Rates from the coalescence of relaxation times

JOURNAL

OF

MAGNETIC

RESONANCE

38, 233-244

(1980)

Rates from the Coalescenceof Relaxation Times JOSEPH B. LAMBERT AND JOE W. KEEPERS Department of Chemistry, Northwestern University, Evanston, Illinois 60201 Received July 5, 1979 A method has been developed for obtaining intramolecular exchange rates in liquids from the coalescence of nonselective 13C spin-lattice relaxation times for the general case in which rI at the two sites can have any relative values. Regression analysis and curve fitting of the relaxation data to the double-exponential solution of the Bloch equations modified for exchange yields the exchange rate k and the true spin-lattice relaxation times for each site (T: and Ty ). The method has been applied to the exchange of methyl groups by amide bond rotation in dimethyl-formamide (DMF). Comparison of the true relaxation times with those obtained by single-exponential analysis (T: (eff) and 7’: (eff)) at several temperatures provides insight into the double-exponential behavior of relaxation times in exchanging systems. A computer simulation model of the relaxation behavior demonstrates that the peculiar temperature dependence of rI (eff) at sites A and B is a superposition of exchange effects on dipole-dipole relaxation. For DMF, the exchange rate constants derived from the double-exponential analysis agree reasonably well with the literature values. All r, measurements are made in the temperature range below the initiation of any lineshape changes.

Intramolecular rate processes have been studied by nuclear magnetic resonance (NMR) lineshape techniques with considerable success (for reviews, see Ref. (1)). The method is generally limited to barriers between 5 and 25 kcal mol-*. NMR relaxation measurements offer promise of expanding the barrier range in both directions. Faster rate processes, whose motional frequency is comparable to or larger than the Larmor frequency wo, include methyl rotation and internal chain segmental motion (2). Slower rate processes, whose motional frequency is less than wo, include many types of hindered rotation. This paper addresses the latter category, which has seen few applications. An early approach to expanding the upper barrier range of observable rate processes was that of For&n and Hoffman (3). Their method involved saturation by double resonance of one site while intensity perturbations are measured at the other site. The exchange rate (k) and the spin-lattice relaxation time (TJ are obtained, provided that the relaxation times at the two sites are equal, T? = T? (4). Successful applications have included ring reversal in cyclohexane (S), [2.2]metaparacycIophane (6), and cis-decalin (4). Fortunately, in these systems the equality of T? and T? was met, but such is not always the case. Other authors have developed relaxation methods for obtaining exchange rates when T? and TB are very different, e.g., 5 T? > T? (7). This requirement is seldom met in ‘H cases and is rarely met in 13C cases (except when the exchanging sites differ considerably in the extent of 233

0022-2364/80/050233-12$02.(K)io Copyright 0 IYXO by Academic Press, Inc All right< of reproduction in any form reserved Printed tn Great Br~rarn

234

LAMBERT

AND

KEEPERS

substitution). In the typical organic system, the relaxation times of exchanging sites are neither equal nor very different, so that a procedure is lacking for quantitative examination of most cases. In 1974, Levy and Nelson (8) described the coalescence of relaxation times for the methyl groups in dimethylformamide (DMF). These observations provided an intriguing analogy to lineshape coalescence (LSC). We wanted to explore whether kinetic data could be extracted from relaxation time coalescence (RTC). To establish the viability of the method, we decided to study amide group rotation in DMF through the temperature dependence of the spin-lattice relaxation times. If success could be established in this system, the method could then be applied to new systems. From this investigation we developed and herein describe a relaxation method for determining the rates of high-barrier intramolecular exchange processes that is independent of the relative magnitudes of Tf and Ty. This method provides not only the exchange rate k but also the individual spin-lattice relaxation times at each site, Tf and T?. The method provides considerable insight into the effects of exchange upon the real and the effective spin-lattice relaxation times at each exchange site as a function of temperature. THEORY

In the tions for displayed sequence

absence of an exchange process, Bloch’s differential and integrated equathe t component of magnetization (M,) (which is the signal normally in pulsed NMR spectra) appropriate for an inversion-recovery pulse (180-t-90”-t’) are given by the equations (9,10)

El where MO is the equilibrium z magnetization, Tl is the spin-lattice relaxation time, and t is the time after the 180” pulse. The form of Eq. [2] lends itself to direct solution by either a nonlinear regression or a linear logarithmic analysis. This singleexponential expression is commonly used to obtain Tl in packaged computer programs. Exchange of nuclear spins requires additional terms in the differential Bloch equation to allow, in the case of site A, for loss of magnetization from A to B and gain of magnetization from B to A. The differential expressions including exchange are given by Eqs. [3a] and [3b], respectively, for the B and A sites, and their integrated solutions are given by Eqs. [4a] and [4b],

--=dA4: (t) dt

04: 0) -M: T?

) -- @ (t) + M? ~ (I) , 78

TA

(3bl

RATES FROM

RELAXATION

:‘J5

TIMES

In these expressions, it&, M,,, Ti and t retain their meanings, T is the mean exchange lifetime (k = l/r = l/r*+ l/&, Al and A2 are the effective time constants governing the relaxation process, and the ci are complex coefficients dependent on several factors. For an inversion-recovery pulse sequence and for the specific case of dimethylformamide, the effective time constants and the coefficients are given by the equations h1,2=t[(l/T?

+1/T:

-4(l/T?TY cl-

c2 =

-2i@(l/T’:

+(1/7)(1/T?

+1/T-A*)+ (AI -AZ)

cg= l/T:

l/T?

+1/T?

+2/7)2

+ l/T:)))“‘], 2h4: 701 -Ad’

2it4$

2kf:(l/T:+l/~--h~) (AI-AZ)

cd=

+2/7)f((l/Tf

-T(A,-AZ)'

[6h']

+ l/~-h~,

[6C]

+l/~-h~.

f6d]

The major contrasting element between exchanging and nonexchanging spin systems is that in the former case the relaxation process is governed by two exponential terms (Eq. [4]). This double-exponential form reduces to a singleexponential form outside the critical kinetic range in which the exchange lifetime is comparable to the actual relaxation times, i.e., when T is either very long or very short with respect to Tf' and Ty. In either extreme, the relaxation behavior no longer depends on T, so that kinetic analysis is precluded. In the following sections we will demonstrate how these equations can be used to produce kinetic information within the double-exponential framework for any relative values of T? and T?. RESULTS

Dimethylformamide (1) was chosen as the subject for this study for three reasons First, it produces a simple one-peak i3C spectrum for each exchange site, under conditions of *H decoupling. Second, reliable kinetic data have been reported, so that our results can be compared with lineshape results (11). Finally, Levy and Nelson (8) made relaxation assignments and observed the coalescence process. No kinetic analysis of the data was made and only three temperatures were studied. Structure 1 shows the assignments made by these authors at 25°C. The difference of a factor of 2 between the methyl relaxation times is as large a difference as is typically found in organic systems. The larger value for the methyl A that is cis to the carbonyl oxygen was attributed to a faster rotation permitted by rotameric interactions or to anisotropic tumbling (8). These same authors observed an average Tl of about 18 set at

72°C.

LAMBERT

236

AND

0 II C

KEEPERS

Tf’ = 17.8 set T? = 10.5 set

(CHLJ)A

/\/

H

N I (C&)B 1

Nonselective inversion-recovery experiments were performed on DMF at 2 1,35, 46, 53,56, 71, and 81°C. According to the usual single-exponential formulation of Eq. [2], we calculated the effective relaxation times by local modifications of the Gerhards-Dietrich program (12). In the absence of exchange effects, the resulting figures would be the actual relaxation times. They are listed in Table 1 as “TI (eff)” and are plotted in Fig. 1 as a function of temperature. The number at ambient and high temperatures are in excellent agreement with those of Levy and Nelson (8). The temperature dependence (Fig. 1) illustrates the relaxation time coalescence brought about by the exchange process. Whereas T? (eff) exhibits the normal increase with temperature (doubling between 21 and 81°C) that is expected for dipole-dipole relaxation, Tf’(eff) shows an unusual sigmoid behavior. It begins to increase between 21 and 46°C as if it were subject to dipole-dipole relaxation, then it decreases between 46 and 56”C, as if spin-rotation relaxation were taking over, but it resumes an increase between 56 and 81°C. The result of this behavior is that the rather larger relaxation time of Tt(eff) at 21°C coalesces with that of T?(eff) at 81°C. Such a temperature dependence is different from any reported before. We felt TABLE

RELAXATION T (“(2 Tt T? Tf Ty

21

(eff) (set) (eff) (~4 (set) (set)

16.7 9.5

7 (set)

MOA MF l/Al (se4 1 /A2 (se4 Cl c2 c3

;OE

(A)

NOE

(B)

0 Error

bars for the

PARAMETERS

35 18.0 11.6 18.7 11.1 77.9 134.9 142.1 9.2 15.9 -171.8 -112.3 -0.0048 -0.041 2.43kO.12 2.43ctO.12

1

FOR DIMETHYLFORMAMIDE TEMPERATURES 46 18.4 13.5 24.0 11.3 24.1 116.8 114.5 4.4 16.2 -35.2 -194.0 -0.0059 -0.11

T, (eff) are given in Fig. 1.

53 17.8 13.6 24.3 11.0 11.9 99.7 96.9 3.9 15.9 -25.9 -168.0 -0.071 -0.12

AS A FUNCTION OF

56 18.1 15.6 25.2 12.0 9.0 117.5 108.2 4.0 16.9 -18.4 -198.0 -0.075 -0.12

71

81

17.8 17.1

19.0 18.6

2.52hO.05 2.58jzO.05

RATES

0.92 /I--2.8

FROM RELAXATION

TIMES

3.2

3.0

3.4

lOOO/ T (OK-’ )

FIG. 1. The single-exponential temperature.

(effective) relaxation

times, 7’;(eff)

and ry(eff),

as a function of

that this behavior was brought about by the exchange process and that these observed relaxation times (Tr(eff)) are not the true relaxation times. In order to demonstrate this hypothesis, we analyzed the data according to the rigorous double-exponential expression of Eq. [4], which takes exchange into consideration. A nonlinear regression procedure (see Experimental) allowed variation of various unknowns to fit the observed data. In a three-parameter fit, we varied 7, T?, and T?. In a five-parameter fit, we additionally varied Mt and Mz (in our previous single-exponential work in nonexchangingsystems, we had found that more accurate results were obtained when both T1 and MO were allowed to be fitted by the data (13)). The three- and five-parameter fits gave essentially identical results. Physically meaningful solutions were obtained only at the intermediate ternperatures, 35,46,53, and 56°C since 7 becomes either much larger or much smaller than T1 below and above the range. The values of the mean lifetime r and the true relaxation times Tf’ and Ty obtained from the five-parameter fit are given in Table 1, along with the values of Mt, Mz, ci, and Ai. Figure 2 illustrates the temperature dependence of the true Tf and T?.

LAMBERT

238

30

AND KEEPERS

3.2

3.1 lOOO/

FIG.

2. The double-exponential

33

T (OK-’ I

(real) relaxation times, rf

and TF, as a function of temperature.

The true (double exponential) relaxation times (Ti) can be converted to the effective (single exponential) relaxation times (Ti (eff)) according to the following procedure. The values of T?, T?, and 7 from Eq. [4], as listed in Table 1, are used to generate calculated magnetization amplitudes at various times r after the 180” pulse. These amplitudes in turn are used to calculate Tf’(eff) and T? (eff) from the single-exponential expression Eq. [2]. The resulting values are essentially identical to those listed in Table 1. The same, nearly linear behavior of TF(eff) and the sigmoid behavior of Tt (eff) are obtained. This exercise serves to demonstrate that the single- and double-exponential fits correspond to the same set of spectral intensities. The rate constants (k = 7-l) from the double-exponential analysis are plotted in Fig. 3. The line in the plot is from reported lineshape measurements (II). It is seen that the rate constants from the double-exponential analysis of the relaxation times in a temperature range in which lineshapes are unperturbed fall close to the line extrapolated from lineshape analysis rate constants.

RATES

FROM

RELAXATION

TIMES

139

-I c

-2.c

T E Y

-3c .

x

-4 c

-5.c 2

3.2

30

34

lOOO/TF’K-‘)

FIG. 3. Rate constants from the double-exponential from the data in Ref. (II).

SIMULATION

analysis as a function of temperature. The fine is

MODEL

We want to be able to say that the shape of the plot of the effective (single exponential) relaxation times as a function of temperature (Fig. 1) is the result of the exchange phenomenon. Unfortunately, in such a plot the exchange effects are superimposed on the normal temperature dependence of dipole-dipole or spinrotation relaxation. Consequently, we carried out a simulation calculation that could isolate these effects. The variables in the complete double-exponential equation (Eq. [4]) are T, Tf, T?, Mt, and A4:. The ci and Ai can be calculated from these parameters (Eqs. [S] and [6]). For 15 points in the temperature range 19 to 8 1°C we calculated values of T from the literature (11). We set Mt = h4,” = 112.8 (these values and their equahty are arbitrary and do not affect the final result). For values of the true Tl we made interpolations from the log Tl vs reciprocal temperature plot (Fig. 2), imposing the linear dependence and negative slope (Arrhenius behavior) required by dip&edipole relaxation or its mathematical equivalent (spin-rotation relaxation would have required a positive slope). For the interpolation, the maximum and rn~~~ values of T, (10 to 12 set for B, 18.5 to 26 set for A) were taken from our earlier

LAMBERT

240

AND

KEEPERS

calculation of the true relaxation times. Thus the assumptions of this simulation model are that the temperature dependence of the log Tr (eff) vs 1 /T plot (Fig. 1) is the result of an exchange phenomenon superimposed on dipole-dipole relaxation or its equivalent. The boundary values for the MO and the T1 impair the quantitative accuracy of the model but not its qualitative utility. From these assumptions and boundary values, Eq. [4] gives a series of AI, as a function of the pulse delay t for each temperature. Each such series can be inserted into the single-exponential expression, Eq. [2], to give relaxation times from a linear fit at each temperature. These values of T1 (eff, sim) are not experimental, like those in Table 1, but are a creature of the simulation model. Figure 4 gives a plot of log T1 (eff, sim) vs l/T for these numbers. With the exchange rate constant set by literature figures and with relaxation assumed to be only dipole-dipole, the resulting

1

061

0 90

28

I

I

3.0

3.2 lOOO/

1

3.4

T(‘K-‘1

FIG. 4. Simulated single-exponential relaxation times, Tf (eff, sim) and Ty (eff, sim), calculated from a model of exchange effects superimposed on dipole-dipole relaxation (see text), as a function of temperature.

RATES

FROM

RELAXATION

TIMES

141

plot closely resembles in form the experimental single-exponential plot (Fig. l), in which the T? (eff) dependence is nearly linear and the T? (eff) dependence is sigmoid. The simulated plot is translated to somewhat lower l/ T1 values, because of the arbitrary assumptions concerning boundary vaIues.

DISCUSSION

OF RESULTS

The five-parameter fit gives rate constants (Table 1, Fig. 3) that are in qualitative agreement with the literature values. Because these measurements were made in the temperature range below any lineshape changes, this method should be applicable to cases in which the barrier is too high to achieve fast exchange. Furthermore, there are no limitations to the relative values of the relaxation times, Tf’ and T?, in contrast to earlier methods (3, 7). Because nonselective, rather than selective, pulses are used, the procedure can be carrried out on a wider range of NMR instruments. For the three- and five-parameter fits, convergence was only obtained in the temperature range 35 to 56°C. Apparently, outside this range 7 becomes too different from the Tl values. Within this range, it is useful to examine the relative magnitudes of the two terms of Eq. [4], which are respectively governed by A, and A 2. As the temperature increases, A1 becomes very large. Consequently, the cl term in Eqs. [4a] and [4b] becomes small. The expressions for Iki,(t) tend toward a single exponential at high temperature, as expected when T becomes much smaller than the TX values. At the same time, A2 remains fairly constant. In fact a quick calculation shows that A2 stays close to the populational average of l/T? and l/T?, i.e., 0.5 (11 T? + l/ T! ). Thus the second term appears to reflect the average behavior of the two relaxation times, and the first term contributes the distinctive character to the relaxation times. When the distinctive character dies out at high temperatures, only the average behavifjr is left. For example, at 56”C, most of the effects of the A 1 exponential die off in about 12 sec. In terms of the usual log M, vs t plot, this complementary behavior

of the distinctive (A 1) and average (A2) exponential terms is seen as curvature at short t, when A1 is still important, and as linearity at long t, when AZ becomes dominant. The nonlinear regression for the complete double-exponential expression of Eq. [4] hence is needed in order to fit the entire range of t. At very high temperatures, A, no longer contributes significantly and an average relaxation time results from singleexponential analysis. At very low temperatures, the distinctive behavior dominates, and the real and effective relaxation times coincide. The next section continues the examination of the low-temperature (large 7) region. It is also useful to examine the preexponential coefficients Ci as a function of temperature. The cl and c2 are of comparable magnitudes (172 and 112) at 35”C, but at 56”C, cl has become much smaller than c2 (18 and 198). Thus the effects of the c, reinforce those of the Ai. The double-exponential analysis (Fig. 2) gives a picture of the relaxation behavior quite different from that of the single-exponential analysis (Fig. 1). The effective behavior (Fig. 1) has Ty smaller and increasing rapidly and Tf’ larger and remaining about constant, so that they approach each other. The real behavior (Fig. 2) shows that TT is smaller and increases only slightly, whereas T? begins larger and

242

LAMBERT

AND

KEEPERS

increases rapidly. The real behavior is the result of normal dipole-dipole relaxation, whereas the effective behavior results from the superposition of the exchange effects on the dipole-dipole effects.

DISCUSSION

OF THE SIMULATION

MODEL

The simulation model answered the question concerning the cause of the unusual temperature dependence of the single-exponential relaxation times. By removal of the exchange lifetime T as a variable through the use of literature values, and with the assumption of dipole-dipole rather than spin-rotation, the temperature dependence could be reproduced qualitatively (Fig. 4). The observed NOE values (Table 1) are essentially the same for A and B and are about 80 to 86% full, suggesting that small contributions other than dipole-dipole may contribute but not alter the Arrhenius temperature dependence. The data generated by the simulation model can be analyzed for further insight into the coupled nature of the relaxation process. We calculated A 1 and AZ for a series of 15 T values with T]: = 10 to 12 set and T? = 18.5 to 26 set: for example, T = 500.6 set (Al = 0.1021 see-‘, A2 = 0.0560 set-‘), 250 (0.1024, 0.0557), 80 (0.1105, 0.0582), 25.2 (0.154, 0.0627), 10.6 (0.257, 0.0639), 2.5 (0.864, 0.0625), 0.9 (2.28, 0.0607). At fast exchange (T = 0.9 set), corresponding to high temperatures, the ratio Al/A* is 38 (for r = 0.01 it exceeds 1000). The relaxation process is governed by the second term almost entirely at these temperatures. Moreover, A2 (0.061) has approached the populationally averaged relaxation rate, 0.5(1/T? + l/ T? ) = 0.06 1, for the values of T1 used. At slow exchange (T = 500.6 set), corresponding to low temperatures, Al/A2 = 1.8. It is of interest that l/hi (9.8) is now very close to T? (10.0) and l/A* (17.9) is close to Tf' (18.5). The similarity of A 1 and AZcan mean either than both terms in Eq. [4] influence the relaxation process of either site or that the relaxation processes have been uncoupled, depending on the values of the coefficients ci. The latter situation occurs when the MF expression is dominated by the Ai term and the Mt term is dominated by the AZ term. Examination of the coefficients in the simulation model confirms uncoupled behavior in this temperature range; i.e., the sites exhibit distinct relaxation that can be characterized by a single-exponential expression. For example, at T = 500.5 set the ratio of the first to the second term in the M: expression (Eq. [4a]) varies from 21.0 for t = 0.1 set to 6.4 for t = 26 set; the ratio of the first to the second term in the Mf expression (Eq. [4b]) varies from -0.040 for t = 0.1 set to -0.012 for t = 26 sec. In contrast, the ratio stays essentially zero at 7 = 0.9 sec. This component analysis of the various terms shows that at fast exchange the “distinctive” Ai term is not important and that relaxation is governed by the “average” AZ term. At slow exchange the values are comparable, but the coefficients (Ci are such that B magnetization is governed by the A1 term (l/A1 has approached T? ) and A magnetization is governed by the AZterm (l/AZ has approached Tf ). The intermediate region is the kinetically important one from which exchange kinetics can be derived through double-exponential analysis.

RATES

FROM RELAXATION

SUMMARY

TIMES

243

AND CONCLUSIONS

Rotation about the amide bond in dimethylformamide influences the methyl relaxation process at temperatures below any perturbation on lineshapes. At sufficiently long mean lifetime r (low temperatures), methyl relaxation is governed by a single-exponential expression. For methyl-B magnetization, the first term in the double-exponential expression (Eq. [4a]) dominates, and for methyl-A magnetization the second term in Eq. [4b] dominates. Thus single-exponential analysis in the slow exchange limit gives the true Tf ( = l/A i) and Tf' ( = l/ A*). At sufficiently short mean lifetime (high temperatures), the first term for either site, governed by the exponential A r, becomes small. Thus both A and B magnetizations are determined by the second (h2) term, in which A2 has reached the populational average of the true relaxation rates, 0.5 (l/ Tf' + l/T? ). As a result, single-exponential analysis in the fast exchange limit produces the average of the true relaxation times. At intermediate lifetimes and temperatures, the magnetization must be expressed as a double exponential for both sites. Single-exponential analysis (Fig. 1) gives linear and sigmoid behavior for the respective temperature dependences of B and A relaxation. We have developed a nonlinear regression method, using Eqs. [4] to [6], for obtaining the mean lifetime 7, the true relaxation times T? and Tf', and the equilibrium magnetizations MC and ME at these intermediate temperatures. The amide rotational rates (k = l/r) are semiquantitative at best but compare favorably with values extrapolated from literature lineshape analyses at higher temperatures. A simulation model shows that the unusual linear/sigmoid temperature dependence of the single-exponential relaxation times at these intermediate temperatures is the result of the superposition of exchange effects on dipole-dipole relaxation or its equivalent. The true double-exponential relaxation times can recreate the singleexponential behavior by back calculation. Regression analysis using the double-exponential expression offers the first procedure for obtaining kinetics in the absence of lineshape perturbations by means of relaxation measurements and at the same time gives the true relaxation times 7’: and Ty without constraint on their relative values; i.e., they need not be equal or very unequal.

EXPERIMENTAL

Dimethylformamide was distilled from BaO and placed in an 8-mm NMR tube, which was subjected to 10 freeze-thaw-pump cycles and sealed under vacuum. Carbon-13 relaxation times were measured as a function of temperature on a Varian CFT-20 spectrometer. The temperature was monitored before and after each experiment with a digital thermometer. For the inversion-recovery pulse sequence ( 180-t-90-AT-5T1), the 90” and 180” pulses were determined to be 14 and 28 psec, respectively. For the double-exponential nonlinear regression, the program ZXSSQ (International Mathematics and Statistics Library) was used. The residuals of m equations in n unknowns are minimized by a modified Marquardt-Levenberg algorithm. The single-exponential analysis was a modified form of the Gerhards-Dietrich program

LAMBERT

244

AND

KEEPERS

(12). Local programs SIMULTl and PREL were wirtten for the respective purposes of the simulation model and data plotting.

This

work

was supported

ACKNOWLEDGMENT by the National Science Foundation

under

Grant

CHE77-08384.

REFERENCES

1. G. BINSCH, 2. 3. 4. 5. 6.

E. S. B. F. S.

7. H. 8. G. 9. T. 10. F.

11. M. 12. R.

Top. Srereochem. 3,97 (1968); “Dynamic Nuclear Magnetic Resonance Spectroscopy” (L. M. Jackman and F. A. Cotton, Eds.), Academic Press, New York, 1975. BREITMAIER, K.-H. SPOHN, AND S. BERGER, Angew. Chem. Znt. Ed. Engl. 14,144 (1975). FOR&N AND R. A. HOFFMAN, J. Chem. Phys. 39,2892 (1963). E. MANN, .I. Magn. Reson. 21, 17 (1976). A. L. ANET AND A. J. R. BOURN, J. Am. Chem. Sot. 89,760 (1967). A. SHERROD, R. L. DA COSTA, R. A. BARNES, AND V. BOEKELHEIDE, J. Am. Chem. Sot. 96, 1565 (1974); S. A. SHERROD AND V. BOEKELHEIDE, .I. Am. Chem. Sot. 94,5513 (1972). STREHLOW AND J. FRAHM, Ber. Bungsenges. Phys. Chem. 79, 57 (1975). C. LEVY AND G. L. NELSON, J. Am. Chem. Sot. 94,4897 (1972). C. FARRAR AND E. D. BECKER, “Pulse and Fourier Transform NMR,” Academic Press, New York, 1971. BLOCH, Phys. Rev. 70,460 (1946). RABINOVITZ AND A. PINES, J. Am. Chem. Sot. 91,1585 (1969). GERHARDS AND W. DIETRICH, J. Magn. Reson. 23,21 (1976).