. Additional set of polarization functions on main
group atoms. Frozen core approximation for inner shells
Relativity:
Relativistic frozen cores and first-order scalar Pauli
Hamiltonian[8]
Vosko-Wilk-Nusair[9] (VWN) LDA; Becke-Perdew86[10,11]
(BP86) GGA
Functionals:
Treatment of radicals: Spin-unrestricted for quantitative calculations;
Spin-restricted for qualitative analysis
Hardware:
The Cobalt cluster[12] (do not miss the Cobalt poster!)
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Prediction of EPR g-Tensors with DFT, CSC’82, 1999
EPR g-tensor in axial MEX4 systems
z
x
y
Due to the high C4v symmetry of the MEX4 complexes,
only two independent g-tensor components are possible in
this system[1]. The isotropic value is then given simply by:
parallel component, field
along the C4 (Z) axis
doubly degenerate perpendicular
component, field perpendicular
to the C4 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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Prediction of EPR g-Tensors with DFT, CSC’82, 1999
Principal MO contributions to g
2*e1 (dxz,dyz)



SOMO
b2 (dxy)

2*e1
-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 TcNBr419
-spin ligand field-type contribution is
typically large and negative, and dominates
g for most MEX4 complexes. It becomes
larger in the 3d4d 5d series
Prediction of EPR g-Tensors with DFT, CSC’82, 1999
Principal MO contributions to g||
b1 (*)



b2 (dxy)
SOMO



b1

b1 ()
Positive -spin contribution increases
both for heavier metals and heavier
ligands. It dominates in complexes of 3d
and 4d metals with heavy ligands.
Negative -spin ligand field-type contribution
dominates g|| in most cases. It increases for
heavier metals and decreases for heavier
ligands.
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Prediction of EPR g-Tensors with DFT, CSC’82, 1999
Molecular geometry and g-tensor
ReOF4
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 giso show qualitatively
different dependence on the structural
parameters in MEX4 complexes: g shows no
dependence on the distance between the metal
and singly bonded ligand (RM-X), while g|| is
insensitive to the position of the doubly bound
ligand (RM=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.
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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 gtensor, 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 ReOBr4). The parallel
component g|| is highly sensitive, while the perpendicular component g is
almost unaffected.
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Prediction of EPR g-Tensors with DFT, CSC’82, 1999
Approximate functionals and g-tensor
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.
Nval
VWN//VWN
BP86//VWN
ave
abs rms ave
abs
rms
giso
15
48
48
66
49
49
67
g||
15
65
65
75
66
66
77
g
14
32
35
56
34
35
57
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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. Nval gives the number of
comparison for each component.
Prediction of EPR g-Tensors with DFT, CSC’82, 1999
Systematic errors: g component
Calculated (VWN)
TcNBr41-
ReOBr4
ReOCl4
Experiment
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.
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Prediction of EPR g-Tensors with DFT, CSC’82, 1999
Calculated (VWN)
Systematic errors: g|| component
TcNBr41-
ReOF4
Experiment
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||.
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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:
gcorr  gcalcd  cgcalcd 
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.
VWN
Nval
Corr. VWN
ave
abs rms ave
abs
rms
giso
15
48
48
66
2
14
20
g||
15
65
65
75
0
9
11
g
14
32
35
56
0
17
25
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Table 2. Empirical corrections
Metal
3d
4d
5d
c(giso)
-7
-27
-99
c(g||)
-26
-52
-112
c(g)
3
-15
-93
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 giso does not
vanish since the c(giso) correction was
computed from c(g||) and c(g) rather
than fitted independently.
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
MEX4 complexes we considered before. It is interesting to
see well we can describe one of the “real world” series MoO(SPh)41-, MoO(SePh)41-, WO(SPh)41-, and WO(SPh)41-:
Experiment[13]
1-,
MoO(SePh)4
C4v
MoO(SePh)41-, C4
g||
g
2.017
1.979
VWN
g||
g
2.024
1.927
2.028
1.977
MoO(SPh)41-
C4v
MoO(SePh)41-
C4
2.072
2.005
2.126
1.979
WO(SPh)41-
C4
2.018
1.903
2.056
1.928
WO(SPh)41-
C4
2.086
1.923
2.139
1.926
C4
So, we can describe qualitative trends due to gross changes
in molecular geometry (C4v vs C4 values for MoO(SePh)41-)
and periodic trends correctly in this series. We are still far
from approaching the experimental accuracy, however.
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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 freeelectron 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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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 31205-AC3). Dr. Georg
Schreckenbach is gratefully acknowledged for making the GIAODFT implementation of the EPR g tensors available to the authors.
References
1. J.E. Wertz and J.R. Bolton, Electron Spin Resonance
(Chapman and Hall, New York, 1986)
2. J.E. Harriman, Theoretical Foundations of Electron
Spin Resonance (Academic Press, New York, 1978)
3. G.H. Lushington and F. Grein, J. Chem. Phys. 106,
3292 (1997).
4. T. Ziegler, Can. J. Chem. 73, 743 (1995)
5. G. Schreckenbach and T. Ziegler, J. Phys. Chem. A
101, 3388 (1997)
6. G. Schreckenbach, Relativity and Magnetic
Properties. A Density Functional Study (Ph.D. thesis,
Calgary, 1996)
7. ADF 2.3.3, http://tc.chem.vu.nl/SCM (Dept.
of Theoretical Chemistry, Vrije Universiteit,
Amsterdam).
19
8. T. Ziegler, V. Tschinke, E.J. Baerends, J.G. Snijders,
and W. Ravenek, J. Phys. Chem. 93, 3050 (1989)
9. S.H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58
1200 (1980)
10. A.D. Becke, Phys. Rev. A 38, 3098 (1988)
11. J.P. Perdew, Phys. Rev. B 33, 8822 (1986); 34, 7406
(1986)
12. Cobalt cluster, see
http://www.cobalt.chem.ucalgary.ca/
13. G.R. Hanson, G.L. Wilson, T.D. Bailey, J.R. Pilbrow,
and A.G. Wedd, J. Am. Chem. Soc. 109, 2609 (1987)
Prediction of EPR g-Tensors with DFT, CSC’82, 1999