Transcript EPR of spatially non-degenerate high

Calculation of EPR g tensors for spatially
non-degenerate high-spin radicals with
density functional theory
I am on the Web:
http://www.cobalt.chem.ucalgary.ca/ps/posters/EPR-HS/
S. Patchkovskii and T. Ziegler
Department of Chemistry, University of Calgary,
2500 University Dr. NW, Calgary, Alberta,
T2N 1N4 Canada
EPR g-tensors of high-spin radicals with DFT
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Introduction
Accurate techniques for ab initio prediction of EPR g-tensors of small spatially nondegenerate 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.
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Effective spin-Hamiltonian
The simplest form of the effective spin-Hamiltonian, with hyperfine terms omitted, is:
Bohr magneton
Effective spin
Zero-field splitting
2.0023…
Magnetic field
Deviation from free electron
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:
Energy level,
Field direction,
For small deviations from the free-electron ge value, energy levels are given by:
In the free-electron limit (p=0) two of the eigenfunctions are given by simple products:
;
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Microscopic Hamiltonian
The corresponding microscopic DFT Hamiltonian, is given by (see[2,7] for the exact forms
of the individual operators):
Scalar field- Spin-Zeeman
Diamagnetic
(free
electron)
free operators
(gauge) terms
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, i0 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
EPR g-tensors of high-spin radicals with DFT
of the effective spin-Hamiltonian treatment.
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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.
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Methods
Theoretical approach: Density functional theory (DFT)
Program:
Amsterdam Density Functional (ADF) v. 2.3.3[8]
Implementation of the EPR g tensors due to Schreckenbach and
Ziegler[2]
Basis set:
Uncontracted triple- Slater on the ns, np, nd, (n+1)s, and
(n+1)p valence shells of metal atoms; 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[9]
Vosko-Wilk-Nusair[10] (VWN) LDA; Relative energies: BeckePerdew86[,11] (BP86) GGA
Functionals:
Treatment of radicals: Spin-unrestricted
Hardware:
The Cobalt cluster[12]
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Results: Main group diatomics
40 8
0
B2,
Ge2
3
O2, 3
BC, 4
SiB, 4
AlC, 4
-4
-20
-8
-8
PH, 3
+, 4 
Si2+, 4
-4
SiAl, 4
0
SeO, 3
C 2 +, 4 
4
S2 , 3 
8
-40
Ge2+, 4
GaAs+, 4
-60
Calculated g  (BP86)
NH, 3
20 4
0
NI, 3
SO, 3
Experimental g, parts per thousand (ppt)
-80
-80
-60
-40
-20
0
20
BP86
VWN
16
16
Average
error, ppt
-1.7
-1.2
absolute
error, ppt
3.3
3.7
RMS
error, ppt
5.9
6.7
Points
Gas-phase
value
40
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
EPR g-tensors of high-spin radicals with DFT
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Main group-transition metal diatomics
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MnBr,
MnI, 7
7
MnO, 6
MnH, 7
CrH, 6
CrN, 4
-20
MnF, 7
NbO, 4
-40
VO, 4
YAl+, 4
CrF, 6
YB+, 4
-60
Calculated g  (BP86)
MnS, 6
MoN, 4
0
VWN
15
15
Average
error, ppt
+4.6
+3.9
absolute
error, ppt
8.0
8.0
RMS
error, ppt
9.8
9.8
Points
MnCl, 7
20
BP86
Gas-phase
value
Experimental g, parts per thousand (ppt)
-80
-80
-60
-40
EPR g-tensors of high-spin radicals with DFT
-20
0
8
20
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Transition metal-transition metal diatomics
20
ZrNb,
4
CrAg, 6 MnAg,
CrAu, 6
7
WAu, 6
0
Mn2+, 12
-20
TiV, 4
-40
-100
V2+, 4
TiNb, 4
-60
ZrV, 4
HfV, 4
-80
WAg, 6
-100
WCu, 6
-80
BP86
VWN
13
13
Average
error, ppt
+37
+38
absolute
error, ppt
37
38
RMS
error, ppt
43
43
Points
Calculated g  (BP86)
40
Experimental g, parts per thousand (ppt)
-60
EPR g-tensors of high-spin radicals with DFT
-40
-20
0
9
20
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3B
u
trans-(CNSSS)22+
g2
g3
21º
16±2º
g1,
ppt
 g2 ,
ppt
 g3 ,
ppt
Exp[14]
-0.11
+14.82 +24.82
VWN
-0.6
+13.4
+21.8
BP86
-0.7
+13.3
+21.4
EPR g-tensors of high-spin radicals with DFT
(CNSSS)22+ 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.
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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 gtensor 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
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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 original
GIAO-DFT implementation of the EPR g tensors available to the
authors.
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EPR g-tensors of high-spin radicals with DFT
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