Prediction of EPR gTensors of Transition Metal Compounds with Density Functional Theory: First Appli

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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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Introduction
Electron Paramagnetic Resonance (EPR) is an
important tool in experimental studies of systems
containing unpaired electrons1. 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
understood2. 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 expensive3. 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 compounds4,
making it the tool of choice for such
calculations. In this work, we apply the recently
developed DFT implementation of the EPR
g-tensors5 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 first-principles theoretical approach to
EPR g-tensors of transition metal compounds.
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Theory
Quasi-relativistic DFT formulation of the EPR
g-tensors used in this work distinguishes between
several contributions to the g-tensor5,6
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
The occ-vir term is usually the most
qualitatively important contribution.
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The contribution is given by
(atomic units)
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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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
The principal contribution to the coupling
coefficient u is in turn given by
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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
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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Methods
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EPR g-tensor in axial MEX4 systems
Due to the high C4v symmetry of the MEX4
complexes, only two independent g-tensor
components are possible in this system1. The
isotropic value is then given simply by
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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Principal MO contributions to ?g?
?-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 TcNBr41-
?-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
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Principal MO contributions to ?g
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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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 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 (RME). 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.
ReOF4
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Relativity and g-tensor
Within the present quasi-relativistic approach,
it is possible to distinguish several sources of
relativistic contributions to ?g, namely
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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.
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Systematic errors g? component
The errors in the g? component are strongly
correlated with the metals 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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Systematic errors g component
The errors in the g component are also strongly
correlated with the metals 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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Empirical corrections to calculated g and g?
Given the strong correlation between the residual
errors calculated g-tensor components and the
metals 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.
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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-
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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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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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 GIAO-DFT
implementation of the EPR g tensors available to
the authors.
References