LINEWIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
with two or more solvent molecules, although I do not know if this structure can explain the kinetic data reasonably. The other possibility is that the ions may exist as an ion cluster with solvent moledes between ions rather than the ordinary one-to-one ion pair.
E. DE BOER(University of Nijmegen, Netherlands). Do you have an explanation for the reversal of the sign of the spin density at the Li nucleus for the Li-fluorenone ion-pair when you change the temperature? N. HIROTA.You have suggested that there are two different mechanisms to produce spin density at metal nucleus, one to produce positive spin density and the other to produce negative spin density, and that the sign depends on the relative importance of two mechanism [Rec. Trav. Chim., 84,609 (1965)l. I
139
think this is another example to support your suggestion. If the Li ion sits on the nodal plane of 2pn orbital of fluorenone, and the vibration of the Li ion perpendicular to this plane is small, the positive contribution is expected to be very small because of the small overlap integral between the 2 p orbital ~ of ketyl ion and the s orbital of the Li ion. Furthermore, if the Li ion separates more from the negative ion as the temperature dependence of CIS splitting seems to indicate, this contribution would quickly decrease a t lower temperatures. The negative contribution, on the other hand, may not be so sensitive to the position and the separation aa the positive contribution and may survive at lower temperatures. However, further investigations seem to be necessary in order to understand fully the reversal of the sign as well as the mechanisms to produce alkali metal splittings.
Line Widths and Frequency Shifts in Electron Spin Resonance Spectral
by George K. Fraenkei Departmat of Chemistry, Columbia University, New York, New York
lOOdY
(Received September dY, 1966)
A general account is given of the most important types of line-width variations which occur in the electron spin resonance spectra of dilute solutions of free radicals. The effects considered include modulations of the isotropic hyperfine splittings and of the anisotropic gtensor and electron-nuclear magnetic dipole interactions. The influence of fluctuating interactions on the positions of the hyperfine lines (frequency shifts) is also discussed. General formulas are obtained either from intuitive arguments or quoted without proof from the results of the relaxation-matrix theory. Several model systems are discussed, and a brief survey is included of applications to structural problems of chemical interest.
I. Introduction In the past few years, the widths of the hyperfine lines in the electron spin resonance spectra of free radicals has been the subject of considerable investigation. At first, the line-width variations which were observed appeared to represent relatively isolated examples, but as the work progressed, it became increasingly evident that many of the phenomena were quite general and of considerable importance. On the one hand, the proper interpretation of many spectra is only possible if t,he line-width effects are taken into account. More interestingly, line-width studies can be used to obtain information about structural problems. Recent work
has provided data, for example, about intramolecular and intermolecular motions, ion-pair and other radicalsolvent interactions, and n-electron spin-density distributions. Since the theoretical formulations2*3are quite general and, concomitantly, complex, a useful purpose would undoubtedly be served at this time by presenting in as simple a form as possible the salient and most frequently applicable aspects of the theory (1) This research was supported in part through National Science Foundation Grant No. NSF-GP-4188. (2) (a) D. Kivelson, J . C h a . P h ~ a .27, , 1087 (1957); (b) D. Kivelson, ibid., 33, 1094 (1960). (3) J. H. Freed and G. K. Fraenkel, ibid., 39, 326 (1963).
Volume 71,Number 1
January 1967
GEORGEK. FRAENKEL
140
along with illustrative examples and indications of the role of line-width studies in the investigation of problems of chemical interest. We shall be concerned with two different types of effects that influence the widths and separations (frequency shifts) of the hyperfine lines. One is the result of fluctuations in the isotropic hyperfine splittings that are induced by intramolecular motions or interactions between the radical and the solvent or nonradical solutes (section 11). A number of examples have been discovered in which these fluctuations are large and produce dramatic alterations in the esr spectra. Fortunately, some realistic situations exist in which the general and rigorous relaxation-matrix theory can be adequately approximated by simplified and intuitive arguments, and wherever possible this type of treatment is employed in the following discussion. Since the intuitive treatment is approximate and is inherently restricted to highly specialized models, the results of the more general theory are also given, but without proof. The other effects considered arise from the influence of molecular tumbling on anisotropic intramolecular interactions (section 111). We discuss only the two most frequently important of these, the anisotropic g-tensor and electron-nuclear magnetic dipole interactions. Intuitive arguments have not been developed for treating these latter effects, and we therefore quote without proof the results obtained from the general theory. The last section is devoted to a brief account of applications. The discussion is restricted to free radicals, i.e., molecules with a single unpaired electron, in not too highly viscous solutions. It is further assumed that the radical concentration is sufficiently low to permit the neglect of all intermolecular effects between radicals.
11. The Effects of Fluctuating Hyperfine Splittings I n this section, we investigate the contributions to the line widths and frequency shifts which result when the isotropic hyperfine splittings are functions of time, a(t). The time dependence can arise from intramolecular motions or interactions of the radical with the solvent or (nonradical) solutes. We assume first that the fluctuations are rapid compared to the magnitude of the frequency changes resulting from the variations in a(t); it can then be shown that the positions of the hyperfine lines in the esr spectrum are determined by ii = (a(t)), the average of a(t) over the fluctuating motions. The variations in the hyperfine splittings cause the instantaneous positions of the hyperfine lines to be distributed over a range of values and result in a contribution r((&z)2)to the line widths where r is a time The Journal
of
Phgsieal Chemiatrg
characteristic of the (inverse) rate of the fluctuations and ((6~)~)is the mean-square variation in the splittings. The discussion of the line widths and frequency shifts will first be carried out for a radical in which there is hyperfine interaction with only a single nucleus (section A), and then radicals with several nuclei are considered (section B). Although many more interesting phenomena arise when there are several nuclei, the singlenucleus treatment is of value because it allows the simplest possible introduction of the basic ideas. Most of the discussion is restricted to rapid motions, but slow processes are briefly touched upon. The quantitative conditions for fast and slow motions are r 2 < \re2((6a>2)]-1 and r ? [ l ~ ~ ) l & l ] -respectively, l, where y e is the magnetogyric ratio of the electron and is used to convert field units (gauss) to angular-frequency units (la1 = /Y,[B). For simplicity, the treatment will also be restricted to motions that are not too rapid; we will only consider. situations for which the characteristic time 7 is greater than the inverse of the resonance frequency YO of the esr experiment (the Larmor frequency), or more ~ wo = ( v 0 / 2 r ) . precisely those for which ( ~ ~>"71,)where This condition corresponds to the neglect of relaxationinduced transitions between the electron-spin states that are parallel and antiparallel to the external magnetic field, the so-called nonsecular eff ects2 The nonsecular terms are easily in~luded,2>~ but they are usually less important than those for which there are no electronspin transitions, called secular effects. A . Single Nucleus. I . Line- Width Variations. For concreteness, let us first consider a particularly simple example of a time-dependent variation in hyperfine splitting: suppose that the radical can exist in only two forms, A and B, and that it makes rapid jumps from one form to the other. A model such as this is a reasonable representation, for example, of the nitrobenzene negative-ion radical when it is prepared by reduction of nitrobenzene with potassium. The ionpair complex between the radical anion and the alkali metal cation can be represented by form A and the uncomplexed radical anion by form B. If form A could be isolated and were not undergoing transitions to form B, a nucleus of spin I with hyperfine splitting4a1 would 1 equal-intensity give rise to an esr spectrum of 21 lines with a constant interline sparing UI. Similarly, form B would give a spectrum of 21 1 lines with spacing UII. Each line in the two spectra corresponds to a particular value of the quantum number 7n specifying the field-direction component of the nuclear spin angular momentum.
+
+
(4) R e use the notation U I ,a11 for the hyperfine splittings instead of for convenience in later applications.
UA,U B
LINE WIDTHS ANI) FREQUENCY SHIFTS IN ESRSPECTRA
141
When the two forms are interchanging rapidly, an average of the two spectra is obtained which consists of 21 1 lines with B mean hyperfine splitting
pendent of m. The quantity Tz(m)is the transverse relaxation time for the line corresponding to the nuclear-spin quantum number m. The shape of each line is Lorentzian and given by (on an angular frequency basis)
+
6 =
+
~ A U I
~
B
~
I
I
(2.1)
where p A and p~ are the probabilities of finding forms A and B, respectively. This result for the mean hyperfine splitting follows from the general relaxationmatrix theory3 and also from the Bloch equations5 as modified to include the effects of exchanges6 I n the Bloch equation formulation, lines which appear a t (angular) frequencies wA(m) and wB(m) in the spectra of the two isolated forms coalesce, in the limit of fast exchange, to an average frequency (w(m))= pAwA(m)
+ PBWB(m)
= ~ A ( U I- g)2
(2.3a)
--
PAPB(UI
-
+
-
~ B ( ~ I I
~ I I ) ~
(2.3b)
The rates of the reactions also affect the widths, and if T A and T B are the lifetimes of forms A and B, respectively, the widths are3s6t9
+
T2-'(m) = y,2T((6a)z)m2 =
T2,0-1
- uId2m2+ T2,o-l
(2.4)
Y~~TPAPB(UI
where
+
(2.5) and T2,0-l, the contribution to the widths from other line-broadening mechanisms, is assumed to be inde7
=
TATB/(TA
TB)
=
+
(T2(m)/7r)(1 T22(m)[w - W(m)I2]-'
(2.6)
I n magnetic field units, the width is A(m) = [I ye/T2(m)]-'. This is the half-width at half-maximum intensity, and is related to the separation of the extrema in the first-derivative spectrum, 6(m),by 6(m) = (2/
(I Ye1 /2Bo)((W2)
(2.29)
and eq 2.19 is obtained. It will be noted that the second-order shifts for the two lines arising from the splitting by a single proton are the same, and the only effect of the shifts is to alter the position of the center of the spectrum. A correction may consequently have to be made in the determination of the g value.26 B. Several Nuclei. When the radical contains several nuclei which cause hyperfine splittings, a number of interesting (and complicating) effects take place. Let us first consider some illustrative examples. The singly protonated form of the p-benzosemiquinone anion radical (which is of course a neutral radical) can exist in two thermodynamically equivalent forms, A and B. 0.
PH
IQ:
0.
OH
A
B
b6 Q : \
I
The hyperfine splittings from the protons at positions 2 and 6 are, by ~ymmetry,~'equal to each other in both forms of the radical, but the values for form A may be different from those for form B. Thus a4A) = aa(A) = a1 and az(B) = ae(B) = U I I , and if the spindensity distribution depends on the position of the OH hydrogen atom, aI # U I I . The protons at positions 2 and 6 are called completely equivalent because a ( t ) = &(t) at each instant; in general, two nuclei i and j are completely equivalent if a((t) = al(t). Similarly, the protons at positions 3 and 5 are completely equivalent, and by symmetry a3(A) = a&i) = a11 and a3(B) = as@) = a ~ .The protons at positions 2 and 6 are said to form one completely equivalent subgroup (subThe Jou?nul of Physical C h m k t r y
group a) while those at positions 3 and 5 are said to form another (subgroup b), but az(A) # as(A) if aI # U I I , and therefore the set of all four protons does not form a completely equivalent group. We shall see in detail in the following that when there is rapid exchange between forms A and B, an average splitting is observed, (az(t)) = (a6(t)) = (a3(t)) = (as(t)) = 6, and the spectrum consists of five lines with relative intensities 1 :4 :6 :4 :1. These four protons are called equivalent (but not completely equivalent), and in general two nuclei i and j are said to be equivalent if (at(t)) = (a,(t)). Thus nuclei are equivalent if they have the same average hyperfine splittings, while they are only completely equivalent if, in addition, their instantaneous hyperfine splittings are equal. As a rule, accidental equivalence of any type must be specially examined. These distinctions are essential to the understanding of the ensuing discussion. As a second example, consider the 3,5-dinitrobenzoate dinegative ion radical.
Suppose a cation forms an intermittent complex with the carboxylate group. This will cause a modulation of the spin densities in a manner that will be symmetric with respect to the protons at positions 2 and 6 and also with respect to the two nitrogen nuclei. As far as this mechanism is concerned, these two protons form a completely equivalent subgroup, and so do the two nitrogen nuclei. On the other hand, if a localized complex is formed with the cation at one or the other of the two nitro groups, the protons at positions 2 and 6 no longer have the same splittings and neither do the two nitrogen nuclei. Thus, if this second mechanism is operative, none of the nuclei are completely equivalent. For both processes, however, under conditions of rapid exchange, these two protons are equivalent, and so are the two nitrogen nuclei. 1. Single Completely Equivalent Group of Nuclei. Line Widths and Frequency Shifts. When the radical contains only a single group of n completely equivalent nuclei, all the instantaneous hyperfine splittings are equal [ai(t) = a ( t ) , for all i ] ,and the hyperfine interaction term in the spin Hamiltonian can be written as a function of the operator for the total nuclearspin angular momentum (eq 2.30). (26) B. G. Segal, M. Kaplan, and G. K. Fraenkel, J. Chem. Phys., 43, 4191 (1965). (27) We amume here that the OH groups are freely rotating (see section I V and ref 95, 96, and 97).
LINEWIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
n
J
(2.30)
XI1
=
i=l
where I, is the nuclear-spin angular momentum operator for the i t h nucleus and the nuclear spins are the same ( I t = I ) . Thus C a l ( t ) I i - S= a(t)CI,*S= a ( t ) J * S
(2.31)
i
z
Similarly, the interaction between the external magnetic field and the nuclear magnetic moments is proportional to
C%LBo= rtJzBo
(2.32)
i
where y t , the magnetogyric ratio for the ith nucleus, is the same for all the completely equivalent nuclei within the set being considered. I n the high-field approximation, the lines in the spectrum are distinguished by the quantum number M
M =
Emf
(2.33)
i=l
where M and m, are the eigenvalues (in units of fi) of the operators J, and I,, for the x components of the total angular momentum and of the angular momentum of the i t h nucleus, respectively. In the limit of rapid exchange there are 2nI 1 equally spaced lines with separation (a(t)) and relative intensities D ( M ) . For nuclei with spin 1 = l/z, D ( M ) is given by the binomial distribution
+
D(M)
=
n!/[(n/2)
- M ] ! [ ( n / 2 )+ M I !
(2.34)
and appropriate but more complicated distributions can be written for nuclei with other values of spin. The possible values of the quantum number J which determines the eigenvalue J ( J 1) of the operator J 2 (in units of 6) are given by the usual rules for the combination of angular momenta. Thus, for two protons, J = 1 and 0; for three protons there is one state with J = 3 / 2 and two degenerate states with J = 1/2; and for four protons, J = 2, 1 (three degenerate states), and 0 (two degenerate states). For two 14Nnuclei (spin I = l ) , J = 2, 1, and 0. I n general, the number of states with a particular value of J , W ( J ) , is
+
W ( J ) = D(M= J ) with W(J,,,)
= D(M,,,)
- D(M= J
+ 1)
(2.35)
= 1.
The linewidth variations for a set of completely equivalent nuclei can be determined by the same procedures used in section 1I.A. 1 for a single nucleus. Of the equations derived there are if the quantum number m specifying the z component of angular
145
momentum of the single nucleus is replaced by the quantum number M for the total z component. Similarly, the second-order shifts9o2l are given by the equations in section II.A.2 if the quantum number I and m are replaced by J and M . Since the shifts depend on [ J ( J 1) - M 2 ] ,a line which is degenerate in the high-field approximation may be split by the second-order shifts. For example, the central ( M = 0) line in the spectrum arising from two 14N nuclei consists of three components of equal intensity corresponding to J = 2, 1, and 0, and these are shifted downfield by -6((a2(t))/2Bo) , -2((a2(t))/2Bo),and 0 gauss, respectively.z8 Similar shifts occur in the central line of the spectrum from four equivalent protons, but the three components are of relative intensity 1:3 :2 corresponding to the degeneracies W ( J ) for J = 2, 1, and 0. The M = 1 lines in both these examples are split into two parts: the part for J = 2 is shifted by -5((a2(t))/2Bo)and the part for J = 1 is shifted by - ((a2(t))/2B0). They are of equal intensity in the nitrogen case and of relative intensity 1 : 3 for the four protons. The nondegenerate -11 = * 2 lines are shifted by -2((a2(t))/2Bo)but are not split. The form of the spectrum depends on the relative magnitudes of the second-order splittings and the line widths. If the splittings are large compared to the widths, there are 25 1 Lorentzian shaped lines for each of the possible values of J . The lines for a particular J all have the same intensity, but the relative intensities for sets of lines with different J are proportional to W ( J ) . The widths, !!'Z-l(M), are given by eq 2.10 and depend only on M . I f the second-order splittings are small compared to the widths, the 2n1 1 lines are of approximately Lorentzian shapegand again have widths given by eq 2.10. Their p o ~ i t i o n s ~ ~ , ~ * are determined by the average over the second-order shiftsgfor each value of M
+
*
+
+
( B ( M ) ) = Bo - M d -
+
((az(t))/2Bo)(J(J 1) - M2)M (2.36) where
(J(J
+ 1) -
M2)M
= nI
D(M)-l J-IMI
+ 1 ) - M 2 ] (2.37) the average ( J ( J + 1) -
W(J)[J(J
For n nuclei of spin I = 1/2, M z ) Mis independentgpZ1of M and becomesg( J ( J 1 ) M 2 ) =~ (n/2). Since all lines for I = '/z are shifted by the same amount, only the g valuez6is affected by the second-order corrections* The average values
+
(28) More generally, eq 2.28 shows that ((aZ(t))/2&) should be replaced by [ ( a 2 / 2 ~+ ~ )( 2 ~ y e \ ) - 1 k ( u o ) ~ .
Volume 71,Number 1 January 1967
GEORGE I(.FRAENKEL
146
(J(J + 1) - M 2 ) Min the case of two nuclei with spins I = 1 are 2, 3, and 8/8 for M = *2, M = *1, and M = 0, respectively (see Table I). For intermediate magnitudes of the splittings and widths, each of the 1 hyperfine lines (corresponding to the 2nI 1 2nI values of M ) consists of a set of n I - IMI 1 overlapping Lorentzian-shaped lines (corresponding to the nl - IMI 1 values of J ) with relative intensities W ( J ) . Since the widths T2-'(M) (eq 2.10) depend
+
+
+
+
nI
only on M , each of the D ( M ) =
W ( J ) components
J=IMI
of a hyperfine line has the same width. 2. Several Di'erent Completely Equivalent Groups of Equivalent Nuclei. We now assume that all of the n nuclei are equivalent [{az(t))= for all i (1 I iI n)], but that nu are in one completely equivalent set
Table I: Average Values of (J(J for Spins Z = I No. of nuclei
M
2
*2
3
WMIa
Zkl
1 2
0
3
1 3
322
1 3
0
4
rt4 A3
6
W(J)"
2 2 1 2 1 0
3
1 1 1 1 1 1 1 1 2 1
2 1
. 2 3
11 5 1
3 3
rtl
0
2 5 1
6 0 3 8 2
3
1
12
2
6
3
2 0
1 4 10
16
19
4 3 2 4
1 1 1 3 1 3 6 1
3
3 2 1
6 6
4
1
3 2 1
3 6 6 3
4
11 3 16
8 2 19 11
5 1 20 12 6 2 0
D ( M ) and W ( J )are the degeneracies with respect to M and From eq 2.37.
J, respectively (cf. eq 2.35).
The Journal of P h y s k d Chemistry
+
+
+
2
2 1 0 4 4
0 a
M1]
7
3
rt2
-
J
2 :It 1
+ 1) - M*)M
[ai.(t) = a,(t), 1 I i I nu],nb in another [az(t) = ab(t) # a,(t), nu < i 5 n b ] , and so on. Examples of radicals containing two different completely equivalent groups were discussed a t the beginning of section 1I.B. a. Line Widths in the Absence of Significant Frequency Shifts. We initially use the two-jump model and apply the modified Bloch equations, as in section II.A.l., to two groups of completely equivalent nuclei [nu nb = n ] . The nu nuclei in the completely equiv% lent group a are assumed to have splittings a,(A) = a1 in form A and a,(B) = a11 in form B. For the nb nuclei in group b we choose ab(A) = a11 and a,@) = UI, so that the hyperfine interactions are interchanged between the two forms. I n order for the two groups of nuclei to be equivalent, it is necessary that PA = p~ = l / 2 , which implies that T A = T B = 27. If the rate of exchange between the two forms is very slow, the spectrum of form A consists of 2nJ 1 lines of relative intensity Dn,(MJ with separation UI each of 1 lines of relative inwhich is further split into 2nJ tensity Dnb(Mb)with separation UII. For example, if both completely equivalent groups consist of only a single 14Nnucleus, and if a11 >> aI, the spectrum consists of three sets of triplets. An identical spectrum would be obtained for the radicals in form B. The (angular) frequencies are (to first order) uA(Mu,Mb) =
WO
W B ( M ~ , M=~wo )
+I +1
?'e/
Ye1
+ AfbaII) (Mba~+ M , ~ I I ) (2.38) (MuaI
When there is rapid exchange between the two forms, the average frequency is (cf. eq 2.2)
u(M)
= c(Ma,Mb) = = w
where M
=
+ IyejME Mu
+
Mb
('/2)
[WA(MuiMb)
+ Wb(Mu,Mb)1 (2.39)
and
E = ('/z>(a1
+
(2.40)
UII)
The mean-square frequency deviation (cf. eq 2.7b) becomes ([6W(Mu,Mb)I2)= re2((6a>2)(Mu
- Mb)'
(2.41a)
where the mean-square deviation in the hyperfine splitting (eq 2.3) is ((sa)z) = (l/P)(UI
- a1d2
(2.41b)
The line widths are therefore (eq 2.7a) TZ-'(Mu,Mb) = ')'e'7((6a)2)(Mu -
f
T2,O-l
(2.42)
It is instructive to consider in detail how the lines from the static forms A and B are affected by the exchange in
147
LINEWIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
two simple cases, one involving two 14N nuclei and the other two protons (nu= nb= 1). The lines arising from the two 14Nnuclei with M u = Mb = 1 ( M = 2 ) have the same position in forms A and B, and when exchange takes place they are unshifted and unbroadened. 2(ye/6and their width Their position is W(2) = wo is Tz-l(1,l)= T2,0-1. Similarly, the lines with Mu = M , = 0 ( M = 0 ) andM, = M b = -1 ( M = -2) are unaffected by exchange, are at positions G(0) = 00 and 4 - 2 ) = wo - 2lyeIa, respectively, and have widths Tz-l(O,O) = T2-l(-1,-l) = Tz,o-l. The lines with M , = 1 and &Ib = 0 ( M = 1) have frequencies ~ ~ ( 1= ~ 0wo) / Y e ( a I and w~(1,O) = wo [ Y e / a I I in forms A and B, respectively, and when there is rapid exchange they coalesce to the frequency W(1) = wo (yela with width Tz-’(l,O) = yez~((6a)z) T2,O-l. A second component is formed a t the same frequency with the same width from states with Ma = 0 and M b = -1, and two components at W(-1) = wo - /yelawith the same width are formed from lines with Mu = -1, Mb = 0 and M a = 0, = -1 ( M = -1). Finally, = - 1 ( M = 0) at frequencies lines with M u = 1and w ~ ( l , - l ) = wo Iyel(a1 - UII) and ~ ~ ( l , -=l 00 ) (yel(u1 - a11) coalesce to W(0) = wo with width Tz-l(1,-l) = 4ye27((6a)2) Tz,,,-l. A second component at the same frequency with the same width arises from lines with M a = - 1,Mb = 1. The five lines in the averaged spectrum thus have different widths. Those with M = f 2 are sharp (unaffected by the exchange mechanism) and those with M = *1 are broad [incremental width proportional to ~((6a)~)l. The central line ( M = 0) has one sharp component and two broad components [incremental width proportional to 47( ( 6 ~ ) ~ ) l . If ye2~((6~)z)> Tz,o-’, the components affected by the exchange mechanism become too broad to be observable, and only three lines appear. They are of the same amplitude and width (Tz,o-l)and have a spacing of 26. In intermediate cases, the M = + l lines are broader than the M = f 2 lines, and the central line has :in amplitude less than three times the amplitude of the end lines. All the component lines are of Lorentzian shape, but since the central line is a composite of lines with different widths, it is not in general of Lorentzian shape. The variation in widths has led to the designation “alternating line-width effect.” Some computed spectra are shown in Figure 1, and experimental examples are discussed in section IV. In the case of two protons, lines Kith Mu = M , = f l / z ( M = f l ) are unaffected by the exchange and occur at frequencies G( f1) = wo f lyejli with widths Tz-’(fl/z, fI/Z) = T2,o-l. Those with Mu = - M , =
+
b
+
+
+
1
+
+
+
1
d Figure 1. Plots of the computed spectra arising from two 14N nuclei illustrating the alternating line-width phenomenon. The mean hyperfine splitting is taken as tiN = 10.000 units, - ~0.500 unit. In and the residual line width as T ~ , o = each diagram, the first derivative of the spectrum is plotted as a function of magnetic field using eq 2.49 and 2.51. It is assumed that j d ( 0 ) = -jaa (0). (a) &(O) = 0; (b) ja,,(0) = 0.100; (c) jaa(0) = 0.300; (d) jaa(0) = 0.600. The ratios of the line amplitudes to the amplitude of the central line for case (a) are: (a) 0.333, 0.666, 1.00; (b) 0.333, 0.463, 0.507; (c) 0.333, 0.260, 0.362; (d) 0.333, 0.133, 0.339 in the order M = =t2, f l , 0, for each case. For large values of jaa(0),the M = f l lines disappear, and the amplitudes of the remaining lines are all equal to 0.333 relative to that of the M = 0 line in case (a).
* 1/2 ( M = 0) give rise to two components at frequency W(0) = wo and are broadened, the expression for the
+
width being Tz-’(=k1’/2, F ~ / z ) = y e z ~ ( ( 6 u ) 2 ) T2,O-l. The relaxation-matrix treatment can also be applied to several different completely equivalent groups of nuclei in the limit of sufficiently rapid modulations [ T ~ ~ ~ ~ { = j b U ( w ) #
(2*45c)
jUQ(U)
Similar relations hold for the correlation functions, gi,(t’) or gab(t’). The spectrum observed corresponds to that resulting from the average hyperfine splitting 6 = (a,(t)) = (ab(t))= . . . , with widths [if w o 2 ~> 2> 11
[Tz‘yM)]-l = T2-’(Ma, Ma, . . .)
=
C j .,. ,(O)MuMu +
up = a.b,.
Tz,o-’ (2.46)
where
C M. . ,
M =
(2.47)
u=u,b,.
I n general there is a different width for each of the possible assignments of the quantum numbers M,, Mb, . . ., and those components of a line with a particular value of M which arise from different combination of M,, Ma, . . . in eq 2.47 and lead to different widths are distinguished by the superscript k. For example, when there are only two different completely equivalent groups
[T2‘k’(M)1-1 = ~ Q Q ( O[Ma2 )
+
Mb2]
+
2jub(O)M&b
= jQa(0>M2
-
2[jQU(o)
( l / ~ )[jua(O)
T2,O-l
-
+ T2,O-l + j u o ( 0 ) 1M2 +
jab(())
=
+
IMaMb
(1/2)[jUQ(0)- jUh(o)l(AfU
- Ma)2 f
(2.48) Tz,o-l
If jQb(w) = jaa(w), corresponding to complete equivalence of all the nuclei in sets a and b, eq 2.48 (and under similar circumstances, the more general expression, eq 2.46), reduces to eq 2.10 with m replaced by M . If jab(o)= -jQa(w), eq 2.48 becomes
[TZ(’)(M)]-’= j a a ( o ) ( M a - Ma)’
+
T2,o-l
(2.49)
which has the same dependence on the quantum numbers as eq 2.42. This relation between the spectral densities results, in particular, when the conditional probabilities that describe the two-jump model for which eq 2.42 is valid are employed to calculate gQn(t’), The Journal of Physical C h a k t r y
(2.504
- 61
=
- [a& - ii]
(2.50b)
prhich is the same as (‘/z)
provided both sets a and b are e q u i ~ a l e n tbut ,~~~~ jik(w)
re2T((Wz>
More generally, if
i.e. jaa(@>
= jQa(0)=
-jab(())
[aa(t)
+ ab(t)l
=
6
(2.50~)
= -jua(a)a Note that eq 2.50b and 2 . 5 0 ~hold for all time and are quite different from the relations which define ii. Equation 2.46 includes the possibility of a wide variety of modulating mechanisms, not only those resulting from simple arrangements amenable to treatment by the Bloch equations. It demonstrates that a model which fits the observed line-width variations may well not be unique; instead, all processes which lead to the proper relations among the spectral densities may be equally satisfactory. Even if an alternating line-width effect is observed, it is not necessary for jub(0)= -juQ(0) in eq 2.48. This follows because if g(0) = (l/2) [j,,(O) jub(0)]# 0, there will be a contribution to the line widths proportional to j ( 0 ) M z as well as one proportional to [j,,(O) - j ( 0 ) l ( M u- Mol2, and an alternating line-width effect will be observed whenever jua(0) > j ( 0 ) and jua(0)> T2,0-1. Only very simple mechanisms such as the two-jump model lead to jQb(0)= -juu(0), although in a variety of situations ju,(0) > j(0). For example, a complex formed a t the carboxylate group of the 3,5-dinitrobenzoate ion produces a symmetric modulation of the splittings which of necessity makes -jub(0)> jua(0)for the two nitrogen nuclei. Many of the realistic models for motional modulations have symmetric in-phase contributions like this. $ 13-15 Results from eq 2.46 for a few simple cases are given in Table 11. Other tabulations will be found in ref 3, 14, and 15. Quite a variety of effects can result for special relations among the jab. One that is sometimes important occurs when j u b ( w ) = 0; Le., the nuclei in the sets a and b have uncorrelated motions. As another special case, consider two nuclei with spin
jU,(w)
+
(29) The relations among the j i j ( w ) in eq 2.44 follow from similar expressions for the ~ i j ( t ’ ) on substituting ai@) = a&) = a,@). Those in eq 2.45a and 2.4513 only hold if a,(t), ab(t) refer to sets of nuclei which undergo identical motions: the only reason the nuclei are not completely equivalent is because of phase differences (either correlated or uncorrelated). In ref 3 (eq 4.27c), it is incorrectly stated that gao(t’) = g&’) whenever groups a and b are equivalent. This is normally the only case of interest. In principle, however, it is possible for a, = a b but iiaZ# a b 2 , so that (a,(t)a,(t t‘)) # (ab(t)ab(t t ’ ) ) and Qaa(t’) # Qbb(t’). (30) In general,3jij(W) = jji(w) for any two nuclei i and j .
+
+
149
LINE WIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
Table II : Line Widths, Modulations of Isotropic Hyperfine Splittings. Nuclei in Two Completely Equivalent Subgroups width,d { [Tz(*)(M)I-l - Tz,a-')-Out-of-phase correlated, General 3ab = -iaa
-Line Nuclear spin. I
No. of
nuclei
-DegeneraciescMa
[Afi,,Mblb
D(~)(M)
D(M)
0 jaa
8
'/2
f 4 f 3
1 8
52
12 16
28
8
56
f l
48 2 32 36
0
1
2
1 2 1 1
0
1
8
70
1
1
2 2
2 3
1
+
a M = Ma Ma, where M, refers t o one completely equivalent subgroup of nuclei and Mb to the other. Brackets indicate that The degeneracy D ( M ) the state specified, Ma, Mb, and also the state Ma, M , obtained by permutation of a and b are both included. is the statistical weight for the hyperfine line with quantum number M; and Dck)(M)is for the kth component, L e . , the component From eq 2.46. with quantum numbers, including permutations, for the row to which it refers.
I = 1. The M = & 2 and f1 lines can have the same widths if ju,(0) = - (*/z)jaa(0), namely 4j,,(O) Tz,o-', while one component of the central line is sharp with width Tz,o-land the other two are broad with width 3ja,(0) T z , ~ - ' . If j,,(O) is sufficiently large, the observable spectrum is only a single central sharp component instead of five lines with relative amplitudes 1:2:3:2:1. When the components of a degenerate line have different widths, the line shape is a superposition of Lorentzians
+
+
[ D ' " ( M ) T z ' k ' ( M ) / ~X ]
IM(w) = k
(1
+ [ T z ' k ' ( M ) ] 2-[ ~ C ~ ( M ) ] ~ ) (2.51) -'
where D @ ) ( M is ) the degeneracy of the kth component of the line with quantum number M , and the summation is over all differentvalues of the width [ T z ' ~ ' ( M]-') (distinguished by the index IC). The Tz'"(M) are given by eq 2.46. This result, that a degenerate line is a superposition of Lorentzian-shaped components, differs from that first obtained by Kivelson2 using the general theory of Kubo and Tomita.81 The earlier work predicts instead a Lorentzian line with an average width shown in oq 2.52).
(Tz-'(M))
=
D(M)-'CD'k'(M) [!Z'z'k'(M)]-' (2.52) k
When the variations among the widths [Tz'"(M)]-' for a particular line are small compared to the average width (TZ-'(M)), the correct theory also reduces to this result for the line shape and width, but except in this limiting case, a Lorentzian-shaped line is not obtained. The difficulties in the earlier work are apparently inherent in the Kubo and Tomita theory when applied to degenerate lines.s2 b. Shifts and Line Widths When Frequency Shifts Are Signijkant. For simplicity, we first consider the two-jump model of section II.B.2.a, but with the further restriction that only one nucleus of spin I is present in each completely equivalent group (nu = nb = 1, n = 2). Equations 2.38 become, to second order, as in eq 2.18
(31) R. Kubo and K. Tomita, J. P h w . SOC.Japan, 9 , 888 (1954), (32) D. Kivelson, J . Chem. Phys., 41, 1904 (1964).
Volume 71, Number 1 January 1967
GEORGEK. FRAENKEL
150
+ I~el(mza1+ m i a r ~ )+ (l~el/2Bo)(ar~[I(I + 1) - mz21 +
wB(m1,md =
-
wo
+
UII~[I(I 1) - mi2]) (2.53b) and lead to an average frequency
+ I YeIMe + /2&) [a2 + ((sa)')] [21(1 + 1) -
G(M) = G(m1,mz) =
(1
uo
Ye/
mI2 -
m2] (2.54)
where si and ((6~)~)are defined in eq 2.40 and 2.41b. These results only hold9tZ1if the difference in hyperfine splittings la1 - a111 is large compared to the secondorder corrections (a,1*/2Bo) and (a11~/2Bo),because it is only under these circumstances that the second-order perturbation treatment is valid. I n the other limit, that la1 - a111 is small compared to the second-order corrections, the nuclei can be treated as if they were completely equivalent and the results of section II.B.1 are applicable, while in the intermediate case, exact solution of the spin Hamiltonian is required. There are also certain other limitations to eq 2.54 which are discussed in the following. As an example of eq 2.54, we take I = 1. The second-order shifts in the magnetic field are then - ((a2(t))/Bo)for M = f2 lines and - (3/2)((a2(t))/Bo) for both components of the M = =t1 lines. The ml = 1, m2 = -1, and ml = -1, m2 = 1 components of the M = 0 line are shifted by -((a2(t))/Bo)and the ml = m2 = 0 component of this line is shifted by -2((a2(t))/ Bo). Let US assume now that there is an appreciable alternating line-width effect, which corresponds approximately to
Ye2T((6a>2) 5 (2T2,0)-'
(2.55)
Then the shift of the central line is determined by the sharp component (ml = mz = 0), and the difference in the intervals between it and the high- and low-field lines ( M = f2) is, on a field scale, (cf. eq 2.23)
&(/MI = 2)
= -(2/Bo)[G2
+ ((sa)')]
(2.56)
Although, as indicated above, the spectrum from a single completely equivalent group of nuclei exhibits dynamic shifts which are always smaller than the line width, such is not the case when there is an appreciable alternating line-width effect. The alternating linewidth phenomenon broadens some of the lines, but because of special phase relatjons it leaves others sharp. A cancellation like this to the width contributions for some of the components does not occur for the dynamic frequency shifts, however, and the shifts, - [ ( ( 8 ~ ) ~ ) / Bof o ]the , narrow lines are of the same magnitude as the shifts of the broad components. Since these The JourmaE of P h y s M Chemistry
shifts may be a significant fraction of the widths of the broad lines, ~)r,l((6a)~), the shifts of the narrow components may be appreciably greater than their widths, and it is primarily these sharp components that determine the positions of the lines in the spectrum. Thus by accurate measurement of the line positions, and ( ( 6 ~ ) ~can ) be evaluated, and by measurement of the widths of the M = *2 and M = =tl lines, Tz,o-' and sa)^) can be determined. In this way, both the lifetime T and the mean-square deviations in hyperfine splittings can be obtained separately. Examples of this type of measurement are discussed in the following. The relaxation-matrix theory of the shifts is formulated in terms of the spectral density krj(u),the Fourier sine transform of the correlation function gij(t') defined in eq 2.43 (cf. eq 2.24). In the example just considered of two nuclei with spins I = 1 under conditions of an appreciable alternating line-width effect, corresponding to -j1*(0) > (1/2)j~1(0) >> ( ~ T z , o ) - 'the , corrections to the first-order field positions are found to be:9 for the M = *2 lines '
- I Ye1 -'kll(uo)
-2(d2/2B0)
for both components of the 11.1 = -3(a2/2Bo) -
('/2)/
(2.57a)
1 lines
f
Ye1
-'kll(wo)
(2.57b)
for the ml = m2 = 0 component of the M = 0 line -4(E2/2Bo) and, for the ml = 0 line
= f1,
- 21 Ye\ -'kii(uo)
(2.57~)
= 'F 1 component of the
M
-2(82/2B~)
- 1 Ye1 - ' h ( w o )
(2.57d)
I n the special case of the two-jump . . model treated above by the Bloch equations, j1~(0)= --j11(0), and kll(w0) = ( ~ ~ 7 ) - ~ j ~=~ (-ye2/uo)((6a)2) (O) = ( ~ ~ e ~ / B o ) ( ( G Thus a)2). eq 2.57 reduce to the relations previously obtained. As before, the relaxation-matrix results confirm the Bloch equation treatment and represent an extension to more general types of motion. When there is no alternating line-width effect so that the two nuclei of spin I = 1 are completely equivalent [j12(u) = jll(u)], the shifts are (see section II.B.l) determined by the total angular momentum quantum numbers J and M , but when there is an appreciable alternating line-width effect, the appropriate quantum numbers are ml and m2. The shifts of the nondegenerate M = =t2 lines are the same in both cases [for equal values of2*((a2(t))/2Bo)],but if j12(u) = jll(u), the M = f1 lines33are both split into two components, one with shift28 -5((a2(t))/2Bo) and the other with shift - ((a2(t))/2Bo), while if there is a large alternating line-
LINE WIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
width effect, both components are shifted by the same amount, -3((a2(t))/2&). Similarly,33there are three components of the central line shifted by - 6 ( ( a 2 ( t ) ) / 2B0), - 2 ( ( a 2 ( t ) ) / B o )and 0 whenjlz(u) = jll(u), as compared to two different shifts, one of -4((u2(t))/2Bo) and the other, for two of the three components, of - 2 ( ( a 2 ( t ) ) / 2 B o ) when , j12(u) = - j l ~ ( u ) . These results hold even when the dynamic shifts [Icl~(w~)] are small, and thus the static frequency shifts depend on the modulating motions. I n other words, the frequency spectrum of the static problem depends on the line widths. This is in contrast to the usual case in which one can completely disregard the fluctuations in the time-dependent processes that determine the widths when evaluating the positions of the lines. These subtleties are suggested by the Bloch equation treatment, but the results can only be justified by the rigorous relaxation-matrix formulation. 9 The relaxation-matrix procedure can be applied to a number of other general situation^,^ but here we quote the result only for the case of small (unresolved) frequency shifts with small width variations34among the components within each hyperfine line (i.e., among those components with the same value of M). It can be shown that the lines are of approximately Lorentzian shape with an average width given by eq 2.52, and with an average position given by28eq 2.36 where ( J ( J + 1) - M 2 ) Mis replaced by 1) -
c(Ju(Ju+
151
calculations shows clearly how these cross terms contribute to the line widths. For simplicity, let us as(33) The average shifts when there is an alternating line-width effect and when the two nuclei are completely equivalent are the same, however. They are% -2((aZ(t))/2&), -3((aZ(t))/2&) and - (s/a)((a2(t))/2&),for the M = f 2 , fl, and 0 lines, respectively. (34) It can be shown (ref 9, section V1I.c) that all lines arising from nuclei of spin I = I/z are shifted by the same amount whenever the second-order shifts are small, and that this result is independent of the magnitude of the variations in line widths. The shift in field is - ( 4 2 )[(ii2/2Bo), (2Lyel)-‘koa(wo) I, where n is the number of nuclei with average splitting a . (35) A number of details enter into the calculation of (Ju(Ju 1) M,~)M (cf. section III.B.3). For n, equivalent nuclei of spin I, the spectrum (to first order) contains 2 n d 1 lines (-nJ 5 Ma 5 n a I ) with degeneracies D ( M a ) . The possible values of J o are in the range lM,I 5 J , 5 n d with degeneracies W(J,) (eq 2.35) and
+
+
+
W ( J a )= D ( M , ) J a = lMa1 8
A line with M = E M + -j, - +j d X (M2 - M3) (m7 - ms>+ T2,o-l (2.64) where MH = M2 + M I , M Z = m2 + me, M B = m3 +m6, and MN = m7 + ms. If the modulations are
= ('/d [ j 2 2 For form B there is a similar expression [ w ~ ( m ~ , m z )TZ-1(M2,M3,m7,m8) ] ('/d [ ~ Z-Z jZ31(Mz - M3I2 with splittings al(B) and a2(B) instead of al(A) and
[j27
("2)
(2.59)
(2.60)
(2.61)
(2.62a)
(2.62b) Substitution of eq 2.61 into eq 2.7a gives the lir_+width variations. More generally, the relaxation-matrix formulation gives3
. ; . .) = C Cjr,a,(o)M,,M,, T2,o-I
T2-1(Mra,f'!rg,. . . ;
Msg,Msh, . *
Tu S g
3
+
(2.63)
where the summations are over all completely equivalent subgroups a, b,. . ., in the equivalent group r, over all completely equivalent subgroups g, h, . . ., in the equivalent group s, etc., for all the equivalent groups r, s,, . .. There is (in general) a different width for each of the possible assignments of the quantum numbers M,,, MT6,. . .; M,,, Ms,,,.. ., etc. For example, the p-dinitrobenzene negative-ion radical contains two different equivalent groups, the two nitrogen nuclei and the four protons. When solvent or cation complexes are formed with the two nitro groups, the two 14N nuclei are not completely equivalent, and neither are the protons a t positions 2 and 3 or 5 and 6. The protons a t positions 2 and 6 form one completely equivalent group and those at positions 3 and 5 form another. Numbering the 14N nuclei in the nitro groups substituted a t positions 1 and 4 by 7 and 8, respectively, the following relations [where we The Journal of Physical Chemhtry
M3)
('/2)
j37
jZ8
j37
( ' / 2 ) [jZ7
(M2
j781MN2
('/2>
[j27
j28
j37
j38lM~
j37
( ' / 2 > [j27
instantaneously symmetric so that the two nitrogen nuclei form one completely equivalent group and the four protons another, jz3 = j Z z= j", j 7 8 = j , = j", so that j27 = j , = j 3 7 = j, = j N H , and
+
+ ~ ~ N H M H+ M T2,o-I N
T ~ - ~ ( M H , M=N~) H H M Hj N i~v A f N z
(2.65)
Thus, if ~ N Hmakes a significant contribution, lines for which MH and MN are of the same sign [e.g.,MH = 1, M N = 1or M H = -1, M N = -11 have a greaterwidth > 0) than those for which these quantum (if "j numbers have opposite signs [e.g., M H = 1, M N = -1 or M H = -1, M N = 13. On the other hand, if an extreme alternating line-width effect exists, j7s = -jn, and if one makes the reasonable assumption as(t) = that this results because (cf. eq 2.50~)a,(t) 2iiN and that the protons behave in a similar way, az(t) a3(t) = 2dH, it follows that36j 2 , = - j z z and j N H = j ~ 7= j , = -jz8 = -j37. Equation 2.64 then reduces to
+
+
T2-l (M2,M3,m7,md jHH(M2
=
- Ma)'
+ jNN(m7 - md2 -k
2jNH(M2
- M3)(m7 - m8)
(2.66)
Examples of cross-term contributions to the line widths are given in section IV.
In. The Effects of Rotational Motions: Anisotropic Dipolar and g-Tensor Interactions Up to now, we have only considered the effects of modulations of the isotropic hyperfine interactions, but there are also anisotropic contributions to the
LINEWIDTHSAND FREQUENCY SHIFTS IN ESRSPECTRA
spin Hamiltonian. These anisotropic interactions affect the line widths of the esr spectra from radicals which are tumbling rapidly in solution, but (to first order) do not alter the line positions. One of the anisotropic contributions is the direct magnetic dipoledipole interaction between the unpaired electron and the nuclear magnetic moments. The magnetic moment of the electron pe = -gp/Pe/S creates a magnetic field Be at a point r Be
=
r - 3 { [3(pe-r)/r21r- p e l
(3.1)
and the energy of orientation of the magnetic moment g N / PNI I of a nucleus in this field is
p~ =
W(O)= -pN.Be = r-3(pe.pN [ 3 ( ~ e . r > ( p ~ . r >I / r (3.2) ~l =
gelPelgNlbN/r-3([3(S.r)(I.r)/r21- S.1) (3.3)
where S and I iire the (vector) angular momentum operators for the electron and nuclear spin, respectively. The quantities ge = (fil r e l / l P e l ) and gN = ( f i r ~ / l @are~ Ithe ) electron and nuclear g factors, /Pel is the Bohr magneton, and I ON( is the nuclear magneton. The expectation value dD’ = (+I W@)l+) of over the electronic wave function must be taken to obtain the contribution of the dipolar interaction to the spin Hamiltonian. This expectation value is zero for an electron in an S state (e.g., for the hydrogen atom), but atoms in states with nonzero angular momentum and molecules with unpaired electrons have nonvanishing contributions. The dipolar interaction is a second-rank tensor with components that transform under rotations like the spherical harmonics of order 2 (the d functions), ie., like zy, yz, zx,x 2 - y2, and 3z2 T * , and in general the esr spectrum of an oriented radical is a function of the direction of the external magnetic field Bo. On the other hand, when the radical is tumbling rapidly in solution, the variation 16wl of the line positions (frequencies) with orientation times the correlation time TR characteristic of the rotational [S- exp(i+) S+ exp(-it$)] sin 0
= {
+
+ S, cos BIBo cos 0
(3.13)
where the explicit dependence of the angles on time has been suppressed, and
s,
=
s, f is,
(3.14)
The term in S,Bo is essentially secular in nature (see section 11) because it does not induce any transitions but merely causes the resonance frequency to change as cos2 e(t) varies with time. The operator S- converts spin states with ms = '/z into those with ms = (41) When the static second-order frequency shifts are large, there are small correction terms in the expressions for the line widths' that are not considered in the following. In the spectrum arising from hyperfine interactions with a single nucleus, Wilson and Kivelson40 have shown that this correction depends on ma. (42) It should be noted that this result only holds when the radical is tumbling rapidly. When the radical is trapped in a solid matrix so that it is fixed in orientation g(e) = [glz sin2 e
+ gl1 cosz eI1h
LINE WIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
while S+ converts ms = - l / z states into ms = states, so these operators induce transitions a t the frequencies fwo the electron spin resonance Larmor frequency. They are therefore nonsecular and of less importance than the secular terms. The line-broadening effects are proportional to the square of the timedependent terms, and in the axially symmetric case they depend on (gl - gll)z. The results of the detailed calculations3give a secular line-width contribution -l/z,
l/z
[T2-1](G) = (8/3)j'G'(o)Boz
(3.15)
where the spectral density j(G)(u)is j(G)(U)
= [7R/(1
+ U27R2)](8e2/20fi2)x [g12
The correlation time TR
7~
+ 92' 4- 93'
- 3g,'I
(3.16)
for molecular tumbling is
= (4r7a3)/(3kT)
(3.17)
This formula applies to a macroscopic sphere of radius a in a solvent with viscosity 77 at temperature T (k is the Boltzmann constant), but the meaning of the viscosity and the radius are not clear for a microscopic system, particularly one which is solvated. For typical molecules in nonviscous organic solvents at room temperature, 7R lom9to lo-" s ~ c The . ~ ~rotational motion can also be formulated for bodies which are ellipsoidal in shape, but the resulting equations for the line widths become quite complex44and have not yet been applied in detail to any experimental data. Equation 3.15 shows that the line widths vary as the square of the magnetic field, and thus the g-tensor contributions to the line widths can be differentiated from other contributions by changing the magnetic field. Of course, as the field is increased, ( ~ 0 7 is~ not ) ~ necessarily small, and nonsecular contributions may have to be included.43 For a radical with axial symmetry
-
Igi2
+ + gz2
- 3gS2I =
~ 7 3 ~
(2/3)(gi
- gli)z
(3.18)
so that, as indicated by our previous qualitative considerations, T2-l is proportional to (gl - g,,)z. A common characteristic of rotational effects such as the g-tensor and anisotropic dipolar interactions which depend on a rotational correlation tirne 7R is that they vary approximately as (?/?'). There is also another type of effect, known as the spin-rotational interact i ~ n , ~which O is not included in the present discussion. It depends on the g-tensor anisotropy but varies as ( T / v )rather than as ( v / T ) . The principal components gl, g2, g3of the g tensor can only be measured accurately by studies of (dilute) single crystals of free radicals, and although a few
155
such investigations have been performed on aliphatic radicals, there are no data for aromatic radicals. Stone38has carried out Hiickel molecular orbital calculations for the gz of aromatic hydrocarbons, but the results are probably only approximate. Thus, except for the average g value gs (which can be directly m e a s ~ r e d ~it~is~ difficult ~ ~ ) , to obtain reliable estimates of the magnitudes of the quantities in eq 3.16. As we shall see in the following, line-width studies have in fact given some information about the gi. Relatively few measurements have been made at different values of the magnetic field so that the contribution to the total width from eq 3.16 is usually not known, and other types of contributions to the line widths are required to evaluate the g-tensor components. One of the useful results of the theoretical calculations of g values for planar aromatic h y d r o ~ a r b o n , ~however, 8 ~ ~ ~ + ~is~ that the out-of-plane component g3 is less than the mean of the in-plane components, g3 < (l/z)(gI 92). This result will be used in the analysis of some of the experimental data (see below). The anisotropic intramolecular dipolar interaction is treated in a manner similar to that for the g-tensor interaction, but it is somewhat more complicated. The contribution to the spin Hamiltonian can be written as the sum of products of two types of factors, one a function of the electron- and nuclear-spin operators, and the other a function of the spin-density distribution and the geometry of the radical. The spin for the ith nucleus are operators Ai(m)
+
where the operators Sf are given in eq 3.14, and the I,, are defined similarly. The other factors are where the Yzm(B',4') are the spherical harmonics of order two (Condon and Shortley definition and sign see, and the esr spectrum is obtained a t (43) If T R N 5 X X-band frequencies (YO N 9200 Mc/sec), (worR)2 N 9, and the nonsecular terms ~ ( w o make ) a contribution j(W0)
= 11
+
~02TR2]-1j(~)
which is about lo% of the [j(o)l contribution, Equations for the nonsecular line-width contributions are given in ref 3. (44) J. H.Freed, J. Chem. Phys., 41, 2077 (1964). (45) M. S. Blois, Jr., H. W. Brown, and J. E. Maling, Arch. Sci. (Geneva), 13, 243 (1960). (46) SI. 3. Stephen and G. K. Fraenkel, J. Chem. Phys., 3 2 , 1435 (1960).
Volume 71. Number 1
January 1967
GEORGE K. FRAENKEL
156
convention4'), and the indices m = =k2, fl, 0 in the Dt(m)correspond to the superscripts 7 2 , 7 1, 0 of the A's. If the spin operators in eq 3.19 are written with etc.) corresponding to primed components molecule-fixed axes, the expectation value over the electronic wave function of Wi(D), eq 3.3, for the ith nucleus (with electron-nuclear distance rt'), can be written (&I,
2
=
-E( - l)"I y e (y&Dt(m)At(-m) (3.21) m= - 2
where y r is the magnetogyric ratio for the ith nucleus. This result, which follows by straightforward algebraic manipulation, is based on the assumption that the electronic wave function can be written as a product of orthogonal one-electron functions so that is just the spatial part of the wave function for the unpaired electron. The products of spin operators in the molecule-fixed (primed) axes must then be transformed to space-fixed (unprimed) axes, just as in the transformations carried out on eq 3.5 for the g tensor, and averages have to be performed over the time-dependent angular functions in the appropriate correlation fuiictions. The spin operators retain the same form in the two axis systems, but of course the dipolar coefficients Di(m) are a property of the radical and must be calculated in the molecule-fixed frame. Their evaluation is discussed in later sections. When there are several nuclei present, the spin Hamiltonian contains a sum over the ,Xi@) for all the nuclei, and the correlation function therefore depends on cross terms between pairs of nuclei. It is found that the spectral density for nuclei 2' and j is3
+
(3.22) where spherical molecular tumbling motions have been assumed, 7 R is given (approximately) by eq 3.17, and48 T.i
=
(1/2~)lYe1 rtfi
(3.23)
The different terms in eq 3.19 produce a variety of effects. First there is the term ILZSz from Ai(') which does not cause transitions and makes only a secular contribution to the line widths. Then there are the nonsecular terms, Le., those which involve electronspin transitions. The Ii& terms from Ai(*') cause transitions at frequencies ~ W O and , although the I,, X Ss terms from A4(0)and the I&+- - terms from At(*') also involve nuclear-spin transitions, the frequencies The remaining terms are slpproximately Ii*S, (from A and - ( 1 6 / d ~ u b ' D ) ( 0 ) . In contrast to these examples for protons, the entries in Table I11 show that two I4N nuclei which are not completely equivalent cause line-width variations of a more complicated type than if they were completely equivalent. Thus the coefficients of juu(D)(0) for the M = *2, f l , and 0 lines are ( 2 2 / 3 ) , ( 1 7 / 3 ) , and ("/9) (with a smaller contribution for the M = f 1 lines than for the M = 0 line), and the coefficients of j,,'D'(0) are 16/3, 0, and - 3 2 / 3 . These are to be compared with for juaCD)(O) when the coefficients (38/3), (17/3), and two nuclei are completely equivalent. When second-order shifts are important, these results have to be modified. The procedure is complicated, however, and formulas have not been developed for specific examples.g 4. Inequivalent Nuclei. When there are inequivalent nuclei, we proceed as in section II.B.3. The nuclei are collected into equivalent groups according to the values of the average hyperfine splittings. The n, nuclei in group r have average hyperfine splitting a,, the n, nuclei in group s have average splitting li, # a,, etc. Each equivalent group is also divided into subgroups ra, r,, . . . ; sg, sh1 . . ., of nuclei which are completely equivalent with respect to the dipolar interaction.04 Quantum numbers J,,,&!,* for eachcompletely equivalent subgroup are employed [section III.C.31, and the relaxation matrix is a function of all these
+
+
LINEWIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
quantum numbers. As in section III.C.3, t'his matrix is not diagonal if there are two or more different completely equivalent subgoups in any of the equivalent groups, and simple results are only obtained by averaging over the different components of each line that occur within each equivalent group. The resulting expression for the line widths is
+
uicv C(Sl,3)jrurU'D'(0)(~~T,JfT,>M,}
C CC U (8/3)j,,sp(D)(~)(~r,~,,)M,M, + T f 8
U
C (16/3)jl(DG)(0)R~fMr + ('/3)j'G'(0>~02f
T2,o-l
(3.53)
7
where the summations are over all the different equivalent groups T , s, . . and over all completely equivalent in each equivalent group. The subgroups u, v, . averages over AI, in the first two terms are given by eq 3.47 to 3.49 with M u , J,, and M replaced by M,,, JT", and M , = EMT, (3.54) u
As before, all combinations of the M,, M,, . . , for which eq 3.54 holds are included in the sums for the averages. The average over M,Ms is ( i ~ f r u ~ l f s J . ~ , ~ ~=~ lD(JI7) l,
IMru7~M,,9 M,,,Msh,
..
-'D(M,)
-'
D(My,,Mr,, . . .) D(Ms,,Msh,.. .) X
M,M,,
(r # s)
(3.55)
As an example of these somewhat formidable expressions, we consider a radical such as the p-dinitrobenzene anion containing two completely equivalent 14N nuclei and four equivalent protons divided into two completely equivalent pairs. All the terms in eq 3.53 except those involving the average over M,M, have been considered in previous sections. Since the two nitrogen nuclei are completely equivalent, there is only one r#s spectral density of the type jrusg(D),
163
The line-width expression53is, from eq 3.44, 3.45, 3.503.53, and 3.57, with j t j written for j i j ( 0 )
+
(TZ-I(MNJ~H))MN,MH = ~xN(~'v(MN) (*/3)jHH(D)raeMH2 ('6//3)jNH(D)~flj~fH t (16/3)&[jN(DG)MN jH(DG)MH]f (3.58,)
+
+
x
where
The only term appearing in eq 3.58a which would not, be present if either the nitrogen nuclei or the protons were the only nuclei with magnetic moments in the radical is the cross term proportional to MNMH. This type of term has been useful in determining the relative signs of hyperfine splitting# and in checking the internal consistency of line-width datalK3 as discussed in section IV.
IV. Applications This section is devoted to a brief account of the experimental data on line-width variations and their applications to problems of structural interest. We again treat the effects of the anisotropic g-tensor and dipolar interactions separately from those of modulations of the isotropic hyperfine splittings, but for historical reasons reverse the order of presentation from that given above. Where appropriate, the combined effects of the two types of interactions are also considered. A . Anisotropic g-Tensor and Dipolar Interactions. Variations in the line widths among the different hyperfine lines in the esr spectra from free radicals in solution were first reported in work with Venkataraman@on the spectrum of the p-benzosemiquinone ion. Further studies also showed that the different lines in this and related radicals saturate a t different rate^.^^,^^ McGarvey was the first to observe line-width variations in the solution spectra of inorganic ionsC9 exhibiting hyperfine splittings from only a single nucleus. McConnel170 interpreted RlcGarvey's results by applying the theory of Bloembergen, Purcell, and Pound4$ ~
~ X (D' H (0)
=
~ N H ~ ( ~= ) (jO~) ~ ~ ( ~ ' (3.56) (0)
where N refers t>othe two I4Nnuclei (positions 7 and S), while Ha and H, refer to the two completely equivalent subgroups of protons (subgroup a: protons at positions 2 and 5; subgroup b: protons at positions 3 and6). Thus (MNII/!Ha)MNMH = ( M N M H b ) M N M H = D(lMH)-'CD(l~Ha1n/IHb)MNMHa = (~/Z)MNMH MHuMHb
(3.57)
at,
(64) For all nuclei i in the equivalent group T , a, = D,(O) = D,(o), jzl(D)(w)= j7v(D)(w), jtcnG)(w) = jr(DG)(w). For all nuclei i , j in the completely equivalent subgroup ra of T , D%(*z)= Dra(i2), j,,(")(w) = j,a,a(D)(w) = j,r(D)(w), and similarly for nuclei k, I in subgroup T b of T , but # Drb(*z),andjlk(D)(w)= j7a,b(D)(w) Z j,?cD)(w). (65) B. L. Barton and G. K. Fraenkel, J . Chern. Phys., 41, 695 (1964). (66) B. Venkataraman and G . K. Fraenkel, J . Am. Chem. Soc., 77, 2707 (1955). (67) B. Venkataramsn, Thesis, Columbia University, New York, N. P.,1955. (68) G. K. Fraenkel, Ann. A'. Y.Acud. Sci., 67,546 (1957). (69) B. McGarvey, J. Phvs. Chem.. 60, 71 (1956). (70) H. M. RlcConnell, J . Chem. Phys., 25, 709 (1956).
Volume 71,Number 1
January 1967
164
to an axially symmetric spin Hamiltonian of the type familiar from earlier investigations on single crystals of inorganic ions, and extensions of the work were then made by J l ~ G a r v e y . The ~ ~ first correct treatment of the single-nucleus case was given by Kivelson,** who used the Kubo and Tomita31 theory and attributed the width variations to the anisotropic g-tensor and electron-nucleus dipolar interactions discussed in section 111. McConnell's theory had employed essentially the same interactions but was restricted to the axially symmetric case and suffered from the inadequacies of the Bloembergen, Purcell, and Pound theory of line widths. Additional experimental work by Rogers and Pake72on the vanadyl ion provided good confirmation of the theory. The eight well-resolved hyperfine lines in the vanadyl acetylacetonate spectrum exhibit large line-width variations, and recently Wilson and K i v e l ~ o nhave ~ ~ shown that the widths can be fitted by a cubic equation41in the quantum number m. It was not clear from the earliest work how to extend the theory to take into account the degeneracies which occur in free-radical spectra, and a number of additional experimental and theoretical investigations were needed before a full understanding could be developed. A careful quantitative study of the line-width variations and saturation behavior of the spectrum of the p-benzosemiquinone ion was undertaken with Schreurs and B l ~ m g r e n , ~ * ~and ' ~ J *a theoretical treatment of the saturation effects was developed with S t e ~ h e n . ~The ~ , ~line-width ~ theory was extended by IGvelson,2b and later was modified in work with Freed3 along the lines presented in section 111. McL a ~ h l a nhas ~ ~more recently also written on the theory. Other saturation studies were performed by Lloyd and Pake,'17 but these were not concerned with variations among the hyperfine lines. The newer developments3 indicated that the theory of saturation for free-radical spectra in solution was in need of modification, and a reinvestigation of saturation behavior has been undertaken by Freed.78J9 All of the original studies were directed toward the experimental determination of the form of the linewidth variations and to an understanding of the proper theoretical interpretation. After it became clear that the anisotropic g-tensor and dipolar interactions could account for the results, other investigations were undertaken in whirh line-width studies were used to obtain structural information. One of the most important of these applications is the determination of the signs of the isotropic hyperfine splittings, and two different effects have been used for this purpose.s0 The first depends on the cross term between the g-tensor and anisotropic electron-nuclear dipolar interactions and The Journal of Physical Chemistry
GEORGE E(. FRAENKEL
the second on the cross term arising from the dipolar interactions of the electron with different nuclei. de Boer and Mackor,sl following the theoretical treatment carried out with Stephen,46determined the sign of the splitting in the a position of the naphthalene negative ion by showing that the high-field 13C lines in an enriched sample were broader than the lowfield lines. Equation 3.26 is applicable here for the pair of 13Clines (m = fl / Z ) arising from the same proton lines, and the difference in the widths of the highand low-field lines results from the linear term in m with coefficient B. Only the local spin density makes an appreciable contribution to the spectral density j C D 0 ( O ) , so that B is given by eq 3.30. Assuming on theoretical grounds that g(O) is negative and p is positive, it follows from the experimentally determined positive value of B that the sign of the splitting constant, - e, is positive, in agreement with the theory of 13Csplittings developed by Karplus and Fraenkel.52 de Boer and Mackor also found that the 13C splitting at the /3 position in the naphthalene negative ion is negative, which is again in agreement with the theory of 13C splittings. The same procedure cannot be used to determine the sign of the 13C splitting from the y positions (9,lO) because the spin density a t these positions is small and of uncertain magnitude. It still might be possible, however, to establish the sign of this splitting by using the complete expressions, eq 3.24, 3.29, 3.32, and 3.36 (see below). The de Boer and Mackor procedure involving the contributions from the local spin density alone has also been applied to determine the signs of some of the 13C splittings in the anthracene positive and negative i0ns,5*,5~ in the benzene negative ionlS2and in several semiquinone ions.*3 The signs of 14N splittings in nitroaromatic negative ions13*MJ3and in nitrogen (71) B. McGarvey, J . Chem. Phys., 61, 1232 (1957). (72) R. N. Rogers and G. E. Pake, ibid., 33, 1107 (1960). (73) J. W. H. Schreurs, G. E. Blomgren, and G. K. Fraenkel, ibid., 32, 1861 (1960). (74) J. W. H. Schreurs and G. K. Fraenkel, ibid.,34, 756 (1961). (75) M. J. Stephen, ibid., 34, 484 (1961). (76) A. D. McLachlen, Proc. Roy. SOC.(London), A280, 271 (1964). (77) J. P. Lloyd and G. E. Pake, Phys. Rev., 94, 579 (1954). (78) J. H. Freed, J . Chem. Phys., 43, 2312 (1965). (79) J. H. Freed, t o be published. (80) The signs of the hyperfine splittings in solution spectra of free radicals can sometimes also be determined by the variations of hyperfine splittings with solvent.24~64~83 Nmr techniques have also recently been employed.88 (81) E. de Boer and E. L. Mackor, J . Chem. Phys., 38, 1450 (1963). (82) J. R. Bolton, private communication. (83) M. R. Das and G. K. Fraenkel, J . Chem. Phys., 42, 1350 (1965).
LINE WIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
heterocyclic anions and cations60~65~84 have been established in a similar manner. These applications all depend on the theroretically determined result that, g"' is negative for hydrocarbon radicals, but the extension to nonhydrocarbon radicals is somewhat less certain. In the case of the p-dinitrobenzene anionj3 and the pyraeine cation,65 the signs were determined using another technique (see below), thus confirming the assumption about gC0) for these radicals. The studies on the semiquinones with Das83showed that some of the 13Csplittings had signs which differed from those determined using molecular orbital calculations of the spin densities adjusted to fit the proton s p l i t t i n g ~ , ~ 'and 5 ~ *demonstrated ~~ that considerable caution must be used in employing such calculations to estimate the spin densities at positions not directly amenable to evaluation from measurements of hyperfine splittings. This work also showed that the 13Csplittings in the carbonyl position can change in sign as well as in magnitude when the solvent is modified. In contrast to these investigations involving 13Cand 14Snuclei at positions with large local spin density, it is quite difficult to evaluate the signs of isotropic proton splittings from the linear g-tensor-dipolar cross term because j c D G ) ( 0(eq ) 3.24) depends in detail on all of the dipolar coefficients Di(m)and also on the g-tensor components, y(7n). The same difficulty exists to a lesser extent for 13C and 14N nuclei at positions with small local spin density, as at the 9, 10 positions in the naphthalene negative ion. There is often considerable uncertainty in the calculated values of the Dr(m), and any errors in the D f ( m )are reflected in the determination of the g'"'. These latter quantities must be evaluated from the linear line-width terms [j(DG)(0) ] involving nuclei with splittings of known sign, and they depend directly on the Di(m). The one determination of this type,86although consistent with the theoretically predicted signs of the hyperfine splittings, is probably nor, reliable. The other method, based on line-width studies, of determining the signs of isotropic hypefine splittings makes use of the cross term resulting from the anisotropic dipolar interaction of the electron with inequivalent nuclei, and while it only provides a means of establishing relative signs, no assumptions have to be made about the components of the g tensor and frequently the signs of proton splittings can be evaluated. It was first used to establish the sign of the I4Nsplitting in the p dinitrobenzene anion.63 The line-width expression, given by eq 3.5Sa, contains a term proportional to the product M N M H which can be written as
165
EN H n l i N n l i H
(4.1)
in the empirical line-width expression, analogous to eq 3.26, when the width is expressed in field units (gauss) as a function of the spectral index numbers (eq 3.28) instead of the quantum numbers Mi.The " is given by quantity E
ai
ENH
(16/31 Y $ ) ~ N E H ~ N H ( ~ ) ( O )
(4.2)
and - E H are the signs of uN and u H , and is defined in eq 3.22. The local spin density makes the largest contribution to the dipolar coefDH(O)1 ficients for the nitrogen, and since I DH(*')l eq 3.27 with eq 3.22 show that
where
j"@)(O)
-eN
-
ENH= ~ ~ N ~ H (0) PXDH
~
(4.3)
where a is a positive constant and PS is the r-electron spin density on the nitrogen atom. For almost all ring protons, the P,. and QXi in eq 3.36b are positive (see Appendix I), and P,. > Q,.. Thus DH(O) is negative unless atoms with large negative spin densities are associated with large Dt(0) in eq 3.32. In the pdinitrobenzene anion, the spin densities at the protonsubstituted positions (position 2, 3, 5 , and 6) are positive so that uH is negative (eH = l),and all calculations indicate that the nitrogen spin density is also positive. is positive The line-width studies53 show that E" and thus E N = - 1 and uN > 0. Similar investigations of the pyraeine cation radicaP5
show that the splitting constants of the 14Nnucleus aN are of opand of the proton bonded to nitrogen posite sign and it was thus concluded that the u-ir parameter QN" in the relation UN" = Q N H ~ P X is negative. In the same investigation, analysis of the gtensor-dipolar cross term in conjunctmionwith this determination of the relative signs of aN and ax" was used to demonstrate that aN> 0 without making theassuniption that g(O) is negative, and it was shown that in fact g(O) < 0. A comparable analysis of the spectrum of the 2,6-dinitrophenolate dianion radical13 has been carried out to establish, under reasonable assumptions, that' the spin density at position 4 is negative. When the local spin density at one of the nuclei (84) J. C. M. Henning and C. de Waard, P h y s . Letters, 3, 139 (1962). (85) G. Vincow and G. K. Fraenkel, J . Chem. Phys., 34, 1333 (1961). (86) E. de Boer and E. L. Mackor, X o l . Phvs., 5 , 493 (1962); Coll. Ampere, 439 (1962).
Volume 7 1 , Number 1 January 1967
GEORGEK. FRAENKEL
166
involved in a cross term does not make the predominant contribution to the dipolar coefficients, the analysis is more complicated and less certain. A study of the toluene negative ions7 has been undertaken from which it was possible to show with a reasonable degree of confidence that the spin density at the para position is positive. This result is important in the theory of radicals with the unpaired electron in a nearly degenerate orbital, and shows that mixing of the symmetric with the antisymmetric orbital is sufficient to overcome the negative spin density of the antisymmetric orbital a t position 4. The opposite sign was found in the pxylene anion radical by using nmr techniques,s0f88 and thus, in agreement with the theoretical calculat i o n ~the , ~ symmetric ~ orbital makes a smaller contribution in this latter radical. The sign determination in the toluene anion is based on a quantitative rather than a qualitative comparison of calculated and experimentally determined quantities because of the explicit dependence of the spectral densities on the dipolar coefficients, and the uncertainty arises because of a lack of internal consistency in the results which is typical of line-width studies obtained up to this time (see below). Unfortunately, it is not possible to use nmr techniques8' to establish the signs in the toluene radical since for this ion the rather special conditions that are required which relate splitting constants to electronexchange rates cannot be satisfied experimentally. Studies of the contributions of the anisotropic dipolar inter:tctions to the line widths have also been employed to assign 13C splittings from nuclei present in natural abundance to the correct molecular posit i o n ~ . The ~ ~ principles involved in this procedure are discussed at the end of section 1II.B and in ref 55. Applications have been made to the anthracene positive and negative ions and to the p-xylene negative i0n.55 The same technique has been employed to assign '9F splittings and proton splittings in fluoro-substituted r a d i c a l ~ , ~and ~ ~ ~it ~is~ ~generally O applicable for distinguishing between splittings from nuclei with the same nuclear spin whenever their dipolar coefficients are sufficiently different. Since the spin densities on the carbon atoms of methyl groups are very small, the widths of the 13Clines from these carbon atoms exhibit particularly small anisotropic g-tensor and dipolar effects, and consequently the splittings from methyl group l3csatellites should in many instances be distinguishable from those caused by I3C nuclei at ring positions.
Nost of the aDDlications based on the contributions from the anisotropic g-tensor and dipolar interactions to the line Widths which we have discussed depend On qualitative or semiquantitative considerations, but A
I
The Journal of Physical Chemistry
it would be possible to obtain additional structural information of value if detailed quantitative experimental and theoretical investigations could be carried out. Of particular interest are the components of the g tensor, g(m), and the a-electron spin-density distribution. The former, which have been estimated by several investigator^,^^*^^ can be compared with Stone's theoretical cal~ulations.~~ Determinations of the spin-density distribution, or at least of some of the spin densities in a radical, by techniques that did not, depend on the use of u-a parameters (as for example Q C H ~in McConnell's relation asEr= Qc"~,), would be of considerable importance, and one approximate study of this type has been made on the 3,5-difluoronitrobenzene anion.b1 The ratio of the spin densities on the fluorine and nitrogen atoms was estimated from line-width measurements in order to evaluate the u--7~parameters for 19Fsplittings. Studies intended to yield either the g-tensor components or the a-electron spin-density distribution may, however, be subject to considerable errors. Recent ~ o r k ~has~ been , ~ performed ~ , ~ ~ in which enough linewidth parameters were evaluated to provide a means of testing the internal consistency of the experimental data and the calculations. This has been done by measuring three or more of the spectral densitiesj,,'D)(0) using line-width studies and then comparing these values with those calculated from the dipolar coefficients. The rotational correlation time T R is treated as an unknown parameter, and it should enter as the same constant of proportionality relating each of the spectral densities to appropriate functions of the dipolar coefficients. Similar consistency checks can be made using the dipolar-g-tensor cross terms j,(Do'(0) when enough data are available. The calculations of the dipolar coefficients require knowledge of the r-electron spin-density distribution, the geometry of the radical, and the form of the a-electron wave functions. I n certain cases, u-orbital spin polarization must also be i n ~ l u d e d . ~In~ ?addition, ~~ spherical shape must usually be assumed for the rotational motion of the radical, nonsecular contributions are often (87) M. Kaplan, J. R. Bolton, and G. K. Fraenkel, to be published. (88) E. de Boer and C. MacLean, MoZ. P h w . 9 , 191 (1965); E. de Boer and C. MacLean, J. Chem. Phys., 44, 1334 (1966). (89) R. L. Flurry and P. Lykos, Spectrochim. Acta, 18, 1378 (1962); W.D. Hobey, Mol. Phys., 7, 325 (1964); T. H. Brown, 31.Karplus, and J. C. Schug, J. Chem. Phys., 38, 1749 (1963); T.H. Brown and M. Karplus, ibid., 39, 1115 (1963); H. M. McConnell and A. D. McLachlan, ibid., 34, 1 (1961); W.D. Hobey and A. D. .Mctachlan. ibid., 33, 1695 (1960); A. AlcLachlan, Mol. Phys., 4,417 (1962). (90) A. Carrington, A. Hudson, and G . R. Luckhurst, Proc. Roy. SOC.(London), A284, 582 (1965). (91) R. J. Cook, J. R. Rowlands, and D. H. Whiffen, Mol. Phys., 7, 31 (1963-1964); P ~ O C . Chem. soc., 252 (1962).
LINEWIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
neglected, and line-broadening mechanisms other than the anisotropic dipolar and g-tensor interactions are assumed to have a negligible effect. In the few studies made,61,33,87 inconsistencies of the order of a factor of 2 or 3 were found for the value of T R needed to relate the spectral densities to the functions of the dipolar coefficients. The reasons for discrepancies as large as this are not understood, but at least part of the difficulty may arise from significant contributions to the line-width variations which result from modulations of the isotropic splittings. Modulations of the isotropic splittings were treated separately from the anisotropic g-tensor and dipolar interactions in sections I1 and I11 above, but the two types of effects can of course occur simultaneously. As a first approximation, it is only necessary to add the line-width expressions in section I1 for the modulations of the isotropic splitting to those in section I11 for the dipolar and g-tensor interactions (but including the constant term Tz,o-l only once). If there are appreciable modulations of the spin-density distribution, however, appropriate averages have to be performed over the dipolar coefficients, and there are also contributions from a cross term between the isotropic splitting and the average value of the g tensor. The formal treatment of these modifications is given in ref 3, but no calculations of this type have been performed for particular examples. B. Modulations of the Isotropic Splittings. Two independent observations were made which first indicated the importance of modulations of the isotropic hyperfine splittings in esr spectra. Bolton and CarringtonS2examined the spectrum of the durosemiquinone cation radical (1,4-dihydroxy-2,3,5,6-tetramethyl benzene cat ion) i.
and discovered that the 13 lines from the methyl group proton splittirigs are alternately sharp and broad, with the central line sharp; Le., if M is the quantum number for the combination of the 12 methyl group protons (-6 5 M 5 6))lines with M even are sharp and lines with odd are broad. Bernal and Rieger93s94 found a similar line-width alternation in the spectrum of the dinitrodurene anion (1,4-dinitro-2,3,5,6-tetramethylbenzene anion)
167
I
J
N&
If M is the quantum number for the two 14N nuclei, lines with M even are sharp, and those with M odd are broad. Both of these studies can be understood in terms of the theory presented in section 11. Bolton and Carrington assumed that the methyl group proton splittings depend on the orientation of t,he hydroxyl groups in the plane of the benzene ring as first suggested by Makig6to explain the observation of isomers in the spectrum of the terephthalaldehyde anion radical, (1,4-benzene dialdehyde anion)
r
CHOT
lQol
Similar phenomena in which the splittings vary with the orientation of substituent groups have also been found in other c ~ r n p o u n d s but , ~ ~the ~ ~ effects ~ ~ ~ ~are not completely understood. Bolton and Carrington gave simple qualitative arguments involving the rapid interconversion between conformations in which the hydroxyl groups are in cis and trans conformations with respect to each other
r
r
1
1
0000 2
L
L
1
and a more complete treatment has been given in ref 3. There are four different states, two cis and two trans, with two different splitting constants in the cis form (a~' and UII') and two in the trans form (aIT and ~ I I ~ ) If . the fraction of radicals in the cis form is much greater than the fraction in the trans form (or vice versa), the problem reduces to the two-jump model discussed in section 11, and there is an alternating line-width effect. In the extreme case of a large line-width contribution from t,he interconversion be~~~
(92) J. R. Bolton and A. Carrington, Mol. Phys., 5, 161 (1962). (93) I. Bernal and P. H. Rieger, results obtained in these laboratories. (94) J. H. Freed and G. K . Fraenkel, J . Chem. Phys., 37, 1156 (1962); J. Freed, I. Bernal, and G. K. Fraenkel, Bull. Am. Phys. Soc., 7, 42 (1962). (95) A. H. Maki and D. H. Geske, J . Am. Chem. Soc., 83, 1852 (1961); A. H . Maki, J . Chem. Phys., 35, 761 (1961). (96) P. H. Rieger and G. K.Fraenkel, ibid., 37, 2811 (1962). (97) N. Steinberger and G. K. Fraenkel, ibid., 40, 723 (1964).
Volume 71, Number 1 January 1967
GEORGEK. FRAENKEL
168
tween the states, so that the broadened lines become too wide to be detectable, the spectrum would appear to consist of only seven lines with M = k 6 , *4, 1k2, 0 and statistical weights 1:36 :225 :400 : 225 : . . . instead of 13 lines with relative intensities 1: 12 :66 :220 : 495:792:924:792:. . . . If the cis and trans forms both contribute to the spectrum and the average splitting in the two forms is comparable, a large linewidth contribution from the interconversion reactions would also result in a spectrum of seven lines, but now the statistical weights would be 1: 18:99 : 164 :99. . . . Since the experimental data were only qualitative, it is not possible to distinguish between these or other more complicated possibilities, and there is no direct evidence from which either the correlation times or splitting-constant variations can be evaluated. Other models have also been considered14 which produce alternations in the widths of the methyl group proton lines, and motions in which the hydroxyl groups are not always in the plane can cause variations in the widths of the hydroxyl group proton lines as well. Large effects for these lines were not observed, but the spectra were too complicated to exhibit small effects. Analysis of the methyl group motions indicates that still other kinds of line-width variations can be observed if the correlation time of their reorientation is slow enough. 15,98 The alternating line widths in the dinitrodurene anion are also most simply understood in terms of the two-jump models of section 11. One nitro group is assumed to be in the plane of the aromatic ring while the other is perpendicular to it, and then their orientations are interchanged. The in-plane nitro group has the larger splitting,g9and the resulting modulation of tha splittings causes the alternation in line widths. A number of different rotational models have also been analyzed,14 and semiquantitative experimental studies have been ~erf0rmed.l~I n addition to rotational motions, fluctuating complexes of the nitro groups with cations or solvent molecules can cause an alternating line-width effect.3~s~13~14 There are four simple structures for these complexes, one in which neither group is complexed, two in which one of the nitro groups is complexed with a solvent molecule, and one with both nitro groups simultaneously complexed X
X
The m-dinitrobenzene anion radical was also found The Journal of Physical Chemistry
to exhibit an alternating line-width eff ect,Io0 and this observation was particularly interesting because a study of a large number of nitro-substituted benzene anion radicalsg9showed that the hyperfine splittings in the m-dinitrobenzene anion were anomalous. An investigation of other nitro-substituted anions13~26~99 indicated that the line-width effects are correlated with the anomalies in the splitting constants, and it was possible to show that the m-dinitrobenzene anion and certain of its derivatives exist in solution primarily as unsymmetrically distorted or complexed species. For example, the m-dinitrobenzene anion might exist in four different rapidly interconverting formslol similar to those for the p-dinitrobenzene anion. The observed hyperfine splittings are not consistent with either of the symmetrical forms (completely complexed or completely uncomplexed), but are in accord with the splittings predicted for an average of the two singly complexed species.99 A major contribution from these singly complexed forms is also required to account for the alternating line-width effect. Anomalous line widths and hyperfine splittings have been observed in a large number of nitro-substituted benzene anions in a variety of solvents. 10~13,25,90,102-106 In some spectra, the alternating line-width effect is so large that not all of the lines are observed, while in others there is direct evidence from the hyperfine splittings that tight complexes are formed with cations. The extent of the alternating line-width effect usually increases as the temperature is lowered, corresponding to a slowing down of the modulating motions (and perhaps also to an increase in the magnitude of the fluctuations in splittings) with decreasing temperature. At low temperatures the fast-exchange limit may no longer be valid and the spectra become quite complex, while in the limit of very long correlation times it should be possible to obtain spectra corresponding to the individual forms which contribute to
(98) J. H. Freed, J. Chem. Phys., 43, 1710 (1965). (99) P. H. Rieger and G. K. Fraenkel, ibid., 39, 609 (1963). (100) J. H. Freed, P. H. Rieger, and G. K. Fraenkel, ibid., 37, 1881 (1962). (101) In work reported prior to the development of the general theory there were some misinterpretations of spectra exhibiting the alternating line-width effect as, for example, in C. A. McDowell, Rev. Mod. Phys., 35, 528 (1963). (102) I. Bernal and G. K. Fraenkel, J. A m . Chem. SOC.,86, 1671 (1964). (103) R. L. Ward, ibid., 83, 1296 (1961); J. Chem. Phys., 32, 410 (1960); 36, 1405 (1962). (104) S. H. Glarum and J. H. Marshall, ibid., 41, 2182 (1964). (105) C. J. W. Gutch and W. A. Waters, Chem. Commun., 39 (1966). (106) L. H. Piette, P. Ludwig, and R. N. Adams, J. A m . Chem. sot., 83,3909 (1961); 84,4212 (1962).
LINE WIDTHSAND FREQUEXCY SHIFTSIN ESRSPECTRA
169
the exchange process provided that there are only a applicable to the nitrobenzene series because the small number of such forms of importance. In most splittings, particularly those of the I4N nuclei, are very sensitive to the composition of the solvent.) instances, however, it has not been possible to observe de Boerloshas shown that there is an oscillation of the an alternating line-width effect in spectra at one temperature and then to obtain the spectra of one or more alkali metal cation between the two pa.irs of protons at A (or at B) as well as a jumping from A to B, and of the individual forms by lowering the temperature. Thus no conclusions can be drawn in such cases about was able to deduce the appropriate assignment of the splittings to the different radical positions in the spectra the validity of models involving jumps between a small number of different forms as compared to models of the individual forms. Another interesting study was based on essentially continuous motions. When the made by Fessenden and Schuler,lo9who concluded that individual exchanging species cannot be isolated, it is the spectrum of the vinyl radical (CH&H-) could not possible to determine the magnitude of the fluctuaonly be explained by a motional modulation of the tions in hyperfine splittings directly, and usually no hyperfine splittings. They postulated that an unreliable estimate of the correlation times can be made. paired u-electron and a hydrogen atom rapidly interThere is consequently considerable arbitrariness in change positions the interpretation of the data. I n the spectra of the H H H dinitrobenzene and related radicals which exhibit a \ / \ large alternating line-width effect, however, measurec-c. c-c . ~ ~ ~ ~ ments of the second-order frequency s h i f t ~(see / / \ section 11) can sometimes be employed to evaluate the H H H mean-square fluctuations in splittings, and these ambiguities can be removed. Second-order shift measureand from arguments based on the splittings in similar ments cannot be used in this way for the splittings radicals were able to estimate the magnitude of the arising from protons, and thus it is not possible to anafluctuations. Cochran, Adrian, and Bowersl'O conlyze the mechanisms involved in the dihydroxydurene firmed this analysis by studying the same radical cation spectra in detail. trapped in one of its forms in a rigid matrix. A number oi observations have been made, however, Bolton, Carrington, and Todd,"' and Harriman and in which the magnitudes of the fluctuations in hyperMaki112 have studied motional modulations of hyperfine splittings can be deduced directly without resort fine splittings in the region of intermediate exchange to second-order shift measurements. de Boer and rates, and have been able t o fit computed curves to 3Iack0r'~' found a line-width alternation in the specthe experimental data. The former authors investitrum of the pyracene anion radical gated the spectrum of the naphthazarin cation
__
A
@ X2 B H,
which they attribute to the modulation effects of the jumping of an alkali metal cation from one site (A) in the radical to another (B). Since the alkali metal hyperfine splitting was resolved in some spectra, it was possible to conclude that the exchange process was intramolecular in nature. The mechanism is essentially a jump process between two thermodynamically equivalent forms, arid in the limit of long correlation times, the spectrum corresponds to that of one of the exchanging forms so that all the splitting constants can be determined. de Boer and Mackor were able to obtain this spectrum by modifying the solvent and adjusting the temperature. (This procedure is not
A
B
over a range of temperatures sufficient to deduce the different splittings for the two conformations shown. (107) E. de Boer and E. L. hlackor, Proc. Chem. Soc., 23 (1963); J. Am. Chem. Soc., 86, 1513 (1964). (108) E. de Boer, Rec. Trav. Chim., T84, 609 (1965). (109) R. W. Fessenden and R. H. Schuler, J . Chem. Phys., 39, 2147 (1963). (110) E. L. Cochran, F. J. Adrian, and V. A. Bowers, ibid., 40, 213 (1964). (111) J. R. Bolton, A. Carrington, and P. F. Todd, Mol. Phys., 6, 169 (1963). (112) J. E. Harriman and A. H. hTaki, J . Chem. Phys., 39, 778 (1963).
Volume 71, Number I
January 1967
GEORGE K. FRAENKEL
170
The major line-width alternation is observed in the lines from the hydroxyl protons, but a small effect of this type is also found for the ring protons. The theory was developed in an approximate way using a four-jump model; A and B are two of the four conformations. Harriman and Maki studied compounds like the anion of bis(p-nitrophenyl) ether
as well as radical anions with other groups between the two benzene rings, and found complex spectra in regions of intermediate exchange rates. Yamazaki and Piette113 have studied the p-benzosemiquinone radical in a flow system at low pH and obtained results that can be interpreted as an alternating line-width effect arising from a dynamic equilibrium between species singly protonated a t the oxygen atoms. The M = = k l lines are not detected at some values of pH and the amplitudes of the remaining three lines are in the ratio 1:4 :1 [see section 1I.B and Table 111. de Boer and Praat114and Iwaizumi and I ~ o b e "have ~ investigated conformational interconversions of hydropyrenes by line-width studies. Observations of similar interconversions in a number of other radicals10gv116 have also been made. Although most of the investigations of the alternating line-width effect and related phenomena have only been qualitative, some quantitative studies have also been attempted.13~24~25~111 They are often complicated by the contributions from other line-broadening mechanisms, particularly the anisotropic dipolar and g-tensor interactions. Further work in this field will undoubtedly produce more reliable results which will provide better tests of the general theory and give sufficient information to specify in some detail the nature of the mechanisms responsible for the spindensity modulations. The interpretations of all the line-width effects considered in the preceding discussions are based on the assumption that intermolecular contributions are negligible. Two types of intermolecular phenomena occur which can cause difficulties (other than an overall broadening of the lines that tends to mask the variations in width from one hyperfine line to another). The first is electron exchange between the radical and a diamagnetic species of similar structure as, for example, an unreduced material and its negative ion. 16s117 The second is Heisenberg exchange between radicals.118-'20 Both effects cause the lines to broaden symmetrically toward the wings of the spect r ~ m ~ p 1 and ~ ~ can ,~~ introduce ~ , ~ ~ errors 1 in the interpretation of the experimental data. The Journal of Physical Chemistry
Appendix Calculation of Dipolar Coeficients. The formulas of McConnell and Strathdee56 can be expressed in terms of the P,. and Q.. of eq 3.34 ff as follows. Let ad = ZJ2d/2aQ
(A. 1) where a0 is the Bohr radius, Z, is the effective charge of the Slater orbital on atom K containing the 2 p ~ orbital, and R,; is the distance between the nucleus of this atom and the nucleus i for which the interaction is to be computed. If RKiis large (usually -2A or greater), the approximate formulas
Pk G [l - (9/~d')]Rd-~
(A.2)
(9/2~J)R,i-~ (-4.3) are sufficiently accurate. For smaller distances, it is necessary to use the exact formulas (R,, # 0) QKi
P,i
=
[l - (9/2a,i2)
Q,i
= [(9/2a,i2)
a,i3
+
+ M,i e~p(-2u,~)]R,i-~ (A.4) - N,i
exp(-2~~)1R,.-~ (A.5)
where
M,i =
4UKi2
+ load + 17
+ (8/aKi) + (g/a,.')
(A.6)
and
N,; = a,i3
+ 3ah2 + 6aKi+ 9
+ (9/uKi) + (9/2a,i2)
(A.7)
The Dk(m) are then obtained from eq 3.36 and the Dp(m)from eq 3.32. For R,. = 0, eq 3.27 for D'"' (local) are used. It is frequently convenient to express the Di(m) in units of sec-' instead of which is accomplished by using the f i defined in eq 3.23
Di(m)
= r i ~ $ ( m ) sec-1
(Am
In terms of the Dr(m), the spectral densities are (113) I. Yamasaki and L. H. Piette, J. Am. Chem. SOC.,8 7 , 986 (1965). (114) E. de Boer and A. P. Praat, Mol. Phys., 8 , 291 (1964). (115) M.Iwaizumi and T. Isobe, Bull. Chem. SOC.Japan, 38, 1547 (1965). (116) W. T. Dixon and R. 0. C. Norman, J. Chem. SOC., 4850 (1964). (117) R. L.Ward and 8. I. Weissman, J. Am. Chem. SOC.,76, 3612 (1954); 79, 2086 (1957); P. J. Zandstra and S. I. Weissman, J. Chem. Phys., 3 5 , 757 (1961). (118) K.H. Hausser, 2. Naturforsch., 14a, 425 (1959). (119) G. E. Pake and T. R. Tuttle, Jr., Phys. Rev. Letters, 3, 423 (1959). (120) J. D.Currin, Phys. Rev.,126, 1995 (1962). (121) J. Marovskis, research performed in these laboratories.
LINEWIDTHSAND FREQUENCY SHIFTSIN ESRSPECTRA
and
171
fluctuations in the magnitude of the g-tensor components (e.g., from solvation) as well as the tumbling motions which average these components, and if both dynamic processes can be treated as statistically independent, then it is found that for the components of the g tensor affected by both processes one has a reduced correlation time T , T , / ( T ~ where T , and T , are, respectively, correlation times for the tumbling motions and fluctuations in magnitude. 2. We have done a simple experiment which rules out the possibility that the anomalous broadening of the benzene anion is due to a g-tensor mechanism. For a sample with a minimum line width of about 0.37 gauss a t 9.4-kMc microwave frequency, we obtain line widths of about 0.78 gauss at 3.5 khlc. If the data are fitted to a simple equation of form: line width = a bHZ, where a is the field independent component and (bH2) the g-tensor component, we find that at X-band, ( b W ) = 30 mgauss, so that a is the main contribution. Kivelson has recently proposed an Orbach-type process as the source of the broadening. Since it is field independent it is not inconsistent with the experimental result.
+ I,)
where
(A. 11)
+
[Xote: (4n2/5) = 7.89568 and ( 2 ~ l f l e ( l O ~ f=i ) 2.25586 X lo6 gauss/sec-'. 1 Values of ( T - ~ ) , needed to evaluate D,'m'(local) have been collected by Morton, et a1.lZ2
Discussion A. CARRINGTON (University of Cambridge). 1. Do any particular difficulties arise when there is modulation of both the principal directions and magnitudes of the g tensor? It seems possible that such a problem might arise in radicals with electronic degeneracy. 2. There is considerable interest in the liquid phase spectra of very slowly tumbling molecules, for example, proteins. The current relaxation-matrix theory is not really applicable to these situations; what progress is being made toward a more satisfactory theory? G. K. FRAENKEL. 1 . In the absence of degeneracy there is no particular difficulty provided that the modulating and tumbling motions are uncorrelated. This problem is discussed in the work with Freed on the general theory of free-radical line widths. When there is degeneracy, the whole treatment may become quite complex, and recently Kivelson has studied the benzene negative ion in detail. 2. These problems can probably be handled by the general techniques developed for the treatment of solid state phenomena as discussed by Abragam, Slichter, Kubo and Tomita, and others.
J. H. FREED(Cornel1 University).
1. If there are both
P. H. RIEQER(Brown University). I would like to add that Allendoerfer and I have measured dynamic frequency shifts for the radical anions of dinitrodurene, dinitroisodurene, aminodinitromesitylene, as well as dinitromesitylene. Our results are in good qualitative agreement with yours in that half the modula' / ~approximately equal to the average tion amplitude ( ( 6 ~ ) ~ ) is splitting with the exception of dinitrodurene for which ((ba)2)1/z is somewhat larger than (a). The correlation time for the dinitroisodurene anion is somewhat longer than for those you mentioned, about 5 X 10-9 sec in DMF, and increases rapidly on the addition of water to the solution.
R. E. COFFMAN(Augsbury College, Minneapolis, Minn.). Does it follow that a g shift is generally to be expected when a line-width alternation exists? G. K. FRAENKEL. The g value is determined by the position of the center of the spectrum in the absence of hyperfine splittings. When there are hyperfine splittings, there is always a correction to the position of the central line which arises from second-order corrections, but its amount depends in detail on the line-broadening mechanisms. (122) J. R. Morton, J. R . Rowlands, and D. H. Whiffen, National Physical Laboratory, Teddington, England, Bulletin No. BPR 13
(1962).
Volume 71,Number 1
January 1967
