Electron Spin Resonance Volume 12A

A Review of Recent Literature to mid-1989

By M. C. R. Symons

The Royal Society of Chemistry

Copyright © 1990 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85186-881-3

Contents

CHAPTER 1 Organic Radicals in Solution By B.J. Tabner, 1,
1 Introduction, 1,
2 Carbon-centred Radicals, 3,
3 Nitrogen-centred Radicals, 13,
4 Oxygen-centred Radicals, 15,
5 Nitroxyl Radicals, 17,
6 Sulphur-centred Radicals, 22,
7 Radical Cations, 23,
8 Radical Anions, 37,
9 CIDEP, 46,
CHAPTER 2 Triplets and Biradicals By A. Hudson, 60,
1 Introduction, 60,
2 Ground and Thermally Excited Triplets, and States of Higher Multiplicity, 60,
3 Photoexcited Triplet States, 66,
CHAPTER 3 Spin Labels: Biological Systems By Ching-San Lai, 74,
1 Introduction, 74,
2 Proteins, 74,
3 Nucleic Acids, 80,
4 Properties of Model and Biological Membranes, 80,
5 Lipid-Protein Interaction, 84,
6 Cellular Membrane Dynamics, 88,
7 Modification of Membrane Functions, 90,
8 Methods, 93,
9 Nitroxide Reduction, 94,
10 Synthesis, 95,
CHAPTER 4 Applications of E.S.R. in Polymer Chemistry By D.J.T. Hill, J.H. O’Donnell, and P.J. Pomery,
104,
1 Introduction, 104,
2 Polymer Degradation, 104,
3 Polymerization, 112,
4 Polymer Structure, 120,
5 Conducting Polymers, 124,
CHAPTER 5 Free Radical Studies in Biology and Medicine By N.J.F. Dodd, 136,
1 Introduction, 136,
2 Tissues, 136,
3 Radiation Effects in Biological Molecules, 143,
4 Radical Reactions of Drugs and Toxic Chemicals, 150,
5 Enzymes, 162,
6 Oxygen Radicals, 164,
7 Other Systems, 167,

CHAPTER 1

Organic Radicals in Solution

BY B. J. TABNER

1 Introduction

In my report for Volume 12A, which covers the period June 1987 to May 1989, I have retained the same general layout as used previously.

It is pleasing to note that the number of papers published dealing with neutral and charged organic radicals in solution has increased slightly compared with two years ago. Together these papers continue to include a very wide range of applications and interest in the technique is obviously maintained. E.s.r. has now celebrated its 40th birthday and commercial spectrometers have been readily available for over 25 years. Nevertheless new, and sometimes quite ingenious, applications are regularly reported. Modern spectrometers are undoubtedly more sensitive than those of 25 years ago but, perhaps more importantly, there have been other significant developments in instrumentation. A recent review claims that “this is an exciting time in EPR; several research labs are currently developing new experimental techniques and new interpretations that will revolutionise the method’s possible applications. Prominent among these developments are new micro-wave sources and resonators that promise much greater signal-to-noise ratios, time-domain spectroscopies that will open a new dimension of spin interactions and dynamics, Fourier transform EPR that will revolutionise spin-label and other organic radical studies, EPR imaging that will yield a spectrum at any point in the sample, in vivo spectroscopy that will make whole biological organisms accessible to EPR, and computation capabilities that will simplify complex simulations we did not dare dream about just a few years ago.”

One of the instrumental developments which has been available for several years is ENDOR spectroscopy and no doubt many readers will find a recent book, 1 ENDOR Spectroscopy of Radicals in Solution’, invaluable. Also invaluable is a new volume in the Landolt-Bornstein series dealing with the Magnetic Properties of Free Radicals. The most recent Volume covering ‘Nonconjugated Carbon Radicals’ is particularly relevant to the subject area of my report. Also now available is the collection of papers presented at the XVth Annual International Conference on ESR in Organic and Bio-organic Systems, held at Cardiff in 1988.

A significant development over the last 10 years has been the study of radical cations formed by ionising radiation in frozen halocarbon solutions. A review covering these developments is, therefore, timely. Other recent reviews cover the way in which the ·CH2 moiety can be used as a ‘spin probe’ in the study of conformational problems, and the methods available for determining and analysing radical self-reactions and additions. Also reviewed is the analysis of dynamic e.s.r. and ENDOR spectroscopy of organic radicals in solution. It is worth noting that the ENDOR technique can be sensitive to dynamic processes in the ‘rate’ range where e. s. r. spectra are unaffected. Also described is hyperfine-selective ENDOR which is based on a pulsed ENDOR scheme. The main advantage appears to be its capability to measure the ENDOR subspectrum originating exclusively from nuclei with a predicted splitting constant.

Although Fourier transform NMR has become widespread the same is not true in e.s.r. However, a detailed description of a two-dimensional Fourier transform e. s. r. spectrometer has now been presented which includes applications of the method.

Unfortunately in a conventional spectrometer it is not possible to sweep the magnetic field rapidly enough to study transient radicals produced in flash photolysis experiments. However, Mclachlan and Stevens describe how this problem can be overcome by producing radicals repetitively in a series of flashes as in flash photolysis e.s.r.

Bond et al. have reviewed the various simultaneous e.s.r.-electrochemical techniques. They cover both stationary and flow-through cells. The same authors also describe a novel bubble electrode which appears to operate reproducibly and allows longterm experiments in the absence of both oxygen and light. A new design of flat cell, which incorporates a long capillary folded within a conventional flat cell, is convenient and can be oriented either parallel or perpendicular to the magnetic field.

Considerable attention continues to be paid to the computational analysis and interpretation of complex hyperfine patterns. Both a maximum entropy method and a novel symmetry development transformation approach have been described. The former is suitable for noisy spectra. Several research groups describe methods of obtaining simulations which give the best possible fit to experimental data. Heikki describes a second-order simulation suitable for the Apple II Plus whereas Beckwith and Brumby deal with the old problem of determining radical concentrations. Finally, a simple and convenient subtraction technique, not limited by noise, has been developed.

2 Carbon-centred Radjcals

I have slightly modified the first part of my report so as to include a small separate section on a-electronic radicals. As in previous reports interests cover the kinetics and mechanisms of radical reactions as well as radical decompositions, additions, isomerisations, and other related topics.

2.1 Nonconjugated Carbon-centred Radicals

2.1.1 Alkyl Radicals. -A common radical reaction is the addition of a small radical across a carbon-carbon double bond. Ozawa et al. have studied the addition of the sulphite radical anion to a range of olefinic compounds to give the corresponding adduct. These reactions have a rate comparable with that of ·OH addition, but the e.s.r. spectra of the resulting radicals indicate that addition is influenced by steric factors to a greater extent. The addition of the hydroxyl radical to alkenenitriles generally favours the least substituted carbon atom but spectra resulting from the alternative addition are sometimes also observed.

The mechanism of rearrangement and isomerisation reactions has attracted considerable attention for many years. This interest is reflected in a number of reports. For example, Asmus and Gilbert et al. have studied the oxidation of 2,3-dimethyl-butane-2-ol using time-resolved pulse radiolysis, complimented by steady-state e.s.r. to identify the radicals involved. Reaction with hydroxyl radicals gives ·CH2CHMeCMe2OH, Me2CHC(OH)(Me)CH2 and Me2CCMe2OH [a(6H) 2.30 and 0.05 mT]. At pH ca. 1 the latter radical is replaced by ·CH2C(Me)=CMe2 [a.(3H) 1.575, 1.245, and 0.30 and a(H) 1.31 and 1.28 mT]. The formation of this latter radical, by acid catalysed loss of hydroxide, provides evidence Electron Spin Resonance for the formation of a radical cation intermediate. The rearrangement of 3-ethoxycarbonylbut-1-enyl to 1-ethoxycarbonyl-but-1-enyl, at 233 K, has also been demonstrated. The former radical has been generated from three different halides at 200 K [a(H) 2.315, 2.20, 2.15, 0.163, 0.113, and 0.05 mT (note that the two β-protons are slightly inequivalent)]. At 160 K the CH2=CHCH(CO2Et)CH2CH2 radical [a(2Hα) 2.235, a(Hβ) 2.36, and a(2H) 0.05 mT] is observed which, upon raising the temperature, rearranges to give the former radical.

A muon spin rotation (µSR) study of the isomerisation of α-carbonyl-, α-carboxyl-, and α-carbamide-substituted alkyl radicals indicates that the activation energy for the isomerisation is not significantly influenced by methyl substitution at the radical centre or by replacement of the ester group by a dimethylamido group. These µSR results indicate a considerable barrier to rotation about the partial double bond between the alkyl radical centre and the substituent and confirm earlier e.s.r. measurements.

Spectral assignments can be complicated by the presence of a chiral centre and such centres can have long-range effects on the magnetic equivalence of β-methylene protons. For example, in the CH2(OH)CH(OH)CHCH2OH radical all six observed hyperfine couplings are non-equivalent implying that the β-carbon chiral centre causes inequivalence of the nonadjacent β-H(CH2OH) protons as well as the adjacent γ-H(CH2OH) protons.

Radicals derived from amines are important in radiation chemistry and in biochemistry and can be readily generated by photolysis of a solution of the amine and di-t-butyl peroxide. The e.s.r. spectrum of the α-aminoethyl radical, CH3CHNH2, has a(Hα) 1.47, .a(N) 0.44, and a(3Hβ) 2.07 mT and two non-equivalent amino protons [a(H) 0.25 and 0.545 mT]. The barrier to rotation was found to be 31.9 kJ mo1-1. The spin-trapping technique (employing n-.t-butyl-a-phenylnitrone) has also been used to study the radicals formed upon photolysis of mixtures of CH4, NH3, and H2O. The trapped radicals [a(N) 1.48 and a(H) 0.36 mT] are assigned to the ·CH2NH2 adduct. The photolysis of di-t-butyl peroxide forms a convenient route to hydrogen atom abstraction from CH2(Ome)2. Both the ·CH(OMe)2 radical [a(Hα) 1.19 and a(H[gama]) 0.0795 mT) and the ·CH2OCH2OMe radical [a(Hα) 1.784 and a(Hγ) 0.078 mT] are formed. The relative proportions of the two radicals appears to be influenced by their stabilities.

The reports covered so far illustrate only some of the common radical reactions. Gilbert et al. have examined the role of electronic and steric factors in determining the ease of abstraction, fragmentation, and ring-opening reactions in order to establish if intermediate radicals formed via 1, 5-shifts are involved in these reactions. For example, the •CHMeOEt radical reacts with butynedioic acid (HO2CC[equivalent to]CCO2H) to give ‘CH(Me)OCH(Me)C(CO2H)=CH(CO2H) [a(3H) 1.25 and a(H) 1.33 and 1.25 mT]. These radicals undergo fragmentation and ring-opening and can trap on a second alkyne molecule. These results show that oxygen-conjugated radicals add particularly rapidly to butynedioic acid and that in most cases the resulting vinyl radical undergoes a rapid 1,5-hydrogen transfer.

The e.s.r. spectra observed following reaction of ‘OH with N-acetylamino acids, in a flow system, indicate the formation of MeCONHCHCO2H from N-acetylglycine and MeCONHC(Me)CO2H from N-acetylalanine. In the former case the MeCONHCH2 radical, formed by decarboxylation, is also observed. When a flow-system is not employed the spin-trapping technique is invaluable for systems which produce a very low steady-state radical concentration. This latter technique has been employed to study alkyl, and other radicals, formed during the photolysis of N-hydroxypyridine-2-thione esters and during the decomposition of diacyl peroxides.

Kinetic parameters for radical reactions have always attracted interest, particularly where they apply to ‘clock’ reactions. Ingold et al. have now reported a secondary alkyl radical reaction, the cyclisation of 1-methy1-5-hexenyl, which is useful in this respect. Several other useful clock reactions reported include the related 3-oxahex-5-enyl, 2-oxahex-5-enyl, and 2,2-dimethylbut-3-enoyloxymethyl radicals all of which undergo cyclisation.

There has been considerable interest in cycloalkylmethyl radicals where the ·CH2 moiety is attached to alkyl rings of various sizes. The smallest member of the series, the cyclo-propylmethyl radical, has been the subject of two studies. Walton has prepared this radical by bromine atom abstraction from cyclopropylmethyl bromide. At 122 K the e.s.r. spectrum can be interpreted in terms of a(2Hα) 2.09, a(Hβ) 0.250 and a(2Hγ) 0.296 and 0.200 mT. At lower temperatures the central multiplet in the spectrum broadens indicating restricted rotation about Cβ-Cα (Ea = 11.5 ± 0.8 kJ mol-1). Newcomb et al. have established that in the temperature range 236 -323 K the ring opens to give the 3-butenyl radical, CH2=CHCH2CH2. This reaction is useful as a radical clock rearrangement. The spectra of cyclohexylmethyl radicals indicate the presence of two conformations, one in which the CH2 moiety is equatorial and the other in which it is axial. The line broadening observed in the axial conformer is attributed to restricted rotation of the ·CH2 moiety (ca. 25 kJ mol-1). This relatively high barrier results from steric interactions with the syn axial hydrogens at positions 3 and 5. The e.s.r. spectra of cycloalkylmethyl radicals with 9-to 15-membered rings all indicate the presence of two conformers. It is apparent that the ·CH2 moiety can provide a very useful “spin probe” since it reveals information on the preferred conformation with respect to the Cβ-Hβ bond and also, in some cases, on the dynamics of ring interconvertions.

2-(Cycloalkenyl)ethyl radicals with C4 C7 rings have also been reported. The e.s.r. spectra of all of these radicals are characterised by a basic triplet of triplets [a(Hα)ca. 2.16 – 2.25 mT]. The negative temperature coefficients for the _a(Hβ) splittings reveal a preferred conformation about the Cα-Cβ bond in which the bond makes an angle of 90° with the plane of the ring. Restricted rotation about the Cα-Cβ bond is observed with the barrier to rotation’ decreasing as the size of the ring increases.

2.1.2 Cyclic Alkyl Radicals.-Three-membered ring radicals studied include 1-substituted cyclopropyl radicals48 and the oxiranyl radical. At 203 K the 1-methylcyclopropyl radical has a(3H) 1.95 and a(4H) 2.1 mT. At lower temperatures line-broadening occurs which is complicated by slow rotation of the methyl group. However, at 92 K the a(4H) splitting becomes a(2H) 2.485 and 1.655 mT. The former value is assigned to the two ring protons, the latter to the two anti ring protons (Ea [??] 13 kJ mol-1). Deuterium substitution has been employed to study the inversion of the oxiranyl radical. In the [2-2H]oxiran-2-yl radical the inversion is frozen at low temperatures and the two β-protons are distinguishable [a(Hβ) 0.523 and 0.474 mT]. In the oxiranyl radical itself the two β-protons remain equivalent (0.506 mT) although some line broadening is observed. A curved Arrhenius plot is obtained for the latter radical from which it is concluded that inversion proceeds via quantum-mechanical tunnelling. The oxiranyl radical has also been observed during the photolysis of CH3CN in oxirane/cyclopropane.

The e.s.r. spectra of 3-methylenecyclobutyl (1) and cyclo-pent-3-enyl (2) have a(Hα) 2.20, a(4Hβ) 3.76, and a(2Hδ) 0.035 mT 0 and a(Hα) 2.12, a(4Hβ) 3.69, and a(2Hγ) 0.046 mT respectively. Both of these radicals have only small couplings of the protons attached to the C=C bond indicating very little unpaired electron density reaches this bond.

Reaction of cyclopentanone with hydroxyl radicals gives two radicals, (3) and (4). Radical (3) predominates and has a(H1) 2.123, a(2H2) 3.493, a(2H4) 0.090, and a(2H5) 3.758 mT. This spectrum displays higher-order effects arising from the interaction of two pairs of nearly equivalent β-CH2 protons.

The methylcyclohexyl radical (5) has been prepared by γ-irradiation of the thiourea canal complex. Spectra have been recorded from 77 to 294 K and interpreted in terms of a pyramidal structure with a CCC angle of about 117°. At low temperatures, two planar conformations can be identified which are related via an umbrella inversion.

The addition of the sulphate radical anion to a variety of substituted uracils, to give a range of C-5 and C-6 hydroxyl adducts, has been studied as has the addition of the CH2OH radical to some α-enones. The spin-trapping technique has been employed to study the radicals produced from 2-aryl-1,3-dithianes during photolysis with benzophenone.

(Continues…)Excerpted from Electron Spin Resonance Volume 12A by M. C. R. Symons. Copyright © 1990 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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Electron Spin Resonance Volume 10B

A Review of Recent Literature to mid-1986

By M. C. R. Symons

The Royal Society of Chemistry

Copyright © 1987 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85186-851-6

Contents

CHAPTER 1 Spin -Spin Interactions in Weakly Interacting Dimers By C.P. Keijzers,
Introduction,
2 Spin-Hamiltonian, 2,
3 Isotropic Exchange, 5,
4 The Zero Field Splitting Tensor, 20,
5 Antisymmetric Exchange, 28,
CHAPTER 2 Transition-metal Ions By J.F. Gibson,
1 Introduction, 39,
2 General, 40,
3 S = 1/2, 61,
4 S = 1, 77,
5 S = 3/2, 77,
6 s = 5/2, 79,
7 S = 3, 85,
CHAPTER 3 Metalloproteins By G.R. Hanson and J.R. Pilbrow,
1 Introduction, 93,
2 Copper Enzymes, 94,
4 Iron Sulphur Proteins, 105,
5 Nickel -Iron -Sulphur and Related Iron -Sulphur Enzymes, 109,
6 Molybdenum Containing Enzymes, 111,
7 Vanadium, 114,
8 Paramagnetic Metal Substituted Enzymes, 114,
9 Mitochondrial Electron Transport Chain, 119,
10 Photosynthesis, 123,
11 Statistical Model for E.p.r. Line Broadening, 127,
12 Relaxation References, 128,
CHAPTER 4 ENDOR Methodology By A. Schweiger,
1 Introduction, 138,
2 Basic Instrumentation and Experimental Techniques, 139,
3 Analysis of ENDOR Spectra, 143,
4 Theoretical Approaches, Mechanisms and Conditions, 155,
5 Advanced ENDOR Techniques, 159,
6 Outlook, 176,
CHAPTER 5 Spin Trapping Free Radical Metabolites of Inorganic Chemicals By R.P. Mason and C. Mottley,
1 Introduction, 185,
2 Carbon Dioxide Anion Radical, 186,
3 Sulfur Dioxide-, Bisulfite-or Sulphite-Derived Radicals, 188,
4 Azidyl Radical, 191,
5 Hydrogen Atom, 192,
6 Oxygen-derived Radicals, 193,
7 Conclusion, 195,
CHAPTER 6 Inorganic and Organometallic Radicals By Martyn C.R. Symons,
1 Introduction, 198,
2 Trapped and Solvated Electrons, 200,
3 Atoms, Atom Clusters and Atom -Ligand Complexes, 202,
4 Diatomic Radicals (AB), 213,
5 Triatomic Radicals (AB) and Related Species, 215,
6 Tetraatomic Radicals (AB3) and Related Species, 218,
7 Pentaatomic Radicals (AB4) and Related Species including Higher Coordinated Species, 219,
8 Other Radicals, 222,
9 Radicals in Inorganic Materials, 224,
10 The Use of Spin-Traps, 228,
11 Transition Metal Carbonyls and Related Species, 230,
12 Radicals in the Gas Phase, 233,

CHAPTER 1

Spin-Spin Interactions in Weakly Interacting Dimers

BY C. P. KEIJZERS

1 Introduction

As such, the subject of “Spin-Spin Interactions” has not beenthe subject of discussion in this series. Under different titles, such as “Transition Metal Ions”, “Triplet Biradicals” or “Inorganic and Organometallic Radicals”, various theoretical and experimental results have been discussed that are related to this subject (see, for instance, reference 1) but an integrated discussion has not been provided. In the past years, several groups have applied themselves specifically to the study of various aspects of the “exchange” phenomenon in order to obtain a better understanding of the physical interactions that are underlying the various terms in the effective spin-Hamiltonian with which the EPR spectra of systems with spin-spin interactions are described. An understanding of the magnetic exchange interactions propagated by multi-atom bridges could give insight into, for instance, the pathways of electron transfer in biological electron transport chains. It could also be used as a guideline for the preparation of new and interesting polymetallic complexes or one-and two-dimensional magnetic exchange systems with magnetic properties that can be predicted, both in nature and in magnitude.

It is not the intention of this contribution to be an all inclusive review of spin-spin interaction studies, this would be impossible in view of the breadth of the field and the vast literature. Instead, the subject is limited to the spin-Hamiltonian

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

which is applied for the description of magnetic exchange in weakly interacting dimers or other discrete (transition) metal complexes. A review is given of the fundamental theory that is necessary for the interpretation of the Hamiltonian (1) and selected papers from the literature are cited in which these interactions are either calculated or experimentally determined. For the experimental work, the attention is focussed mainly to EPR which means that various other techniques which are relevant to the subject (like for instance susceptibility, NMR, Mossbauer, optical spectroscopy) are not discussed. Also lineshape and linewidth studies in (low dimensional) magnetic systems are not discussed. This extensive, complicated but very interesting subject would warrant a separate contribution in this series. For the time being, we refer to some reviews and especially also to the extensive work of Soos (for instance references 5 and 6).

Many reviews, textbooks and conference proceedings are available that have a bearing on the subject of this contribution. Especially the last one will be referenced often: it contains contributions covering a wide range of experimental and theoretical topics in this field.

2 Spin-Hamiltonian

The spin-Hamiltonian (1) is the usual Hamiltonian for the description of the interaction of two S = 1/2 ions in zero magnetic field. The tensor[??] can be decomposed into its trace, a symmetric tensor and an antisymmetrical tensor:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

(One has to pay attention when “J”-values from different authors are to be compared, because for the isotropic part also the formulations [FORMULA NOT REPRODUCIBLE IN ASCII] and [FORMULA NOT REPRODUCIBLE IN ASCII] are used). The effect of J is a separation of the four spin-functions into a singlet and a triplet. [??]s splits the triplet into a doublet and a singlet (in case of axial symmetry) or into three singlets.[??], finally, mixes the singlet with all three triplet functions. The essential requirement for the antisymmetric term,[??].[??]1x[??]2 is the absence of a centre of symmetry between the magnetic sites containing S1 and S2. If an inversion centre would exist, [??]1 and [??]2 would interchange under the inversion operation and [??]1x[??]2 would change its sign. Since the Hamilton operator must be invariant for any symmetry operation of the system, this means that[??] must change sign as well. Hence,[??] = 0.

In the principal axes system of[??]s and in the basis of the eigenfunctions of [??]s, the Hamiltonian matrix is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where X, Y, z are the eigenvalues of Ds (X + Y + Z = 0), and the basisfunctions are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

For [??]s and/or [??] equal to zero, the energies and eigenfunctions of (3) are:

[ILLUSTRATION OMITTED]

In this diagram, the functions |ψi >are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The diagram shows that the effect of [??], the antisymmetric exchange interaction (AEI), is an increase of the apparent isotropic exchange interaction (IEI) and, as far as the triplet manifold is concerned, it is equivalent to an axial zero field splitting (ZFS) tensor [??]s, but only for the energies. The AEI mixes also the triplet functions with the singlet and, therefore, the EPR transition probabilities within the triplet manifold are influenced. It is questionable, however, whether this effect can be distinguished from the symmetrical ZFS tensor in an actual EPR spectrum where also other interactions (like nuclear hyperfine coupling) will influence the line intensities. A second consequence of the singlet-triplet mixing is that new transitions, between |ψ1 > and |ψ2-4>, are allowed but even the observation of these transitions is not a direct proof of the presence of an antisymmetric contribution in the spin-spin interaction. The reason is that d exists only for low symmetry dimers (see paragraph 5). For such systems, the g-tensors of the two spins will be different and also this causes singlet-triplet transitions to be allowed (|S0 > is not an eigenfunction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The symmetry aspects of the antisymmetric exchange and a general formulation for its matrix elements for any pair of S1, S2 values were discussed by Bencini and Gatteschi. These authors conclude also that” … at the present stage of knowledge the meaningful quantitative determination of the antisymmetric exchange term seems to be impossible in general cases”. Their expectation that the misalignment between the g and D axes can give the key to recognize the presence of this exchange term is not justified because even in high(er) symmetry systems g and D do not necessarily coincide as is found in many dimers like, for instance, Cu-Cu and Ag-Ag dimers in [Zn(et2dtc)2]2.

For the case |J | [much greater than] [??]s, [??] (i.e. S is a good quantum number) Scaringe et al. derived expressions for the g-tensor, the ZFS and the nuclear hyperfine interaction tensors of a dimer in relation to the monomer tensors both, for = S1 = S2 and for S1 ≠ S2. The validity of these relations was shown in many examples as, for instance, Cu-Cu, Cu-Mn and Cu-Zn pairs in [Cu(PyNO)Cl2 (H2O)]2 (PyNO = Pyridine-N-oxide).

An application of the relations between monomer and dimer tensors could be the determination of the monomer tensors from the dimer parameters if the monomers themselves do not show an EPR signal. Gatteschi and Bencini discussed the calculation of the g-tensor of Ni(II) ions from the spectra of Cu-Ni pairs and knowing the tensor of the Cu-ion. However, care should be exercised in applying these relations because sometimes they do not hold and the reason for that is not yet found although magnetostriction and/or the presence of antisymmetric exchange have been mentioned.

3 Isotropic Exchange

3.1 Spin-Hamiltonian. -Although the isotropic exchange interaction between magnetic atoms or molecules is usually represented by the Heisenberg Hamiltonian

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

in principle also higher order terms are possible, like for instance biquadratic exchange [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. However, multiple exchange interactions between the atoms or molecules are negligible, then the Hamiltonian can be truncated after the bilinear term. Anderson derived with a perturbation treatment that (a) these higher order terms only appear for ions with more than one electron and (b) that they are of the order of 1% of the bilinear term. Herring used a modified Heitler-London method to prove the validity of the Heisenberg Hamiltonian assuming only that the real Hamiltonian does not contain spin variables and that the separated monomers have no orbital degeneracy. As Anderson, he concluded that the higher order terms are in principle present but that they are negligibly small. The quantitative estimate of Anderson was experimentally verified for a number of systems: Gudel et al. found (with inelastic neutron scattering) that the Lande-interval rule (E(S) -E(S-1) -2JS), which is the result of the Heisenberg Hamiltonian, is not satisfied for dimers of Mn2+ (S = 5/2) ions in CsMnxMg1-xBr3. The magnitude of the biquadratic exchange which they derived was ~0.003% of the bilinear term. A similar ratio between J and j was found in mono and bis(µ-hydroxo)-bridged Cr3+ (S = 3/2) dimers with luminescence and absorption spectroscopy and in a tris (µ-hydroxo)-bridged Cr3+ dimer from a fit of the magnetic susceptibility Gudel warned that if larger j-values are obtained experimentally, they are likely to be artifacts or the result of other physical effects, as e.g. magneto striction. Also an ab-initio calculation of the exchange interaction between O2 molecules showed that the coupling between the monomer triplets can be very well fitted with the Heisenberg Hamiltonian and that the Lande-interval rule is perfectly obeyed.

Drillon and Georges and Leuenberger and Gudelpointed out that the Heisenberg Hamiltonian is appropriate only for ions in orbital singlet states. For the coupling of ions in orbitally degenerate states, as Ti3+ in a trigonal ligand field, Hamiltonians are to be used which contain orbital parameters that cannot be collected in an overall J: for instance for the description of the exchange interaction between two Ti3+ ions, also the local trigonal distortion, spin-orbit coupling and covalency effects are introduced in the Hamiltonian. A study of Ti2x3-9 (X = C1, Br) units showed that the exchange parameters are in agreement with a simple molecular orbital calculation. The most uncertain parameter in the Hamiltonian is the (trigonal) ligand field.

3.2 Analytical Expressions for J. -In this section, the discussion will be limited to orbitally non-degenerate states of the monomers. For systems with orbital degeneracy, Hamiltonians were derived by Fuchikami in collaboration with Tanabe and with Block.

For non-degenerate systems, various analytical expressions for J were derived by a number of authors. All expressions agree in so far as that they contain a ferromagnetic and an antiferromagnetic contribution and that the ferromagnetic contribution (positive in the formulation (6) of the Heisenberg Hamiltonain) is, in first order, a two-electron exchange integral. However, the expressions for the antiferromagnetic contributions are different for different derivations. The reason for this disagreement is that the description of a dimer (the discussion is restricted to the interaction between two monomers) is not straightforward. It is well known that a molecular orbital (MO) treatment of a dimer as a “super molecule” leads to an incorrect asymptotic behaviour of the wave function for large monomer-monomer distances (It contains covalent and ionic configurations with equal weight). But also a localized, Heitler London (or Valence Bond, VB), approach does not lead fast enough to a correct asymptotic result. For that reason, Herringdeveloped a modified Reitler-London method with functions that resemble, in their localization, the free atom functions but which, at the same time, approximate as closely as possible the exact eigenfunctions at large separations of the monomers. However, this approach found little application, as far as we know.

For MO as well as VB descriptions, improvement is to be expected from configuration interaction (CI), but one is forced to a limited CI if an analytical expression for J is to be derived. (However, also the ab-initio calculation of J, as the difference between the energies of different spin states of a dimer, is impossible without approximations for extended systems, because of the very large number of two-centre integrals and the enormous number of possible configurations).

Usually, the first step in a calculation of J is obtaining a wave function of the magnetic monomers which are surrounded by various diamagnetic groups. In this step, the exchange effects of the other magnetic ion(s) are excluded. That these exchange effects in weakly interacting dimers do not disturb the wave function is shown experimentally by the agreement of hyperfine interactions with ligand nuclei in dilute and in concentrated paramagnetic dopes of the same diamagnetic lattice. The next step is to construct dimer functions from the monomer fragments. The most serious question that arises here is that concerning the orthogonality of the monomer wave functions. Already Andersonrealized the importance of this point and more recently Kahnand others drew the attention to the fact that the construction of orthogonal monomer functions does lead to an easier but certainly not to a better description of the dimer. The orthonormalized functions fail to be eigenfunctions of any physical system. Therefore, they cannot be expected to describe reasonably correct the unperturbed states of the monomers. For that reason, a perturbation expansion may have to be carried through to a higher order in order to obtain acceptable values for the parameters in the spin-Hamiltonian.

3.2.1 Localized Method. -In the VB method, the orbital parts of the ground state singlet and triplet functions are constructed with the functions ψA and ψB of the monomers A and B:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where “+” holds for the singlet and “-” for the triplet; SAB is the overlap integral . If the Hamiltonian is defined

H = h(l) + h(2) + hint (8)

where h(i) are the one-electron Hamiltonians, including the intramonomer interactions hA(l) and hB(2) (of which ψA and ψB are eigenfunctions) and the interactions with the nuclei of the “other” monomer, and hint is e2/r12, then the energies of ψS,T are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

and where it is assumed that the monomers A and B are identical. The resulting singlet-triplet splitting is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

This localized method, but with orthogonalized orbitals, was used by Anderson and by Hay, Thibeault and Hoffmann. Essentially they followed the same procedure but Anderson applied it for an infinite lattice whereas Hay et al. calculated J for a dimer. Kahn rephrased the Anderson approach for the interaction between two identical single-ion doublet states. The first step is to calculate the two highest singly occupied (sometimes called magnetic) molecular orbitals ψb (bonding) and ψa (antibonding) of the triplet state. In a weakly interacting dimer these MO’s are approximately:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

The next step is to determine the orthogonalized magnetic orbitals:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

ψA and [spi]B have metal and ligand character, but they are essentially centered on the monomers A and B, respectively (In the Anderson treatment these steps were 1. calculation of Bloch functions and 2. construction of Wannier functions). Another way of obtaining orthogonal localized orbitals is Lowdin orthogonalization [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], resulting in:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

(Continues…)Excerpted from Electron Spin Resonance Volume 10B by M. C. R. Symons. Copyright © 1987 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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