ELECTRON RESONANCE IN ANISOTROPIC SOLVENTS . . . . . . . . 63 . . . . . . 66 * . . . . . 68 . . . . 75 Meldola medal lecture G. R. Luckhurst, B.Sc., Ph.D. , . . . * . Dept of Chemistry, Tbe University, Southampton SO9 5NH * . Liquid crystals The anisotropic spectrum . . . . . . . . Examples . . .. . . . . .. . . The sign of the isotropic coupling constant, 68 The sign of the spin density, 71 Radical geometry, 72 The g-factor shift . . Radicals with more than one unpaired electron Conclusion . . Acknowledgments . . . . .. . . 74 * . . I . . .. . . 83 . . . . 83 .. . . .. . . .. . . . . . . . . . . . . . . 1 . References . . . . . . . . . . . . . . .. . . 83 Any filled molecular or atomic orbital contains two electrons which have exactly the same energy, but opposite spins.When there is only one electron in an orbital, this electron can be in either of the two spin states which are equal in energy. States of equal energy are said to be degenerate. In the presence of a magnetic field, the degeneracy of the two spin states of an unpaired electron is removed-the state in which the electron spin is anti- parallel to the applied magnetic field decreases in energy and becomes favoured over the state in which the electron spin is parallel to the field. The simplest electron resonance experiment consists of inducing transitions between these two spin levels, the resulting absorption of energy being observed as a single line in the spectrum. When the unpaired electron interacts with nuclei which also possess spin the electron resonance spectrum contains more than one line.This splitting of the single line is produced by smaller magnetic fields generated by the nuclei either adding to or subtracting from the applied field. If there is just one nucleus, with spin I, then the spectrum has 21 4- 1 equally spaced lines with the same intensity. The spacing between adjacent lines is called the nuclear hyperfine coupling constant. There are two mechanisms responsible for the electron-nuclear coupling. The first is the Fermi-contact term which occurs when the electron has a finite probability of being found at the nucleus. This is only true for electrons in orbitals with some s-character and the interaction is therefore isotropic- the same in all directions.The dipolar interaction between the electron and nuclear spins is responsible for the other coupling which is anisotropic- direction dependent. The Fermi-contact term has no classical analogue while the dipolar coupling is comparable to the interaction between two bar magnets. 1 Measurement of both components of the interaction is important because it leads to the determination of certain features of the molecular structure. Luckhurst 61 The isotropic coupling constant gives the s-character of the molecular orbital containing the unpaired electron, while the anisotropic hyperfine coupling is determined by the average electron-nuclear separation. Provided the radical can be incorporated in a crystalline lattice both components of the hyperfine coupling may be determined.Experimentally the crystal is mounted in the spectrometer and the spacing between the hyperfine lines measured as a function of the relative orientation of the crystal and the magnetic field. But for the majority of organic radicals, it is difficult to grow single crystals which are magnetically dilute and therefore they have only been studied in fluid solution. In a typical solvent, such as benzene, the dipolar or anisotropic hyperfine splitting is averaged to zero by the rapid rotational diffusion of the solute. Analysis of the line positions in a solution electron resonance spectrum yields only the isotropic coupling constants. Indeed, even the sign of this coupling cannot be determined directly. Solution studies therefore result in a consider- able loss of structural information, especially as other magnetic interactions within a radical are also anisotropic.When the molecular motion is slow, details of the anisotropic interactions can be gleaned from the widths of the hyperfine lines.2~3 This technique is not generally applicable and a method of obtaining the anisotropic interactions from the line positions is desirable. A liquid differs from a crystal in two important respects. In a crystal the molecular motion is usually completely quenched, and the molecules are macroscopically ordered. In a liquid the motion is rapid and there is no molecular ordering. Clearly, a series of intermediate states exists between these two extremes.For example, in a polycrystalline sample there is no molecular motion to average the anisotropic interactions, but the molecules are still distributed isotropically. As a result the lines in the electron resonance spectrum are very broad and the components of the hyperfine interaction can only be determined when they are large.4 The technique is not generally applicable. Another state is obtained by retaining the rapid molecular motion which is responsible for the narrow hyperfine lines and removing the isotropic character of the motion. The result of partially aligning the radical is to retain some dependence of the line positions on the anisotropic interactions. An early study of a partially aligned radical in a liquid was made by Ohnishi and McConnell.5 They measured the spectrum of the chloroproma- zine cation bound to DNA while flowing the solution through a narrow tube.The spectrum was found to depend on the orientation of the tube with respect to the magnetic field because the large size of DNA, together with viscous forces, results in partial alignment of the radical. Although the experiment is intriguing the technique is of limited applicability. The problem of partial alignment is encountered in nuclear magnetic reson- ance, where the anisotropic interactions include the nuclear dipolar coupling and the quadrupole coupling.6 If the molecule possesses an electric dipole moment then it can be aligned in an electric field. The extent of the alignment is extremely small, even with the maximum fields attainable, and it is only because the anisotropic interactions are enormous in comparison with their isotropic components that the effects of alignment can be discerned in the ~pectrum.~~ In electron resonance the anisotropic interactions are comparable R.I.C. Reviews 62 to their isotropic components and the alignment resulting from the applica- tion of an electric field would be quite incapable of altering the positions of the lines in the electron resonance spectrum. Fortunately, the use of liquid- crystalline solventsg has provided a more satisfactory and general solution to the problem of aligning solute molecules in magnetic resonance spectroscopy.lOJ1 LIQUID CRYSTALS The tendency for the planes of benzene molecules to lie parallel to one another in the bulk fluid only extends over several molecules.This situation occurs in the majority of liquids and can be described by saying that the local order or angular correlation is small. There is a class of compounds, known as liquid crystals, in which the angular correlation extends over many thousands of molecules. Liquid crystals are usually solid at room temperature and on heating they melt sharply to give the liquid-crystalline phase. Further increase in the temperature reduces the degree of local order until there is a second first-order phase transition to an isotropic fluid. There are three distinct types of thermotropic liquid crystal each characterized by a different mole- cular arrangement in the liquid-crystalline mesophase. In certain cases a compound may adopt more than one molecular arrangement before finally passing to the isotropic phase.The order-disorder phase transition in liquid-crystalline systems is a result of the anisotropy in the intermolecular potential function.l2 Indeed the temperature at which the transition to the isotropic phase occurs increases directly with this anisotropy. Any compound whose molecules deviate from spherical symmetry should exhibit liquid-crystalline behaviour. Obviously, most compounds are not liquid crystals; this is because, on cooling, the isotropic fluid freezes before the transition to the ordered fluid can be reached. Of the three liquid-crystalline forms the nematic mesophase is most commonly used as a solvent in magnetic resonance experiments.Compounds which give a nematic mesophase have essentially rod-like molecules and hence a highly anisotropic intermolecular potential. For example, 4,4'-dimethoxy- azoxybenzene is a yellow solid which melts at 118 "C and has a nematic- isotropic transition point at 135 "C. In the nematic mesophase, between 118 "C and 135 "C, the long axes of the molecules tend to be arranged parallel to one another as in Fig. 1. Even though the mesophase is highly ordered the high angular correlation in a given region has only a transitory existence and the rate of molecular motion is comparable to that in a normal liquid. Providing no constraint is applied to the mesophase the regions of high local order are randomly oriented with respect to one another, and the system is still isotropic in the macro- scopic sense. Application of a magnetic field greater than about one kilogauss13 aligns the molecules with their long axes parallel to the direction of the magnetic field.This ordering is a result of the anisotropy in the magnetic susceptibility, enhanced by the high local order, interacting with the magnetic field. A magnetic field is, of course, an integral part of a magnetic resonance experiment, and it is natural to use this to align the liquid-crystalline solvent. Luckhurst 63 5 Fig. I (above). The molecular arrange- ment in a nematic mesophase, e.g. 4,4’- dimethoxyazoxybenzene. Fig. 2 (right). Possible local molecular arrangements in a smectic mesophase, e.g.ethyl 4-azoxy- benzoate. This procedure has slight limitations because the direction of the alignment cannot be varied. However, electric fields can also align the molecules in the mesophase with their long axes parallel to the electric field. When the two fields are in competition an electric field of about 5 kV cm-1 is sufficient to overcome the magnetic field of 3 kG typically found in an X-band electron resonance spectrometer.14J5 The use of an electric field to align the sample does not have the inflexibility of a magnetic field alone. When the attraction between the sides of neighbouring molecules is large the high degree of local order extends in two dimensions;g two possible molecular arrangements are shown in Fig.2. The resulting mesophase is called smectic and is again formed by rod-like molecules. The crystal-Iike structure of the smectic mesophase implies a high viscosity and the molecular motion is very inhibited. As a result, molecules in the mesophase are not aligned by magnetic fields, although an ordered state is said to be obtained on cooling the isotropic phase below the smectic-isotropic transition point in the presence of a magnetic field.16J7 Alternatively, if the compound, for example 4,4’-di-n-heptyloxyazoxybenzene, passes through a nematic phase before reaching the smectic phase, the nematic phase can be aligned by a magnetic field and this alignment is frozen into the sample on passing into the smectic mesophase.18 The technique is valuable because the direction of R.I.C.Reviews 64 Me Me I Me Me 0 Pr \C-0 // s Fig. 3. The helical structure of the cholesteric mesophase, e.g. cholesteryl propionate. alignment, with respect to the magnetic field, can then be altered simply by rotating the sample tube. The third type of thermotropic liquid crystal, known as cholesteric, is formed by esters of cholesterol but not by cholesterol itself, which does not exhi bit any liquid-crystalline properties. The molecules in the cholesteric phase are arranged with their long axes parallel as in a nematic mesophase. However, on passing from one molecular layer to the next there is a contin- uous and constant change in the direction of alignment.The resulting helical arrangement (Fig. 3) is responsible for the unusual and technologically important optical properties of the cholesteric mesophase. The factors respon- sible for the helix are the asymmetric centres in the cholesteryl esters. Indeed, if the nematogen has an asymmetric centre, both the d- and the I-enantio- morphs are found to be cholesteric,lg while the racemate is nematic. The helical arrangement in the ordered regions reduces the anisotropy in the diamagnet- Luckhurst 65 50 ism and so a magnetic field of several kilogauss is unable to align the choles- teric mesophase. However, at high magnetic fields the helix is unwound to form a nematic mesophase which is then aligned. If two cholesteric liquid crystals with opposite rotations are combined to form an optically inactive mixture, then this is nematic and aligned by a magnetic field.20 Such mixtures have been used to orient solute molecules in nuclear magnetic resonance, but not in electron resonance.The addition of a solute to a nematic liquid crystal must affect the properties of the mesophase. In fact, the addition depresses the nematic-isotropic transition point, often below the melting point because the entropy changes at the two transition points are quite different. Although the range of the mesophase may be reduced by the presence of the solute the ordering proper- ties of the solution are comparable to those of the pure solvent.21 The solute will also be aligned on application of a magnetic field, not because of direct interaction of its anisotropic diamagnetic susceptibility with the magnetic field but because of interaction with the anisotropic potential generated by the macroscopically ordered solvent.The sense of the solute-solvent inter- action can often be inferred from the shape of the solute molecule and its preferred orientation can then be deduced. For example, a rod-like solvent will orient a planar solute with its plane parallel to the long axis of the rod, and hence to the direction of the magnetic field. THE ANISOTROPIC SPECTRUM The electron resonance spectrum of a partially aligned radical, which will be called the anisotropic spectrum, can only be interpreted fully if the spin hamiltonian is known. The most elegant derivation of this spin hamiltonian is obtained by writing the hamiltonian in terms of irreducible tensor operators and transforming it under rotation with Wigner rotation r n a t r i ~ e s .~ ~ ~ ~ The resulting equations can be rationalized using arguments which emphasize the physics of the problem. Consideration of a specific example simplifies the arguments still further. The electron resonance spectrum of di-t-butylnitroxide, whose structure is shown in Fig. 4, consists of three equally-intense lines caused by the interaction of the unpaired electron with the nitrogen nucleus (I = 1). Provided the radical tumbles rapidly in solution the separation between adjacent hyperfine lines is the isotropic nitrogen coupling constant a ( N ) .If the radical is fixed in a crystal with the magnetic field along the 3 axis, shown in Fig. 4, the spacing between the lines is now the sum of the isotropic splitting and the anisotropic coupfing, splitting is found to be a ( N ) plus A’(:) and an identical spacing of a ( N ) + A’\:) appropriate for this axis. Similarly, when the field is along axis 2 the is found when the field is along axis I (i.e. A’(;) = A’(:)). Such a situation is R. I. C. Reviews 66 described by saying that the anisotropic hyperfine interaction has cylindrical symmetry about axis 3. In a nematic mesophase di-t-butylnitroxide will tend to be aligned with the 1-2 molecular plane parallel to the direction of the would be a ( N ) + A'(:). However, the molecular motion prevents the align- magnetic field.The anisotropic coupling constant for complete alignment ment from ever reaching completion and the observed spacing is a(N) plus some fraction of the anisotropic coupling A'(:). The value of this weighting fraction is readily determined. If the magnetic field makes an arbitrary angle with the molecular axis system, the spacing between the hyperfine lines is where l3 is the cosine of the angle between the symmetry axis 3 and the magnetic field. When the radical moves rapidly from one orientation to another the observed splitting is obtained by averaging equation 1 over all orienta- - tions. In a nematic mesophase the anisotropic nitrogen coupling constant, a ( N ) , is where the ensemble average is denoted by the bar.It is convenient to denote the weighting fraction by the symbol 03, where 3 occurs twice in the subscript because it is found in both direction cosines, I,. Although the average p3 is a measure of the alignment of the 3 molecular axis with respect to the mag- netic field, @,, is a more logical indication of the degree of alignment. For example, when the motion is isotropic all values of l3 are equally probable and so 03, is zero. If axis 3 is parallel to the magnetic field 033 is unity. The values of Oil and 02, are given by expressions analogous to that for C33. It is also possible to define quantities which involve the direction cosines of different axes and in general The 0 values can be arranged in a square array which, not unexpectedly, is called the ordering matrix.24 The properties of the direction cosines mean that the sum of the diagonal elements of the matrix (its trace) is zero.The ordering matrix notation may be included in equation 2 to give This can then be written in a more symmetric form25 by using the traceless property of the anisotropic hyperfine interaction, Luckhurst 67 Fig. 5. The structure and axis system for d i pheny Initroxide. Equation 5 has been derived for a cylindrically symmetric hyperfine tensor. In general the shift in the coupling constant 6 - a is given by (7) i,i 6a = 3 C Aij 8ij where i, j , . . . represent molecular axes.25 When the radical is cylindrically symmetric, as for example perinaphthenyl, the alignment of any axes set in the molecular plane will be equal.The elements of the ordering matrix el1 and 8,, will also be equal and because the matrix is traceless 011 == 822 = - 83312 (8) Substitution of this result into equation 7 gives an equation of the same form as 5, (9) EXAMPLES 6a = A& 833 The coupling constant shift, 6a, is obtained experimentally by measuring the spectrum of the radical dissolved in the liquid crystal both above and below the nematic-isotropic transition point. As we shall see this shift can then be used to provide information about the anisotropic hyperfine ten~or.269~7 Alternatively, if the components of A’ are known then the elements of the ordering matrix can be determined from the shift. This is important because the form of the anisotropic solute-solvent intermolecular poten- tia1,28>’9 which determines the degree of alignment, can be obtained from 0.Indeed, it is also possible to investigate the properties of the pure liquid crystal given the ordering matrix for the solute.29y30 We now illustrate the structural information which can be obtained from the coupling constant shifts with three examples. Tlie sign of the isotropic coupling constant The first example is diphenylnitroxide, whose structure and molecular axis system are shown in Fig. 5. The isotropic spectrum of the radical dissolved in 4,4’-dimethoxyazoxybenzene has three groups of lines : the spacing between these is the nitrogen coupling constant of 28.17 MHz. Below 135°C the solvent exists in its nematic mesophase and there is a dramatic change23 in the electron resonance spectrum (Fig.6). The nitrogen coupling constant decreases to 18.18 MHz, demonstrating quite clearly the partial alignment of the radical. The anisotropic spectrum also exhibits a linewidth variation, as shown by the digerent heights of the first derivative line shapes illustrated R.I.C. Reviews 68 10 gauss Fig. 6. The isotropic and anisotropic electron resonance spectra of diphenylnitroxide dissolved in 4,4'-d imet hoxyazoxybenzene. in Fig. 6. The differences in the linewidth contain important information concerning the dynamic structure of the mesophase, but we shall not explore this aspect here.31~32 Although the separation between the hyperfine lines gives the magnitude of the nitrogen coupling constant it does not yield its sign.This information can be obtained from the nitrogen shift adN) but Luckhurst 69 6 Fig. 7. The axis system for the X-Y fragment. Y before this can be done we must look into the theoretical expressions for the anisotropic hyperfine tensor. We shall only discuss n-radicals in this article and expressions for the anisotropic hyperfine tensors in such systems have been derived by McConnell and Strathdee.33 They calculated the hyperfine tensor resulting from the dipolar interaction between a nucleus Y and an electron contained in a Slater 2pz orbital on a nucleus X. The axis system for the fragment X-Y is given in Fig. 7. The principal components of the tensor are where p is the spin density on atom X.The quantities P and Q are related to the internuclear separation Y and the effective nuclear charge Z for atom X by In these equations yy is the magnetogyric ratio for nucleus Y and a is Zr/1.058. These results can also be used to estimate the hyperfine tensor for Y when the electron is in a 2pz orbital on Y by taking the limit of equation 10 as r tends to zero. The resulting components of the tensor are The spin density on the nitrogen atom in diphenylnitroxide is predicted to be both large and positive by all molecular orbital calculations. According to equation 13 the nitrogen hyperfine tensor will approximate closely to cylindrical symmetry about the’3 axis normal to the molecular plane and so the nitrogen shift will be given by equation 5: The tensor component A’$;) is positive because of the positive spin density on the nitrogen and so to calculate the sign of the shift we need to know the sign of 033.Since diphenylnitroxide is essentially planar the 3 axis will tend R.I.C. Reviews 70 to be aligned perpendicular to the direction of the magnetic field. The average value of 1; will tend to be small, O,, will be negative, and the nitrogen shift constant the must therefore is found be to negative. be larger Experimentally than the anisotropic isotropic coupling; nitrogen ldN)I > coupling v[. Since both splittings have the same sign the experimental shift, a ( N ) - a(N) can only be negative if the isotropic splitting is positive.The liquid crystal experiment therefore confirms the theoretical prediction of a positive nitrogen coupling constant. Before leaving this example it is appropriate to comment on the assertion that the isotropic and anisotropic coupling constants have the same sign. The experimental values of a and are usually found to be comparable and in order for the coupling to have changed sign the shift would need to be twice the isotropic splitting. Theoretically the shift is C A;j oij and even for a i, i completely aligned radical this sum is unlikely to be 2a. For the remainder of the article the coupling constant is assumed to retain its sign on partial alignment. The sign of the spin density The isotropic proton coupling constants in an aromatic radical are propor- tional to the mpin density on the adjacent carbon atom: (14) The proportionality constant, Q, in the McConnell relationship, which provides a useful method for testing r-molecular wave-functions, is equal to about -80 MHz for a neutral radical.Because molecular orbital theory can calculate the absolute magnitude of the spin density it is important to be able to measure this quantity in order to provide a complete test. However, as the absolute magnitudes of the coupling constants are not obtained from the isotropic spectrum, equation 14 cannot be used to determine the signs of the spin densities. As we shall see measurement of the anisotropic spectrum can lead to a sign determination. a(i) = Qp(t) The first such determination was made for the perinaphthenyl radical whose spectrum was measured above and below the nematic-isotropic transition point of 4,4’-dimethoxyazoxybenzene.35 The structure of the radical and its axis system are shown in Fig.8. Both spectra are readily analysed in terms of a large septet splitting from the six equivalent /3-protons, and a smaller quartet splitting from the three equivalent a-protons. The isotropic coupling constants are 5.10 MHz for the a-protons and 17.6 MHz for the protons whereas the corresponding anisotropic values are 5.75 MHz and 17.6 MHz. Only Fig. 8. The structure of the perinaphthenyl radical. P a Luckhurst 71 the a-protons exhibit a significant shift, 6 a ( a ) , on alignment. The ordering matrix for perinaphthenyl must be cylindrically symmetric because of the circular shape of the radical.The proton shifts are therefore given by equation 9 6a = A;, O,, where 3, the symmetry axis, is perpendicular to the molecular plane. The sign determination is based on calculating values for both sides of this equation. On the one hand 6a can be measured from the isotropic and anisotropic spectra and on the other hand we can calculate the hyperfine tensor A’ and estimate the degree of alignment 8. In order to determine the sign of the spin density both sides of the equation are calculated for either choice of sign. We then seek to eliminate one of these by seeing if the signs obtained for 6a and Aj.3 03, are opposite. Clearly, if both 6a and A;, 0 3 3 have the same dependence on the spin density this procedure must fail.Fortunately, the sign of the isotropic proton coupling constant, and hence 6a, depends only on the spin density at the adjacent carbon atom, whereas the anisotropic proton hyperfine tensor is determined by a sum of contri- butions from all the spin density in the radical. In practice the technique works in the following way. If p(a) is positive the isotropic coupling constant for the a-protons will be negative, and so the shift will also be negative since ~p[> la(a)l. The /?-proton splitting of 17.6 MHz ensures that the spin density p(P) must be positive. Calculation of the hyperfine tensor element, A;:), with positive spin density at both a and /3 positions gives a value of -2.76 MHz.As for diphenylnitroxide the element 0,, of the ordering matrix will be negative and so the term A’s“) 03, is found to be positive. This result is inconsistent with the sign of 8a@) and so the a spin density cannot be positive. Clearly the choice of a negative sign should be consistent with the a shift. If p ( ~ ) is negative then the isotropic splitting is positive and the shift is also positive. The tensor element A’%\ calculated with p(a) negative and p(B) positive is still negative, although equal to -2.17 MHz. The term A’$) 8,, is positive and agrees with the sign of 8 a ( a ) . The a spin density is quite clearly negative in agreement with self-consistent-field molecular-orbital calcula- tions.36 in 4,4‘-dimethoxyazoxybenzene,23 Radical geometry The use of liquid-crystalline solvents in nuclear magnetic resonance has been valuable for measuring the bond lengths and bond angles of the solute.6,7 The method is capable of high accuracy because the anisotropic nuclear dipolar coupling can be calculated exactly from the internuclear separation.In principle a similar analysis should be possible in electron resonance. In practice the inaccuracy of the 2pz wave-function used in calculating the hyper- fine tensor limits the use of the coupling constant shift in determining the geometry of the radical. However, an indication of the potential of liquid- crystal experiments is given by a study of the triphenylmethyl radical dissolved As usual the spectrum was measured in both phases but in this case the R.I . C. Reviews 72 Table. Temperature variation of the coupling constants in the triphenyl- methyl radical Coupling constant (MHz) Temperature ("C) meta ortho para I25 (isotropic) I20 3.187 7.18 7.78 3.310 7.10 7.66 3.335 7.08 7.64 3.372 7.08 7.64 3.391 7.09 7.65 3.422 7.11 7.67 3.488 7.13 7.69 I I 5 I10 I05 I 00 95 anisotropic spectrum and hence the proton shifts were determined as a func- tion of temperature. The radical possesses cylindrical symmetry so the anisotropic proton coupling constants are given by equation 9 : ii = a -+ A;, O,, where axis 3 is perpendicular to the molecular plane. The anisotropic coupling will either increase or decrease with decreasing temperature, depending on the relative signs of a and A;, 03,, because the degree of alignment increases at lower temperatures.For the majority of solutes the temperature dependence of 0 is found to be in qualitative but not quantitative agreement with theory.29 The results for triphenylmethyl, given in the Table, are not in complete agreement with those of other solutes. The anisotropic meta splitting does increase with decreasing temperature, whereas both the ortho and para splittings first decrease, pass through a minimum and then increase. The structure of triphenylmethyl is thought to be like a symmetrically pitched propeller with the phenyl groups inclined at an angle of about 25" to the horizontal plane. In the nematic mesophase the radical is subjected to a highly anisotropic potential whose magnitude increases with decreasing temperat~re.~g As the anisotropy increases we might expect the radical to be forced into a more planar configuration.This squashing of the molecule is thought to be responsible for the unusual temperature dependence of the anisotropic proton coupling constants. Small distortions of tetramethylsilane and neopentane dissolved in 4,4'-di-n-hexyloxyazoxybenzene have already been observed by nuclear magnetic resonance. The increasing planarity of the radical will effect a" by changing both the isotropic coupling a and the tensor component A;,. According to molecular orbital theory the spin density at the ortho and para positions will increase while that at the meta position becomes more negative as the radical tends to a planar conformation.23 The three isotropic coupling constants will therefore increase in magnitude. The change in both the spin distribution and the geometry will also be reflected in Aj303, because of modifications in A j 3 .Although quantitative calculation of the hyperfine tensor is unreliable22J3 i t is certain that 4, is negative for all three protons. Since the element 03, of the ordering matrix is also negative the anisotropic contribution to ii is Luckhurst 73 positive. In contrast the isotropic contribution a is negative for the ortho and para protons but positive for the meta. Thus, the anisotropic meta coupling constant is a sum of two positive terms both of which should increase with decreasing temperature.On the other hand, a' for the ortho and para protons is a sum of two terms with opposite signs. As the magnitude of both terms increase with decreasing temperature, the anisotropic coupling constant might well pass through a minimum. The observed trends are therefore in accord with a model based on molecular deformation by the anisotropic potential of the liquid crystal. THE G-FACTOR SHIFT The value of the magnetic field HO at which the centre of an electron resonance spectrum occurs, is determined by the g-factor for the radical. If the frequency of the microwaves produced by the klystron in the spectrometer is V , gP (16) hv ffo = -- where h is Planck's constant, and ,8 is the electron Bohr magneton.For a free electron which has no orbital angular momentum the g-factor is 2.0023 19 and isotropic. The orbital angular momentum is largely quenched in the majority of organic radicals, although some is retained because of spin-orbit coupling. As a result the g-factor exhibits a small departure from the free- spin value. Furthermore, different molecular axes in the radical have different g-factors. The components of this anisotropic g-tensor constitute another source of structural information, but, like the hyperfine tensor, cannot usually be determined for organic radicals. The use of liquid-crystalline solvents changes this s i t ~ a t i o n . ~ ~ ~ 2 3 ~ ~ 8 W e n radicals are aligned, the centre of the anisotropic spectrum will depend on the total g-tensor. The value of g determined from equation 15 is now the average g which is related to the components of the anisotropic tensor g' by g = g + C g,>Bij i,i This relationship is analogous to the equation for the hyperfine shift and similarly reduces to Sg = g;3 033 (17) when either the g-tensor or the ordering matrix possesses cylindrical symmetry about the 3 axis.The sign of the g-shift can be determined directly from experi- ment because the g-factor is always positive. Provided the components of the g-tensor are known then the sign of the ordering matrix can be derived without making any assumptions. This derivation is important because intuitive ideas about the direction of the preferred orientation for the solute can be tested.g-Shifts have been determined for a number of radicals dissolved in various liquid crystals. In general, analysis of the shifts yields information about the anisotropic g-tensor and often all the components of the tensor can be determined. For example,22 the isotropic g-factor of perinaphthenyl dissolved R. I.C. Reviews 74 in 4,4’-dimethoxyazoxybenzene is 2.00261 and at 100°C the g-shift is 10.9 x 10-5. Both g and 0 are cylindrically symmetric because of the circular shape of the radical. The component gj, of the anisotropic g-tensor could be determined from the g-shift if the degree of alignment O,, was known. Although 03$ can be obtained from the shift for the a-protons the possible inaccuracies in the theoretical proton hyperfine tensor makes the procedure unreliable.Fortunately the 13C hyperfine tensors can be calculated more precisely; measurement of the 13C shifts for perinaphthenyl22 gives a value for 03, of -0.31 at 100°C. The value of g;3 is therefore -35.3 + 10-5 and the component of the g-tensor perpendicular to the molecular plane is 2.00226. Theoretically the in-plane components of the g-tensor are expected to vary from one radical to the next, whereas the component perpendicular to the plane is predicted to be independent of the radical and to have the free-spin val~e.~9740 Clearly the result for perinaphthenyl is in good but not complete agreement with theory. RADICALS WITH MORE THAN ONE UNPAIRED ELECTRON Until now we have been concerned with radicals possessing a single unpaired electron. The presence of a second electron introduces another important magnetic interaction.As well as coupling with the spins of magnetic nuclei the two electron spins can couple with each 0ther.l The discussion of this electron-electron interaction is simplified when the biradical does not contain magnetic nuclei. The interaction has both an isotropic and anisotropic component, like most forms of magnetic coupling. The isotropic coupling, known as the exchange interaction J , separates the four electron-spin orienta- tions into three degenerate levels: which form the triplet state and the singlet state The triplet and singlet states are separated in energy by an amount J. When J is positive the ground state of the system is a diamagnetic singlet, but if J is negative the ground state is a paramagnetic triplet.We shall ignore the singlet state for the moment. The degeneracy of the triplet levels is removed by a magnetic field and electron resonance transitions can be induced between the spin levels.1 The two allowed transitions (Fig. 9) occur at the same field value, and so the electron resonance spectrum contains a single line. In an isotropic solvent the only interactions not averaged to zero by rapid tumbling are the isotropic couplings and so the spectrum of a biradical is identical to that of the com- parable monoradical. In organic biradicals the anisotropic electron-electron coupling results from the dipolar interaction between the electron spins.This anisotropic interaction is known as the zero-field splitting tensor D Luckhurs f 75 Fig. 9. The energy levels and allowed electron resonance transitions for a triplet state dissolved in the isotropic and anisotropic phase. because it partially removes the degeneracy of the triplet levels even in zero magnetic field. For example, when a biradical is aligned in a nematic meso- phase the relative energies of the triplet levels are: and The effect of a magnetic field on these new energy levels is also shown in Fig. 9. The presence of the unaveraged zero-field splitting has removed the degeneracy of the two allowed transitions and the anisotropic spectrum contains two lines separated by 4 C Dgj Oij.The quite different forms of the i , i anisotropic spectra for monoradicals and biradicals should provide a simple means of distinguishing between the two species. The biradical bi~galvanoxyl~~1 whose structure is shown in Fig. 10, is an excellent example. Bisgalvanoxyl is prepared by oxidation of the parent diphenol using lead dioxide ; the oxidation rarely proceeds to completion and the product is contaminated with the monoradical. The isotropic spec- trum (Fig. 10) of the product dissolved in 4,4’-dimethoxyazoxybenzene at 139°C consists of an intense signal from the monoradical superimposed on a barely discernible broad line from the biradical. However, on alignment in R. I. C. Reviews 76 c 0” Vk& 0 T= 109°C m loo gauss I f( Fig.10. The isotropic and anisotropic spectra of bisgalvanoxyl dissolved in 4,4’-dimethoxy- azoxybenzene. the nematic mesophase, the monoradical spectrum is essentially unaltered whereas the biradical line is split into a doublet of separation 395 M H z . ~ ~ The experiment clearly demonstrates the presence of two unpaired electrons in one of the oxidation products. The description of the electron resonance spectra of biradicals is more complicated when they contain magnetic nuclei. The nitroxide biradicals of the form: are well characterized43 and we shall limit the discussion to these species. The unpaired electrons interact predominantly with the nitrogen nuclear spins, and the system may be regarded as containing two electrons and just two nuclei.Because the electrons are coupled to the nuclei as well as to each other, there is a competition for the spin of a particular electron and the form of the isotropic electron resonance spectrum depends critically on the result of this competition. When the electrons are coupled more strongly to each other than to either Luckhurst 77 * 10 gauss gl utarate. Fig. I I . The isotropic and anisotropic spectra of bis(2,2,6,6,-tetramethylpiperidin-4-yloxyl-l) nitrogen nucleus, that is when J>> a ( N ) , the spectrum contains five lines, J : separated > a ( N ) the by spectrum a(N)/2 with is characteristic intensities 1 of : 2 two 3 : equivalent 2 : 1. In other nitrogen words, nuclei. if The only effect of alignment in a nematic mesophase is to split each of the five lines into a doublet separated by 4 C Dij oij, as before.44 Bis(2,2,6,6- i,j L-/ tetrarnethylpiperidin-4-yloxyl-l)glutarate, that iswithX equal to OC(CH&CO, provides an example of this behaviour.The isotropic spectrum at 142°C (Fig. 11) consists of five equally spaced lines. The heights of the lines are not in the expected ratio of 1 : 2 : 3 : 2 : 1 because the intramolecular motion modulates the value of J which produces an alternating linewidth effect.45 On alignment in the nematic mesophase of 4,4'-dimethoxyazoxybenzene at 98"C, each line is split into a doubleP4 but, because of overlap, only nine of the expected 10 lines can be seen (Fig. 11). At the other extreme, when J < a(N), each electron is unaware of the other's presence in the biradical.The electron resonance spectrum is therefore indistinguishable from that of the appropriate monoradical and contains three lines separated by a ( N ) . For example, the isotropic spectrum (at 141 "C) of the terephthalate (X = OC-(--\--CO) has only three lines, and gives 78 R. I . C. Reviews & 10 gauss Fig. 12. The spectra of bis(2,2,6,6-tetrarnethylpiperidin-Cyloxyl- I ) terephthalate dissolved in the isotropic and nematic mesophase. no indication that the radical has two unpaired electrons;44 but in the nematic mesophase of 4,4'-dimethoxyazoxybenzene at 95°C each of the three lines is apparently split into a doublet44 (Fig. 12). The successful synthesis of a biradical is demonstrated by this additional splitting in the anisotropic spectrum.In principle the anisotropic spectrum should be more complex but the large linewidth results in a deceptively simple spectrum.46 i,i The isotropic electron resonance spectrum for an intermediate coupling scheme is complicated because the triplet and singlet levels are mixed by the hyperfine term. However, the complexity is valuable since electron-electron coupling J can be obtained by analysing the spectrum. In contrast analysis of the simpler spectra for the two extremes only allows a h i t to be placed on J . The spectral analysis does not yield the sign of J , but this can be deter- mined from the anisotropic spectrum. In the nematic mesophase certain lines are split into doublets separated by 4 C Dij oij whereas the spacing between other lines now depends on both J and C Dzj 023.The signs of the i , j components of D can be obtained from a point-charge model for the electrons and the sign of 8 can be inferred from the shape of the biradical. Thus the sign of the sum C Dij Oij can be calculated and that of J obtained. For the carbonate i,i Luckhurst 79 (X = CO) J is positive and so its ground state is a singlet, but the separation from the triplet is small and the latter is thermally pop~lated.47y4~ The use of liquid-crystalline solvents is important when the number of unpaired electrons in the paramagnetic species is not known with cer- taint~.*~$5O The formula for certain conjugated binitrones can be written with diamagnetic or paramagnetic structures, for example 0- -0 can be reformulated as the biradical:5* 0 I* 0 * I 0 The intensities of the electron resonance spectra increase with increasing temperature, suggesting that this triplet state is thermally accessible ;51 but the isotropic and anisotropic spectra in 4,4'-dimethoxyazoxybenzene (Fig.13) failed to confirm the existence of a biradical.50 The only effect on passing into the mesophase is to decrease the nitrogen splitting. The observation of the nitrogen shift of 2.8 MHz is important since it shows that the radical is aligned. Clearly, the absence of any splitting must be because the radical contains just one unpaired electron. The paramagnetic species is now thought to be the mononitroxide : OH formed from the binitrone by the abstraction of a hydrogen atom, most probably from the solvent.50 The chemist's ingenuity in linking any number of monoradicals together to form a polyradical makes it important to have a procedure for testing the success of the synthesis.The use of liquid-crystalline solvents in electron resonance spectroscopy provides such a technique. Although the form of the R.I.C. Reviews 80 - 5 gauss Fig. 13. The isotropic and anisotropic spectra of the radical derived from the binitrone. anisotropic spectrum of a polyradical is complicated by any magnetic nuclei certain lines will behave as if the nuclei were absent. If the polyradical contains n unpaired electrons, then on alignment the single line in the iso- tropic spectrum will be split into an n-line multiplet with a binomial distri- bution of intensities. This behaviour is completely analogous to the nuclear magnetic resonance experiment with a solute containing n equivalent nuclei of spin 4.6 The triradical: 52 illustrates the technique.The widths of the hyperfine lines for the triradical are rather large and in order to resolve the triplet splitting expected on align- Luckhurst 81 189°C of I M°C Fig. 14. The spectrum the triradical dissolved in the isotropic and aniso- tropic phase of C(p-meth- oxy benzy 1 id en e) am i noazo- benzene. ment, 4-(pmethoxybenzylidene) aminoazobenzene was used as solvent because it had a high nematic-isotropic transition point (180 0C).53 The isotropic spectrum exhibits the alternating linewidth effect 54 found in nitrox- ide biradicals but we need only be concerned with the three intense lines shown in Fig.14. In the anisotropic spectra these lines are each split into a triplet whose separation increases at lower temperatures because of the increased alignment. The paramagnetic species must contain three unpaired electrons. v 82 20 gauss R.1. C. Reviews CONCLUSION The use of isotropic solvents in electron resonance spectroscopy is restrictive because it results in the loss of all knowledge of any anisotropic interaction. This limitation is removed by using liquid-crystalline solvents which can partially align the solute.Analysis of the anisotropic spectrum can then yield the signs of coupling constants, spin densities and the exchange interaction in biradicals as well as the geometry of the radical, the magnitude of the g-tensor and the number of unpaired electrons in the radical. The way is now open for the investigation of anisotropic interactions, even in the most complicated radicals. There is, of course, another side to the story. The aniso- tropic couplings are already known for certain species and so the form of the anisotropic spectrum can be used to study the fascinating and important properties of the liquid-crystalline state. ACKNOWLEDGEMENTS My work has undoubtedly benefited from collaboration with my friends Drs David Chen, Howard Falle, Andy Hudson, Janet Ockwell and Mr Peter James: it is a pleasure to acknowledge their collaboration. I owe my greatest debt to Professor Alan Carrington for his continuing inspiration and guidance.REFERENCES 1 A. Carrington and A. D. McLachlan, Introduction to magnetic resonance. New York: Harper and Row, 1967. 2 A. Hudson, A. Carrington and G. R. Luckhurst, Proc. R. SOC., 1965, A284,582. 3 A. Hudson and G. R. Luckhurst, Chem. Rev., 1969, 69, 191. 4 P. W. Atkins and M. C. R. Symons, The structure of inorganic radicals. Amsterdam: Elsevier, 1967. 5 S. Ohnishi and H. M. McConnell, J. Am. chem. SOC., 1965,87,2293. 6 G. R. Luckhurst, Q. Rev. chem. SOC., 1968, 22, 179. 7 A. D. Buckingham and K. A. McLauchlan, Prog. NMR Spectrosc., 1967,2,64. 8 C.W. Hilbers and C. Maclean, Molec. Phys., 1969, 16, 275. 9 G. W. Gray, Molecular structure and the properties of liquid crystals. London: Academic, 1962. 10 A. Saupe and G. Englert, Phys. Rev. Lett., 1963, 11,462. 11 A. Carrington and G. R. Luckhurst, Molec. Phys., 1964,8,401. 12 W. Maier and A. Saupe, 2. Naturf., 1959,14a, 882. 13 W. Maier and A. Saupe, 2. phys. Chem., 1956,6,327. 14 D. H. Chen and G. R. Luckhurst, Molec. Phys., 1969,16,91. 15 C. F. Schwerdtfeger and P. Diehl, Molec. Phys., 1969,17,417. 16 G. Foex, Trans. Faraday SOC., 1933, 29, 958. 17 C. S. Yannoni, J. Am. chem. Soc., 1969,91,4611. 18 P. D. Francis and G. R. Luckhurst, Chem. Phys. Lett., 1969, 3, 213. 19 G. W. Gray, Molec. Crystals, 1969, 7, 127. 20 E. Sackmann, S. Meiboom and L. C. Snyder, J. Am. chem. SOC., 1967,89,5981. 21 D. H. Chen and G. R. Luckhurst, Trans. Faraday SOC., 1969, 65, 656. 22 S. H. Glarum and J. H. Marshall, J. chem. Phys., 1966, 44,2884. 23 H. R. Falle and G. R. Luckhurst, to be published. 26 G. R. Luckhurst, Molec. Phys., 1966, 11, 205. 50,258. 24 A. Saupe, 2. Naturf., 1964, 19a, 161. 25 G. R. Luckhurst, Molec. Crystals, 1967, 2, 363. 27 H. R. Falle, G. R. Luckhurst, A. Horsfield and M. Ballester, J. chem. Phys., 1969, 28 A. Saupe, Molec. Crystals, 1966, 1, 527. 29 D. H. Chen, P. G. James and G. R. Luckhurst, Molec. Crystals, 1969, 8, 71. 30 H. C. Longuet-Higgins and G. R. Luckhurst, Molec. Phys., 1964, 8, 613. 31 H. R. Falle and G. R. Luckhurst, Molec. Phys., 1967, 12,493. Luckhurst 83 32 S. H. Glarum and J. H. Marshall, J. chem. Phys., 1967,46,55. 33 H. M. McConnell and J. Strathdee, Molec. Phys., 1959, 2, 129. 34 H. M. McConnell, J. chem. Phys., 1956, 24, 764. 35 H. R. Falle and G. R. Luckhurst, Molec. Phys., 1966,11,299. 36 L. C. Snyder and T. Amos, J. chem. Phys., 1965,42,3670. 37 L. C. Snyder and S . Meiboom, J. chem. Phys., 1966,44,4057. 38 K. Mobius, H. Haustein and M. Plato, 2. Naturf., 1968, 23a, 1626. 39 G. G. Hall and A. Hardisson, Proc. R . Soc., 1964, A278, 129. 40 A. J. Stone, Molec. Phys., 1964, 7, 31 1. 41 E. A. Chandross, J. Am. chem. Soc., 1964, 86, 1263. 42 H. R. Falle and G. R. Luckhurst, unpublished results. 43 A. R. Forrester, J. M. Hay and R. H. Thomson, Organic chemistry of stable free radicals. London : Academic, 1968. 44 H. R. Falle, G. R. Luckhurst, H. Lemaire, Y . Marechal, A. Rassat and P. Rey, Molec. Phys., 1966,11,49. 45 G. R. Luckhurst, Molec. Phys., 1966, 10, 543. 46 H. Lemaire, A, Rassat, P. Rey and G. R. Luckhurst, Molec Phys., 1968, 14,441. 47 H. Lemaire, J. chim. Phys., 1967, 64, 559. 48 S. H. Glarum and J. H. Marshall, J . chem. Phys., 1967,47, 1374. 49 I. Agranat, M. Rabinovitz, H. R. Falle, G. R. Luckhurst and J. N. Ockwell, J. chem. SOC. (B), 1970, 294. 50 A. R. Forrester, R. H. Thomson and G. R. Luckhurst, J . chem. Soc, (B), 1968, 1311. 51 M. Colonna and P. Bruni, Gazz. chim. Ital., 1964, 94, 1448. 52 A. L. Butschatschenko, V. A. Golubev, M. B. Neiman and E. G. Rosantsev, Dokl. Akad. Nauk SSSR, 1965, 163, 1416. 53 G. R. Luckhurst and E. G. Rosantsev, Izv. Akad. Nauk SSR Ser. Khim., 1968, 8, 1708. 54 A. Hudson and G. R. Luckhurst, Molec. Phys., 1967,13,409. 84 R. I. C. Reviews