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Internal rotation and nonexponential methyl nuclear relaxation for macromolecules
JOURNAL
OF MAGNETIC
RESONANCE
11,299-313
(1973)
nternal Rotation and NonexponentialMethyl Nuclear Relaxation for Macromolecules* LAWRENCE G. WERBEL~W
AND ALAN
6. MARSHALL
Department of Chemistry, University of British Columbia, Vancouver 8, Canada Received January 24,1973 The transverse and longitudinal nuclear magnetizations for an internally rotating methyl group attached to a symmetric top molecule have been calculated; crosscorrelation between the motion of two of the methyl protons and motion of the third has been taken into account, by extending an existing treatment by Hubbard to a range of rotational correlation times which are greater than or equal to the reciprocal nuclear Larmor frequency. For macromolecules in liquids, both the transverse and longitudinal magnetizations show a time-dependence which is more nonexponential than is possible for small molecules in liquids, but less nonexponential than for rigid solids. Immediately following a magnetic field pulse to prepare the spin system, the initial magnetization decay is approximated by the result obtained when crosscorrelation is completely neglected. Transverse relaxation is more nonexponential than longitudinal, except in the extreme narrowing limit where their behavior is identical. Finally, nonexponential -CHs or -CFB relaxation in liquids will be difficult to observe experimentally for small molecules because of a competing spinrotation mechanism; the optimum system for experimental detection of nonexponential relaxation should be a hindered methyl group on a large molecule. INTRODUCTION
Despite an extensive literature on the subject (see below), there is at present some confusion as to the extent, occurrence, and importance of nonexponential nuclear magnetic relaxation for a methyl (or trifluoromethyl) group in liquid media being relaxed by the intramolecular dipolar relaxation mechanism. While nonexponential relaxation has been predicted (I) and observed (2-8) in solids, nonexponential effects have not been convincingly observed for small molecules in solution1 Macromolecules in solution exhibit motional properties intermediate between solids and liquids, and nonexponential relaxation might thus be anticipated for such species. However, the detailed relaxation behavior of a -CM, or -CF3 group attached to a macromolecule in solution has never been worked out, in spite of the many attempts to study nuclear relaxation for big molecules (whether by covalent “labelling” (9, 10) or by use of small molecules which 0 0 /I /I bind reversibly to the macromolecule (II)) using-C-CM, or -C-CF, groups because of * Work supported by grants (to A.G.M.) from the National Research Council of Canada (A 6178) the Committee on Research, University of British Columbia (21-9675). * See Fig. 7 and related discussion in Ref. (6) for possible effects in liquids. Also, note that nonexponential elects can arise due to the interference of various competing relaxation mechanisms (a closely related phenomena to the one considered in this paper) and nonexponential relaxation has been seen in this instance in liquids. (J. S. Blicharski, ActaPhysica Polonica A 42,223 (1972).) and
Copyright 0 1973 by Academic Press. Inc. All rights OF reproduction in any Form reserved. Printed in Great Britain
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the high sensitivity and lack of complicated scalar coupling. In this paper, previous work is briefly reviewed, and Hubbard’s cross-correlation treatment of intramolecular dipole-dipole relaxation for a group of three equivalent spin one-half nuclei (12, 13) is extended to the case of large molecules in solution (204, z, 2 1). Finally, the quenching effect of spin-rotation is considered qualitatively, and the interpretation of methyl relaxation rates in terms of molecular rotational motion is discussed. In the absence of magnetic field gradients and chemical exchange, the principal relaxation mechanisms for spin one-half nuclei in liquids are inter- and intramolecular dipole-dipole and spin-rotation interactions. Intermolecular relaxation can readily be identified by experiments conducted at varying degrees of dilution in a nonmagnetic solvent. The remaining interactions depend on the rate and symmetry of molecular rotation; in the Theory section we treat intramolecular dipole-dipole relaxation, and the effect of spin-rotation is considered in the Discussion. There have been a number of derivations of nuclear relaxation times T1 and r, for the intramolecular dipole-dipole interaction of two spins, where the two spins are coupled to each other and to a lattice, but not coupled to 2 third spin. Anisotropic molecular reorientation has been treated, for motional models ranging from rotational diffusion (14) to random large-angle jumps between fixed angular positions (1.5) to collision-interrupted free rotation (16). The effect of chemical exchange which preserves (17), randomizes (28), or partly randomizes molecular direction is known (19). There have been a number of attempts to define general rotational motional models which reduce in particular limits to any of the above cases (X-24). The effect of internal rotation has been calculated (25,26). For problems involving three or more spins, the simplest case is that of Nequivalent (i.e.? having the same Larmor frequency) nuclei situated at the vertices of a regular polygon or polyhedron. Provided that the motion of any two nuclei is uncorrelated with that of a third (so-called “cross-correlation functions” are zero), it is readily shown that the total relaxation rate is just the sum of all the individual pairwise dipole-dipole contributions. However, for a methyl group (or for any rigid frame containing at least three nuclei the motion of the third proton is clearly determined by the motion of the other two, and cross-correlations may not in principle be neglected. Various attempts to account for such cross-correlations will now be listed briefly. Working within the framework of semiclassical density matrix theory of relaxation (27), Hubbard treated the problem of either three or four equally spaced spin one-half nuclei placed at the corners of an equilateral triangle (as for a methyl group or a 1,3,5substituted benzene) or at the corners of a regular tetrahedron. Assuming spherically symmetric rotational diffusion, and an “extreme narrowing” condition, oOz, e 1, the longitudinal relaxation was represented by the sum of two exponentials, but the resultant differed only very slightly from the simpler calculation in which cross-correlations were completely neglected (28). It was this work which led Abragam to state in his monograph (29) that inclusion of cross-correlations was of theoretical interest but of no practical importance. In a later paper (30), Hubbard extended his four-spin calculation to times longer than the Larmor period, and computed both the longitudinal and transverse relaxation. In general both decays were represented by the sum of three exponentials, reducing to two exponentials in the limit oOz, 6 1, and (for longitudinal relaxation) in the limit
NONEXPONENTIAL
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w,,z, > 1, However, again the theory predicted only very slight deviations from the un-correlated treatment. Next, Hilt and Hubbard (1) considered an equilateral triangle of identical spins which rotate about a crystal-fixed axis perpendicular to the plane of the triangle. They considered various orientations of the triangle with respect to an applied external magnetic field direction, and also treated a polycrystalline model. It was shown that the longitudinal decay is given by the sum of four exponentials, and that the predicted decay was markedly nonexponential and hence significantly different from the uncorrelated result. The most recent work follows an approach intermediate between the above limits of a methyl group rigidly bound to a rotating sphere and a methyl at the end of an infinitely long rod. First, Hubbard considered a methyl group rigidly attached along the symmetry axis of a symmetric top molecule undergoing anisotropic rotational diffusion (formally equivalent to a spherical top molecule with an internally rotation methyl group) (12). Finally, the treatment was extended to an asymmetric top with a rotating methyl group attached at an arbitrary angle with respect to the principal molecular axis (13). Wowever, all explicit calculations were restricted to the extreme narrowing limit, so that only the longitudinal decay was computed; in the Theory section, we extend these calculations beyond the extreme narrowing limit, so that the longitudinal and transverse decays must be treated separately. Apart from the calculations of Hubbard, other authors have also considered the problem of cross-terms between various pairwise dipolar couplings in the relaxation of multi-spin systems. Kattawar and Eisner (31) have treated the special case of three identical spin one-half nuclei at the corners of a 30”-120” isosceles triangle. The actual numerical results in this paper are in error, as pointed out in Refs. 32 and 38, doubtless due to the difficulty in solution of 14 simultaneous differential equations, many of which are linearly dependent. Zeidler (32) has carried out a calculation on a similar case where two spins and a non-identical third spin form a triangular arrangement. Richards (33) has derived the interesting result that in the extreme narrowing limit, a system with any number of identical spins must have indentical (though not necessarily a single exponential) time-behavior for both transverse and longitudinal decay, even when cross-correlations are considered. In the absence of cross-correlations, this statement reduces to the well-known fact that T, = T2 under extreme narrowing. Runnels (34), in a rather general calculation, has proved for a system of three equivalent spins one-half describable by a spin-temperature, the inclusion of cross-correlation always acts to make the relaxation less efficient (slower decay). Fenzke (3.5), in a calculation generalized to any number of equivalent spins, independently arrives at many of the same conclusions as Runnels demonstrated. Another example of cross-correlation is a paper by Noggle (36), which shows that for 13C in a methyl group, cross-correlations between the 13C-H’ and H’-H” or 13C-H”-relaxation vectors fortuitously cancel out (to first order), even though the order of magnitude of the spectral density for cross-correlation is the same as for autocorrelation. Kuhlman et al. (37) have come to a similar conclusion from their consideration of Overhauser effects for 13CH3 groups. It should be mentioned however, that the exact behavior of the relaxing four-spin system, 13CH3, has not been carried out in detail, although, as these last two papers might indicate, this would provide a meaningful, albeit tedious, calculation. Recently Pyper (38, 39) has applied Liouville representation formalism
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to the cross-correlation problem and has re-derived previous results to serve as compelling examples of this approach. The other main source of knowledge about cross-correlations between dipole-dipole interactions in magnetic resonance comes from a series of five papers by Schneider (4044) and Blicharski (43, 44), again based on Redfield’s semiclassical density matrix formalism (24). Schneider has derived the longitudinal and transverse relaxation behavior for almost every conceivable arrangement of 3 or 4 spin one-half particles, where the particles are not necessarily equivalent in the molecule and need not even be “like” nuclei. (It should be noted that for calculations treating “unlike” nuclei, it would in general be expected that cross-relaxation effects would lead to nonexponential relaxation even for a two-spin system. Therefore, inclusion of cross-correlation effects for a multispin system, composed of dissimilar nuclei, leads to a very complicated relaxation expression, and one that should be interpreted with care.) For certain configurations of “like” spins, Schneider treats all ranges of the quantity, o,,z,. Furthermore, his treatment can be adapted to motional models includingisotropic oranisotropic rotational diffusion or internal rotation. In cases where the two treatments deal with identical molecular rotational dynamics, Schneider’s results may be seen to reduce to Hubbard’s. For various configurations of the three or four spins, both the longitudinal and transverse relaxation are, in general, the resultant of between three and seven exponentials. Among all the above-listed examples, the one which most closely approximates the criteria of being physically reasonable while at the same time parametrically tractable is that of a symmetric top molecule to which is attached (at arbitrary angle) a methyl group which may rotate independently of the molecule as a whole (12, 13). Unfortunately, the existing literature is limited to the extreme narrowing approximation, and is thus restricted to small molecules in liquids. For presently typical magnetic fields of 15-75 kG in NMR, the extreme narrowing limit will be violated by molecules with molecular weight greater than about 5000 in aqueous solution (proteins, enzymes, membranes, nucleic acids, synthetic polymers), as well as by small molecules in viscous media or by solids near the melting points; the behavior of a methyl group on such molecules is thus unknown. In the Theory section, we treat the dipole-dipole intramolecular relaxation for a methyl group on a symmetric top molecule of any size, including the possibility of internal rotation. THEORY
From a semiclassical model [quantum mechanical spin system coupled to a classical lattice] (27), Hubbard has derived an expression for the longitudinal relaxation of a methyl group (12) :
x(t) = a(t) -- a’; o(t) is a reduced density operator which is obtained from the complete density matrix for the system by averaging over lattice degrees of freedom, and C? is the equilibrium value for this operator. The expectation value of any spin operator A, is given by (A) = Tr[cr(t) A ]. M
NONEXPONENTIAL
METHYL
RELAXATION
303
In this section, we derive the expectation values of 1, and I,, using as basis functions the eight spin states of a group of three spin one-half particles, expressed in terms of the respective eigenvalues of P, Z:,, and I,, where I:, = (I1 + 12)2, I2 = (II, + J#, and Iz = I=, + Iz2 + Iz,. If la) = II,, ZM), then notation will be based on the convention,
11)= I1 W 12) = 13) = 14) = 15) = 16) = 17) = 18) =
3/G,
11 312 l/2), j I 3/2 - l/2), 11 3/2 - 3/2), 11 l/2 l/2), 11 l/2 - l/2), IO l/2 l/2), 10 l/2 -l/2).
r31
Letting L41 and introducing two other combinations of matrix elements,
= U/2)(x55-t x77)- W) (X66 + A!*& Y30)= Y2W-t (l/2) (x22- X33)? Y2@>
where
Xijs
r.51
PI VI
it is readily shown that (10)
~Y,W~~= -PJ&%) + 8J,P%)+ 2JcbJ)lYI(~) + WJ~(O,) + 24~~(2~,)1Y,(~) + ~~K(WJ - ~~w,k4t~, d~,wdt = (113) [J,(o,) - 4md - ~~(4 + 4~~(2~~)1k(t) - [35,(O) - Ja(wJ + 2J,(2%) - 3JCCO)+ Jc(aJ - 2Jc(2adlY,(0 + (l/3) [6J,(O) - 10J,(oO) + 4J,(2co,) - 6J,(O) + lOJ,(w,) - 4wdi Ydt), dY&)ld~ = EJakkJ - 2Ja@%) + JC(%)lYl(~) + UP) [9Jh”) + ~J,(w,) + 1w~d~~w - c9u~d - a4 + 4~,(2~,)1 y3(t).
PI
191 DOI
A unique solution to these three coupled differential equations may be found from the initial conditions resulting from application of an rf-pulse along the y-axis in a rotating frame, causing the equilibrium magnetization to rotate by 8”, where relaxation effects during the pulse are ignored. It then follows that ~~(0) = [COSe - mmT, ~43 = W)YI(~), Y3Kn = (1/4)YI(O).
u11
In order to solve Eqs. [&-lo], it is necessary to choose a molecular system and define a model for its rotational dynamics. For the general (but still moderately tractable) system of a symmetric top molecule with the symmetry axis of a rotating methyl group
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attached at angle /3with respect to the symmetry axis of the top, Hubbard has shown that there are just two kinds of spectral densities, J,(W) and J,(o) (13) : (1/4)(3cos2~-
1)‘6DI
+ 3sin2/3cos2/?(5D1 + D,,)
(603 + (km,)’ (5D, + D ,;)” + (kcoo)2 _L (3/4) sin”P(2D, + 40 ,,) + (9/S) sin”P(GD, f 4Di”t) ’ (20, + 40 !,)’ + (key
(60, + 4Dj,J2 + (kqJ2 + (j/2) sin’P(1 + COS”~) (5D, + D !, + 4Dint) (5D,+ D 11+ 4Dint)’ + (k~,)’ + (3/S) [(I + COS’/?)’ + ~COS’~] (20, + 40 11+ 4Dint) , (20, + 40 I/ + 4Di”t)’ + (koo)’ 9 (9/8 sin4P(6D, + 4D,,,) Jc(kwo) = J’(k0o) - @ (60, + 4Di,J2 + (k~,)~
Cl21
+ (3/2 sin’/?(l + cos2~)(5DI + D ,, t 4Di,J (5D,+ D,, + 4Dint)’ + (LOO)’ + (3/X) [(I + COS2p)2 + ~COS’~] (20, + 401, + 4Dint) (20, + 40 11+ 4Dint)’ + (k~,)~ ’ In these equations, D, is the diffusion constant for rotational diffusion about an axis perpendicular to the symmetry axis of the symmetric top, D,, is the parallel diffusion constant, and Dint is the diffusion constant for internal rotation of the methyl group with respect to the molecular frame. By the usual methods, the three simultaneous, first-order, linear homogeneous differential equations [S-lo] with constant coefficients may be solved determina~tall~7 by Laplace transform methods. The form of the solution for the expectation value of I, is given by, (I,) - (Iz)T = (I,)‘(cos i3- 1) [A, e-‘.lf + A2 em+ + A, e-“3t], /I41 for an initial preparation of the system by a Q-pulse. Initial conditions dictate that Ci Ai = 1. The pre-exponential and exponential factors would be exceedingly bulky for listing in closed form, but the longitudinal magnetization is readily computed numerically and is in any case most suitably displayed in graphical form (see Figs. 3-4). For the fast-motion limit, (6DJ2 s 4w& considered previously (12), yl(t) and yz(t) are no longer coupled to y3(t), and the time-dependence for the transverse and longitudinal magnetizations become identical. Since (I,)’ = 0 (there is no equilibrium magnetization in the x-y plane), x(t) = o(t), and we seek (I,) =Tr[a(t)l,]. El151 On setting up the density matrix equations using the same basis set as for the iongitudinal calculation, it becomes evident that the appropriate parametrization is in terms of three linearly independent combinations of the matrix elements of 0.
+
&> =
(l/2)
(l/2) (056
93(f) = 92(t) +
632
k.56 + g65 + d87 + ‘?,I, + G65 + fl87 + a78)~ +
023.
81161 iI71 PSI
NONEXPONENTIAL
METHYL
RELAXATION
305
With these definitions, and using Eqs. [14-201 of (II), the equations for time rate of change of ql, q2, and q3 may be obtained. 4lW~
= l-(3Jm + 3Jc(O)) + (65J,(%) - Jc(%>> + (-2Ja(2%) + 6(Jcb.J + JcG’wJ) q&) + 6(Jc(O) - J&d) dt),
+ 2JcGh3))191(~~ f191
4?,(0/~~= -[Ja(%>- J,(Qh)19&) + Fw,(%) - Jc(aJ>> - 2(Jach) - Jc@%))192(~) + L-Va@> - J,(O))+ (J&4 - JC(wJ)l93@)? PO1 &,(W = -(J,(Q-d - Jc(%>>41(t) + [3(Ja(%) + Jc(wJ)+ ~Jcw%)192~0 + [-3(J,(O) - JdQN - (4J&%) + 2Jc(f%)) - (2~,(2~~) + 4~(2~,))1 h(t).
L-31
‘The initial condition of the spin system is again taken as the instant following a Q-pulse counterclockwise about the y-axis, so that a(f = 0) = e-isry cT e+i6r,. Now for the high-temperature
approximation,
124
Fu, 4 kT,
e-(WkT) (J‘T=
Tr[e’-hE’kT)]
-- (I - (tzE/kT))/Tr[I]
L233
where I denotes the unit matrix. Substituting Eq. [22] into [23], and making note of the fact that E = --cooI,, o(t = 0) = (l/S) + (Aoo/8kT) e-ielyIZ efi61y,
~41
but
eeiexyI, e+isry= I, sin Q + 1, cos 6.
E251
Therefore, cr(t = 0) = (l/S)
x
10000000 01000000 00100000 00010000 00001000 00000100 00000010 00000001
300
010 00-10 0 0 0 000 000 000 000
w,f3cos$ + 16kT
IO
3+ 0 3+ 0 2 0 2 0 0 0 3+
0 0 0 \O
0 0 0 0
0 0 0 0
0 0
0 0 -3 0 0 0 0
0 0 0 0 34 0 0 0 0 0 0 0100 0 1000 0 0001 0 0010
00 00 00 0 0 10 0 -1 00 00 0 0 0 0
00 00 00 0 0 00 0 0 10 0 -1 0 0 0 0
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APU’D MARSHALL
and since
/27]
Wt=o = WdO) f,l, ql(0) = sin 19(1,>~,
Lw
do) = (l/6) 41(O), q&a = (v9q1(0).
pcq c303
In these calculations, the second order Zeeman correction (which is of the order of the linewidth if the reciprocal Larmor frequency is of the order of the correlation time of the bath) is neglected. For longitudinal relaxation, this omission has no effect, whereas for tranverse relaxation, inclusion of the correction leads to a small modulation of the time decay of the magnetization envelope, since each of the three decaying exponentials will precess with slightly different frequency. Fraenkel (J. Chem. P&s. 42,4275 (1965)) describes in great detail the effect of this small Zeeman correction, and ref. 28 illustrates the complexity in both calculations and results when the correction is included. In our judgment for the present case, the trivial analytical difference to be expected between the results described here and the “exact” calculation does not warrant the additional computational difficulty required. Solution of the system of differential equations, Eqs. [19-211, subject to the initial conditions, Eqs. [28-301, yields a solution of the form, (I,)* = sin d(1,)’ [B, e+lt + B, emyzf+ B, ecYst].
:311
Again, the B’s are constrained, the sum being equal to unity. This solution is adapted to the rotating frame in which the rapid time dependence has been suppressed (i.e., the true solution of Redfield’s equations yields a result where the r.h.s. of Eq. [31] is multiplied by a factor of exp(--iwO(t + t’)) where t’ is an initial phase factor). As with the longitudinal relaxation, it is not worth listing the pre-exponential and exponential factors analytically; the graphical results are analyzed in the Discussion. It may be pointed out that in the limit of extreme narrowing, both Eqs. [14] and [31] reduce to the sum of two exponentials, and the time-dependence of both equations becomes identical, reducing to Hubbard’s result (12, 13). RESULTS
AND
DISCUSSION
Disregarding cross-correlations (i.e., setting all J,(o) = 0), it can be seen from Eqs. [g-10] and [19-211 respectively that (CL> - WT)/(cos
tJ - l)(L>T = exp[(-2J,(a)
- 8Jat2~d
and (I,)/sin
Q (I,)T = exp[(-3&(O) - 5J,(c0,,) - 2J,(200)) t].
rl,
1321 WI
Woessner et al. (1.5) have previously considered the (uncorrelated) spin-lattice relaxation for axially symmetric ellipsoids with internal motion. It may be verified that Eqs. [32-331 are identical to the result of Ref. (15). Since analytical expressions for the relaxation would be too bulky for ready presentation, we have chosen to express all results graphically in Figs. l-4. For all plots, it was assumed that the methyl group axis was coincident with the symmetry axis of the symmetric top to which the methyl was attached; it was found in general that the nonexponential character of the relaxation was most exaggerated for this case, and in any
NONEXPONENTIAL
METHYL
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307
case, explicit display of this angle-dependence would be unprofitably lengthy. For each plot, the magnetic field was chosen as typical for high-resolution NMR experiments : CC)*= 2n x 108sec-I. Finally, for each plot, an interproton distance of 1.8 &I was assumed, since X-ray and microwave values average close to 1.77 III 0.03 A. Figure I shows plots of log(magnetization) versus time for three values of D,, the diffusion constant for rotational diffusion about an axis perpendicular to the symmetry axis of the symmetric top: the upper plot corresponds to an extreme-narrowed case, 60,* 2c00, and in this limit the transverse and longitudinal magnetizations have the same time-dependence so only one plot is required; the middle plots show the longitudinal (left-hand plot) and transverse (right-hand plot) magnetization for a case where 6D, LZ20,; the bottom plots give the longitudinal (left-hand plot) and transverse
FIG. 1. Plots of Iog(magnetization) versus time, following a &pulse along the y-axis of a rotating frame, for a methyl group attached along the symmetry axis of a symmetric top molecule. For plots lb and Id, g(t) =~; for plot la, longitudinal = transverse magnetization = C(t). For all plots, WC,= 2n x lO’sec-I, rHMH= 1.8 A. For plot la D, = 10’0rad2sec-1. , for plots lb, lc, D, = 108radZsec-1; for plots Id, le, DI = 10’ rad’sec-I. Gor plot la, curves P, Q, . ., V correspond to respective values of [Dint + D ,,I of I, (7/4), 4, 6, 10, 20, and 100 x 10LcradZ/sec; for plots b and c (or d and e), curves P, Q, , . ,, V correspond to the same [(Dint + D,,)/D,]-ratios as for plot a. See the text for further explanation and discussion.
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(right-hand plot) magnetizations for a methyl group on a very large molecule in solution (or small molecule in very viscous solution), 6DL c 2~0,. The extreme-narrowed case (Fig. la) is identical to Hubbard’s Fig. 1 (12) except for a change in notation of the abscissa scale. From comparison of curves P-V for each graph, for a symmetric top of any size, relaxation (whether longitudinal or transverse) is most nonexponential when the top is most prolate and/or exhibits fastest internal motion. For large molecules (bottom two plots), the longitudinal magnetization decays very nearly according to a single exponential, while the transverse decay is markedly nonexponential. Two general properties proved by Runnels (34) are manifest in Fig. 1: first, that inclusion of crosscorrelations always retards the relaxation; and second, that the initial slope of each
FIG. 2. Resolution of a plot of log(longitudina1 magnetization) versus time into its (three) component exponentials. Curve denotes plot V of Fig. lb. Dotted line denotes linear result obtained by neglecting cross-correlation effects. The three solid lines (in order of increasing slope) are plots of Iog(O.125exp [-0.678t]), log((0.125 + 0.704) exp[-5.34t)), and log(exp[-12.5t]) vesus f. See Eq. [34] and accompanying discussion.
curve approaches the slope of the single exponential decay obtained when cross-correlation is neglected (see Fig. 2). Finally, it should be noted that in Fig. 1 the abscissa is in specific units of seconds rather than the more general “reduced time” variable as seems to be the common convention (12,31). This is because, unless working within the eonfines of the limit wOz, < 1, the explicit field dependence detracts from the usefulness of this form of presentation. Although the scale as shown in Fig. 1 is specific for the case ofan isolated methyl group in a 23.5 kG field (w, = 100MHz); the graphs are still rather general in their presentation. For example, if the gyromagnetic ratio were smaller (as for 19F), the shapes of all curves would remain the same, but the time scale would be contracted (slower relaxation). If r were smaller, the shapes of the curves would again remain the same, and the time scale would be expanded (faster relaxation). For diEerent values of LOO,the curves will again retain the same shapes, provided that the rotational diffusion constants are changed by the appropriate factor. In Fig. 2, curve V of Fig. lb is reproduced (curved line), and resolved into its three component exponentials (the three solid lines), and compared to the result obtained when cross-correlations are neglected (dotted line). The actual equation of the curve is, ((I,) - (I,)‘)/(cos
6 - 1) (I,)= = (0.125) e-0.678t + (0.171) e-12.5t + (0.704) e-5Q34t-~ D41
NONEXPONENTIAL
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309
Note that the y-intercepts of the three solid lines are: log(O.125), log(O.125 + 0.704) and log(1.) ; this seems to be the most useful way to show the limiting behavior at long times. From this plot, it is clear that the initial slope of the curve approaches that ofthe dotted line. The most general display of the present calculations is provided by Figs. 3 and 4, which show the relative contributions and time constants of the three exponentials for the longitudinal (Fig. 3) and transverse (Fig. 4) magnetizations. The pre-exponential factors are shown in the upper three plots of each figure, and the relative time-constants in the lower two plots. This manner of presentation serves the same purpose as an ex-
7
8
9
10
11
LogiD~~Dt)
Log(D,, +Di) FIG. 3. Nonexponentiality of time-decay for longitudinal magnetization, displayed as contour plots, where each contour is a line of constant magnitude for either preexponential factors (AJ or exponential ratios Aj/?,k from Eq. [14], as a function of D, and (D,, + DJ. See text for further details.
tensive digital table or a 3-dimensional plot depicting the mutual effects of size, shape, and internal flexibility on the relaxation behavior of the spins. These two figures are simply contour plots and are interpreted as one would use a topographical map. For example, say one wishes to determine the relaxation behavior of an isolated, flexible methyl group attached to the backbone of a large molecule in solution. Furthermore, for the purpose of serving as an example, assume the large molecule is roughly spherical in shape and the system can be characterized by the three diffusion constants, D, = D,, = 10grad2/sec, and Dint = 101°rad2/sec. Referring to Fig. 3, it is found that 3b2z A, E 4i., and the preexponentials for the two distinctive terms are (AZ + AX) sz0.92 and A 1 z 0.08. Of course, the same relaxation would be expected if lOD, = D ,, = 1O1O
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rad2/sec and Dint = 109rad/%ec (this is due to the fact that for these plots, fi = 0”; therefore, (D,! + Dint) is effectively a single variable). Figures 3 and 4 are most informative when used in conjunction with Fig. 1. By referring to Fig. 1, it is now possible to conclude how small the ratio of &/& (and/or &/,I,) and also how comparable AI and A2 (and/or AS) must be in magnitude in order that relaxation be markedly nonexponential. Also, only the ratios of the exponential factors are shown in Figs. 3 and 4. Therefore, relative, not absolute behavior is depicted in these figures. Again, Fig. 1 shows some absolute plots for fixed choices of Dint, D ,,, and DL. With the aid of Figs. 3 and 4, it is
7
8
9
10
7
8
9
10
J
8
9
10
11
Log [ D,,‘Di]
FIG. 4. Nonexponentiality of time-decay for transverse magnetization, displayed as contour plots, where each contour is a line of constant magnitude for either preexponential factors (I?,) or exponential ratio yj/yr factors from Eq. [31], as a function of D, and D ,, + Of). See text for further details. (Note the trivial change in notation: D, in Figs. 3 and 4 corresponds to I&,, in the text.)
possible to make quite precise qualitative interpolations or extrapolations. In order for nonexponential behavior be manifest, it is necessary botlz that at least two of the pre-exponential factors (A’s or B’s) be of the same order of magnitude and that the ratio of the corresponding exponentials differ appreciably from unity. From Figs. 3 and 4, it is seen that nonexponential relaxation is most evident for very prolate tops and/or when internal rotation is fast compared to DI. In the extreme narrowing limit (upperright hand region of each plot), A, and B, approach zero and/or &/I,,+1 ; in either event, the result reduces to the two exponentials obtained by Hubbard (12). It should also be mentioned that the regions of the plots where D, > (a,,, + D ,,) have little physical meaning since (even for an extremely flattened ablate top) D, doesn’t differ appreciably from the value for D ,,.
NONEXPONENTIAL
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A final intriguing (but unexplained) feature of the transverse magnetization is that for a [(D,, $ &)/DL]-ratio of 7/4, the relaxation approaches that of a single exponential, i.e., B,+ 1.OOin Eq. [31], as shown in Fig. 4. This fact can be shown analytically for the extreme narrowed case (22) and appears simply to be an accident of the algebra. Extension of the treatment to -CF, groups follows directly. The interfluorine distance is typically 2.21 & 0.04 1$, and the lgF magnetogyric ratio is (25179/267§3) that for protons; thus for the same value of w0 and identical molecular dynamics, the 19F dipole-dipole relaxation rate will be down by a factor of [2.52/2.6714 [1.77/2.2116 z (l/5) that for a proton methyl group. From Figs. 3 and 4, it is evident that the best system in which to observe nonexponential decay in magnetization should be a molecule possessing a methyl group whose internal rotation rate is fast compared to reorientation of the molecule as a whole. For small molecules in liquid state, this means that for the effect of cross-correlations to be noticeable, Dint will be restricted to cases where the methyl group rotation approaches the “free diffusion” limit where Dint is of the order of (kT,lmethvi)a z 1Oi3 rad2/sec, where Imcthy, is the moment of inertia for the methyl group. Just such an example is provided by liquid acetonitrile [the molecule for which cross-correlation was first manifested in the solid state (2)]; by comparing proton and nitrogen relaxation times in. CH,CN, it may be shown that the rotation about the C-N axis is about ten times as fast as reorientation of that axis itself (45). However, the proton relaxation was observed to be exponential. This incongruity is resolved by a study of the temperature dependence of T,, which shows that a spin-rotation interaction is present and provides the dominant intramolecular relaxation mechanism above 25°C. Moreover, it has recently been argued (46,47) and demonstrated (48) that the magnetic field due to internal rotation can also produce a spin-rotation interaction; thus even for methyl groups on Iarge molecules, the presence of free or nearly free internal rotation should introduce a large (exponential) spin-rotation relaxation which should mask any nonexponential effects due to dipole-dipole interactions. There is however one situation in which nonexponential decay might be observable in liquids. The spin-rotation interaction generally has the form (46), P51 (l/T&pin-rotation = (213) kTF2 I, C,z rSR, where I, is the moment of inertia of the internal rotor, C, is the corresponding internal rotation spin-rotation constant, and rsRis the correlation time for changes in magnitude of the spin-rotation interaction. rsR i s distinguished in two ways from the average correlation time, rDD, for changes in magnitude of the dipole-dipole interaction: first, rsRmay be thought of as the average time between “collisions” which change either the principal axis direction or rate of the internal rotation, so that rsR becomes sJzorter as the rotational motion becomes more hindered (this may be contrasted with rDD, which may be regarded as the average time it takes for the dipole-dipole axis to reorient of the order of a radian, so that rDD becomes loizgev as the rotation becomes more hindered; thus the dipole-dipole and spin-rotation interactions generate opposite temperature dependence for relaxation, which provides a means of resolving their relative contributions); and second, rsR is in general shorter than t,,,,. Thus, in order to emphasize the dipole-dipole interaction compared to the spin-rotation, one need simply locate a .molecular system with relatively hindered internal motion; subject to this constraint,
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one couid then reasonably expect to observe nonexponential relaxation, provided that the (hindered) internal motion were still fast compared to reorientation of the molecule as a whole. While the importance of spin-rotation contribution to relaxation is notoriously difficult to assessin advance (49), the form of the relaxation is well established for rotational diffusion of spherical (50,51) and symmetric (52,53,54) tops, and recently for a more general motional model for symmetric tops (55). In any case, there is well documented evidence that spin-rotation is an important contribution to relaxation for both -CH, groups (50,54) and -CF, groups (48, SO), on small molecules in liquids. CONCLUSIONS
It has been shown that intramolecular dipole-dipole relaxation for an isolated methyl group can be markedly nonexponential, especially for internally rotating methyl (or trifluoromethyl) groups on large molecules in solution. However, unless the internal motion itself is appreciably hindered, the spin-rotation relaxation will contribute and probably mask any nonexponential effect experimentally. Thus, to detect nonexponential decay in magnetization, the methyl group internal rotation should be fast compared to rotation of the molecule as a whole, but slow compared to a “free diffusion” model. For -CF, groups, the effect will probably never be observable in liquids, because the dipole-dipole relaxation rate is slower by a factor of five, and the spin-rotation coupling is larger than for protons. For -CF3, then, nonexponential relaxation would be expected only when the internal rotation of the -CF, group is very “hindered” However, in order that the nonexponential effect be observable, one would have to invoke a macromolecule whose rotational anisotropy differs by an order of magnitude from a sphere, which is highly improbable in practice. There is one other way in which fluorine magnetic relaxation might be nonexponential, and that is due to possible crosscorrelation effects between competing relaxation processes. This problem is formally similar to the present calculation and will be treated in a later paper. Finally, it has been shown that a plot of either the transverse or longitudinal magnetization versus time gives an initial slope which is the same as would be obtained by neglecting all cross-correlation effects, further compounding the experimental difficulty of detecting nonexponentiality. Nevertheless, an internally rotating methyl group on a large molecule in solution should show nonexponential relaxation. The fact that none has been reported for such systems should signal the likelihood that other relaxation mechanisms are operative (principally spin-rotation and intermolecular dipoledipole interactions); there is at least one example for intermolecular contributions to relaxation for a macromolecule (57). REFERENCES I. 2. 3. 4. 5. 6. 7. 8.
R. L. HILT AND P. S. HUBBARD, Phys. Rev. 134, A392 (1964). M. F. BAUD AND P. S. HUBBARD, Phys. Rev. 170,384 (1968). S. ALBERT AND J. A. RIPMEESTER. J. Chem. Phys. 57,264l (1972). J. L. CAROLAN AND T. A. SCOTT, J. Magn. Resonance 2,243 (1970). K. VAN PUTTE AND G. J. N. EGMOND, J. Magn. Resonance 4,236 (1971). K. VAN PUTTE, J. Magn. Res. 5,367 (1971). L. J. BURNETT AND B. H. MULLER, Chem. Phys. Lett. 18,553 (1973). G. A. DEWITAND M. BLOOM, Can. J. Phys. 47,1195 (1969).
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9. E. W. BITTNER AND J. T. GERIG, J. Amer. Chem. Sac. 92,5001 (1970). 10. W. H. HUESTIS AND M. A. RAFTERY, Biochemistry 10, 1181 (1971); Biochemistry 11, 1648 (1972). II. B. D. SYKES AND M. SCOTT, Ann. Rev. Biophys. Bioeng. 1,251(1972). 12. P. S. HUBBARD, J. Chem. Phys. 51,1647 (1969). 13. P. S. HUBBARD, J. Chem. Phys. 52,563 (1970). 14. W.. T. HUNTRESS, JR., J. Chem. Phys. 48,3524 (1968). 15. D. E. WOESSNER, B. S. SNOWDEN, JR., AND G. H. MEYER, J. Chem. Phys. 50,719 (1969). 16. FL G. GORDON, J. Chem. Phys. 44,183O (1966). 17. J. E. ANDERSON AND P. A. FRYER, J. Chem. Phys. 50,3784 (1969). 18. A. G. MARSHALL, J. Chem. Phys. 52,2527 (1970). 19. H. SILLESCU, J. Chem. Phys. 54,211O (1971). 20. R. D. MCCLUNG, J. Chem. Phys. 57,547s (1972). 21. R. I. CUKIER AND K. LAKATOS-LINDENBERG, J. Chem. Phys. 57,3427 (1972). 22. 14. KIVELSON AND T. KEYES, J. Chem. Phys. 57,4599 (1972). 23. T. KEYES, Mol. Phys. 23,737 (1972). 24. P. A. EGELSTAFF, J. Chem. Phys. 53,259O (1970). 25. D. WALLACH, J. Chem. Phys. 47,525s (1967). 26. A. G. MARSHALL, P. G. SCHMIDT, AND B. D. SYKES, Biochemistry 11,3875 (1972). 27. A. G. REDFIELD, IBM J. Res. Develop. 1, 19 (1957); P. S. HUBBARD, Rev. Mod. Phys. 33: 24a (1961). 28. P. S. HUBBARD, Phys. Rev. 109, 1153 (1958); Phys. Rev. 111,1746 (1958). 29. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Clarendon Press, Oxford, England, 1961, p. 293. 30. P. S. HUBBARD, Phys. Rev. 128,650 (1962). 31. G. W. KATTAWAR AND M. EISNER, Phys. Rev. 126,1054 (1962). 32. M. D. ZEIDLER, Ber. Bunsenges, Physik, Chem. 72,481 (1968). 33. P. M. RICHARDS, Phys. Rev. 132,27 (1963). 34. L. K. RUNNELS, Phys. Rev. 134, A28 (1964). 35. D. FENZKE, Ann. Physik 16,281(1965); D. FENZKE AND H. SCHNEIDER, Ann.Physik 19,321(1967). 36. J. H. NOGGLE, J. Phys. Chem. 72,1324 (1968). 37. K. F. KUHLMANN, D. M. GRANT, AND R. K. HARRIS, J. Chem. Phys. 52,3439 (1970). 38. N. C. PYPER, Mol. Phys. 21,l (1971). 39. N. C. PYPER, Mol. Phys. 22,433 (1971). 40. H. SCHNEIDER, Ann. Physik. 13,313 (1963). 41. H. SCHNEIDER, Ann. Physik 16,135 (1965). 42. H. SCHNEIDER, 2. Naturforsch. 19a, 510 (1964). 43. J. S. BLICHARSKI AND H. SCHNEIDER, Ann. Physik 22,306 (1969). 44. H. SCHNEIDER AND J. S. BLICHARSKI, Ann. Physik 23,139 (1969). 45. D. E. WDESSNER, B. S. SNOWDEN, AND E. T. STROM, Molecular Phys. 14,265 (1968). 46. A. S. DUBINAND S. 1. CHAN, J. Chem. Phys. 46,4533 (1967). 47. C. T. SCHMIDT AND S. I. CHAN, J. Magn. Resonance 5,151 (1971). 48. T. E. BURKE AND S. I. CHAN, J. Magn. Resonance 2,120 (1970). 49- C, DEVERELL, Molecular Phys. 18,319 (1970). 30. D. K. GREEN AND J. G. POWLES, Proc. Phys. Sac. 85,87 (1965). 51. P. S. HUBBARD, Phys. Rev. 131,1155 (1963). -52. H. J. BENDER AND M. D. ZEIDLER, Ber. Bunsenges. Physik. Chem. 75,236 (1971). 53. C. H. WANG, D. M. GRANT, AND J. R. LYERLA, JR., J. Chem. Phys. 55,4674 (1971). 54. C. H. WANG, J. Mugn. Resonance 9,75 (1973). :ip. R. E. D. MCCLUNG, J. Chem. Phys. 57,5478 (1972). 56. R. G. PARKER AND J. JONAS, J. Magn. Resonance 6,106 (1972). 57. G. NAVON AND A. LANIR, J. Magn. Resonance 8,144 (1972).
12
OF MAGNETIC
RESONANCE
11,299-313
(1973)
nternal Rotation and NonexponentialMethyl Nuclear Relaxation for Macromolecules* LAWRENCE G. WERBEL~W
AND ALAN
6. MARSHALL
Department of Chemistry, University of British Columbia, Vancouver 8, Canada Received January 24,1973 The transverse and longitudinal nuclear magnetizations for an internally rotating methyl group attached to a symmetric top molecule have been calculated; crosscorrelation between the motion of two of the methyl protons and motion of the third has been taken into account, by extending an existing treatment by Hubbard to a range of rotational correlation times which are greater than or equal to the reciprocal nuclear Larmor frequency. For macromolecules in liquids, both the transverse and longitudinal magnetizations show a time-dependence which is more nonexponential than is possible for small molecules in liquids, but less nonexponential than for rigid solids. Immediately following a magnetic field pulse to prepare the spin system, the initial magnetization decay is approximated by the result obtained when crosscorrelation is completely neglected. Transverse relaxation is more nonexponential than longitudinal, except in the extreme narrowing limit where their behavior is identical. Finally, nonexponential -CHs or -CFB relaxation in liquids will be difficult to observe experimentally for small molecules because of a competing spinrotation mechanism; the optimum system for experimental detection of nonexponential relaxation should be a hindered methyl group on a large molecule. INTRODUCTION
Despite an extensive literature on the subject (see below), there is at present some confusion as to the extent, occurrence, and importance of nonexponential nuclear magnetic relaxation for a methyl (or trifluoromethyl) group in liquid media being relaxed by the intramolecular dipolar relaxation mechanism. While nonexponential relaxation has been predicted (I) and observed (2-8) in solids, nonexponential effects have not been convincingly observed for small molecules in solution1 Macromolecules in solution exhibit motional properties intermediate between solids and liquids, and nonexponential relaxation might thus be anticipated for such species. However, the detailed relaxation behavior of a -CM, or -CF3 group attached to a macromolecule in solution has never been worked out, in spite of the many attempts to study nuclear relaxation for big molecules (whether by covalent “labelling” (9, 10) or by use of small molecules which 0 0 /I /I bind reversibly to the macromolecule (II)) using-C-CM, or -C-CF, groups because of * Work supported by grants (to A.G.M.) from the National Research Council of Canada (A 6178) the Committee on Research, University of British Columbia (21-9675). * See Fig. 7 and related discussion in Ref. (6) for possible effects in liquids. Also, note that nonexponential elects can arise due to the interference of various competing relaxation mechanisms (a closely related phenomena to the one considered in this paper) and nonexponential relaxation has been seen in this instance in liquids. (J. S. Blicharski, ActaPhysica Polonica A 42,223 (1972).) and
Copyright 0 1973 by Academic Press. Inc. All rights OF reproduction in any Form reserved. Printed in Great Britain
299
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the high sensitivity and lack of complicated scalar coupling. In this paper, previous work is briefly reviewed, and Hubbard’s cross-correlation treatment of intramolecular dipole-dipole relaxation for a group of three equivalent spin one-half nuclei (12, 13) is extended to the case of large molecules in solution (204, z, 2 1). Finally, the quenching effect of spin-rotation is considered qualitatively, and the interpretation of methyl relaxation rates in terms of molecular rotational motion is discussed. In the absence of magnetic field gradients and chemical exchange, the principal relaxation mechanisms for spin one-half nuclei in liquids are inter- and intramolecular dipole-dipole and spin-rotation interactions. Intermolecular relaxation can readily be identified by experiments conducted at varying degrees of dilution in a nonmagnetic solvent. The remaining interactions depend on the rate and symmetry of molecular rotation; in the Theory section we treat intramolecular dipole-dipole relaxation, and the effect of spin-rotation is considered in the Discussion. There have been a number of derivations of nuclear relaxation times T1 and r, for the intramolecular dipole-dipole interaction of two spins, where the two spins are coupled to each other and to a lattice, but not coupled to 2 third spin. Anisotropic molecular reorientation has been treated, for motional models ranging from rotational diffusion (14) to random large-angle jumps between fixed angular positions (1.5) to collision-interrupted free rotation (16). The effect of chemical exchange which preserves (17), randomizes (28), or partly randomizes molecular direction is known (19). There have been a number of attempts to define general rotational motional models which reduce in particular limits to any of the above cases (X-24). The effect of internal rotation has been calculated (25,26). For problems involving three or more spins, the simplest case is that of Nequivalent (i.e.? having the same Larmor frequency) nuclei situated at the vertices of a regular polygon or polyhedron. Provided that the motion of any two nuclei is uncorrelated with that of a third (so-called “cross-correlation functions” are zero), it is readily shown that the total relaxation rate is just the sum of all the individual pairwise dipole-dipole contributions. However, for a methyl group (or for any rigid frame containing at least three nuclei the motion of the third proton is clearly determined by the motion of the other two, and cross-correlations may not in principle be neglected. Various attempts to account for such cross-correlations will now be listed briefly. Working within the framework of semiclassical density matrix theory of relaxation (27), Hubbard treated the problem of either three or four equally spaced spin one-half nuclei placed at the corners of an equilateral triangle (as for a methyl group or a 1,3,5substituted benzene) or at the corners of a regular tetrahedron. Assuming spherically symmetric rotational diffusion, and an “extreme narrowing” condition, oOz, e 1, the longitudinal relaxation was represented by the sum of two exponentials, but the resultant differed only very slightly from the simpler calculation in which cross-correlations were completely neglected (28). It was this work which led Abragam to state in his monograph (29) that inclusion of cross-correlations was of theoretical interest but of no practical importance. In a later paper (30), Hubbard extended his four-spin calculation to times longer than the Larmor period, and computed both the longitudinal and transverse relaxation. In general both decays were represented by the sum of three exponentials, reducing to two exponentials in the limit oOz, 6 1, and (for longitudinal relaxation) in the limit
NONEXPONENTIAL
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w,,z, > 1, However, again the theory predicted only very slight deviations from the un-correlated treatment. Next, Hilt and Hubbard (1) considered an equilateral triangle of identical spins which rotate about a crystal-fixed axis perpendicular to the plane of the triangle. They considered various orientations of the triangle with respect to an applied external magnetic field direction, and also treated a polycrystalline model. It was shown that the longitudinal decay is given by the sum of four exponentials, and that the predicted decay was markedly nonexponential and hence significantly different from the uncorrelated result. The most recent work follows an approach intermediate between the above limits of a methyl group rigidly bound to a rotating sphere and a methyl at the end of an infinitely long rod. First, Hubbard considered a methyl group rigidly attached along the symmetry axis of a symmetric top molecule undergoing anisotropic rotational diffusion (formally equivalent to a spherical top molecule with an internally rotation methyl group) (12). Finally, the treatment was extended to an asymmetric top with a rotating methyl group attached at an arbitrary angle with respect to the principal molecular axis (13). Wowever, all explicit calculations were restricted to the extreme narrowing limit, so that only the longitudinal decay was computed; in the Theory section, we extend these calculations beyond the extreme narrowing limit, so that the longitudinal and transverse decays must be treated separately. Apart from the calculations of Hubbard, other authors have also considered the problem of cross-terms between various pairwise dipolar couplings in the relaxation of multi-spin systems. Kattawar and Eisner (31) have treated the special case of three identical spin one-half nuclei at the corners of a 30”-120” isosceles triangle. The actual numerical results in this paper are in error, as pointed out in Refs. 32 and 38, doubtless due to the difficulty in solution of 14 simultaneous differential equations, many of which are linearly dependent. Zeidler (32) has carried out a calculation on a similar case where two spins and a non-identical third spin form a triangular arrangement. Richards (33) has derived the interesting result that in the extreme narrowing limit, a system with any number of identical spins must have indentical (though not necessarily a single exponential) time-behavior for both transverse and longitudinal decay, even when cross-correlations are considered. In the absence of cross-correlations, this statement reduces to the well-known fact that T, = T2 under extreme narrowing. Runnels (34), in a rather general calculation, has proved for a system of three equivalent spins one-half describable by a spin-temperature, the inclusion of cross-correlation always acts to make the relaxation less efficient (slower decay). Fenzke (3.5), in a calculation generalized to any number of equivalent spins, independently arrives at many of the same conclusions as Runnels demonstrated. Another example of cross-correlation is a paper by Noggle (36), which shows that for 13C in a methyl group, cross-correlations between the 13C-H’ and H’-H” or 13C-H”-relaxation vectors fortuitously cancel out (to first order), even though the order of magnitude of the spectral density for cross-correlation is the same as for autocorrelation. Kuhlman et al. (37) have come to a similar conclusion from their consideration of Overhauser effects for 13CH3 groups. It should be mentioned however, that the exact behavior of the relaxing four-spin system, 13CH3, has not been carried out in detail, although, as these last two papers might indicate, this would provide a meaningful, albeit tedious, calculation. Recently Pyper (38, 39) has applied Liouville representation formalism
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to the cross-correlation problem and has re-derived previous results to serve as compelling examples of this approach. The other main source of knowledge about cross-correlations between dipole-dipole interactions in magnetic resonance comes from a series of five papers by Schneider (4044) and Blicharski (43, 44), again based on Redfield’s semiclassical density matrix formalism (24). Schneider has derived the longitudinal and transverse relaxation behavior for almost every conceivable arrangement of 3 or 4 spin one-half particles, where the particles are not necessarily equivalent in the molecule and need not even be “like” nuclei. (It should be noted that for calculations treating “unlike” nuclei, it would in general be expected that cross-relaxation effects would lead to nonexponential relaxation even for a two-spin system. Therefore, inclusion of cross-correlation effects for a multispin system, composed of dissimilar nuclei, leads to a very complicated relaxation expression, and one that should be interpreted with care.) For certain configurations of “like” spins, Schneider treats all ranges of the quantity, o,,z,. Furthermore, his treatment can be adapted to motional models includingisotropic oranisotropic rotational diffusion or internal rotation. In cases where the two treatments deal with identical molecular rotational dynamics, Schneider’s results may be seen to reduce to Hubbard’s. For various configurations of the three or four spins, both the longitudinal and transverse relaxation are, in general, the resultant of between three and seven exponentials. Among all the above-listed examples, the one which most closely approximates the criteria of being physically reasonable while at the same time parametrically tractable is that of a symmetric top molecule to which is attached (at arbitrary angle) a methyl group which may rotate independently of the molecule as a whole (12, 13). Unfortunately, the existing literature is limited to the extreme narrowing approximation, and is thus restricted to small molecules in liquids. For presently typical magnetic fields of 15-75 kG in NMR, the extreme narrowing limit will be violated by molecules with molecular weight greater than about 5000 in aqueous solution (proteins, enzymes, membranes, nucleic acids, synthetic polymers), as well as by small molecules in viscous media or by solids near the melting points; the behavior of a methyl group on such molecules is thus unknown. In the Theory section, we treat the dipole-dipole intramolecular relaxation for a methyl group on a symmetric top molecule of any size, including the possibility of internal rotation. THEORY
From a semiclassical model [quantum mechanical spin system coupled to a classical lattice] (27), Hubbard has derived an expression for the longitudinal relaxation of a methyl group (12) :
x(t) = a(t) -- a’; o(t) is a reduced density operator which is obtained from the complete density matrix for the system by averaging over lattice degrees of freedom, and C? is the equilibrium value for this operator. The expectation value of any spin operator A, is given by (A) = Tr[cr(t) A ]. M
NONEXPONENTIAL
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In this section, we derive the expectation values of 1, and I,, using as basis functions the eight spin states of a group of three spin one-half particles, expressed in terms of the respective eigenvalues of P, Z:,, and I,, where I:, = (I1 + 12)2, I2 = (II, + J#, and Iz = I=, + Iz2 + Iz,. If la) = II,, ZM), then notation will be based on the convention,
11)= I1 W 12) = 13) = 14) = 15) = 16) = 17) = 18) =
3/G,
11 312 l/2), j I 3/2 - l/2), 11 3/2 - 3/2), 11 l/2 l/2), 11 l/2 - l/2), IO l/2 l/2), 10 l/2 -l/2).
r31
Letting L41 and introducing two other combinations of matrix elements,
= U/2)(x55-t x77)- W) (X66 + A!*& Y30)= Y2W-t (l/2) (x22- X33)? Y2@>
where
Xijs
r.51
PI VI
it is readily shown that (10)
~Y,W~~= -PJ&%) + 8J,P%)+ 2JcbJ)lYI(~) + WJ~(O,) + 24~~(2~,)1Y,(~) + ~~K(WJ - ~~w,k4t~, d~,wdt = (113) [J,(o,) - 4md - ~~(4 + 4~~(2~~)1k(t) - [35,(O) - Ja(wJ + 2J,(2%) - 3JCCO)+ Jc(aJ - 2Jc(2adlY,(0 + (l/3) [6J,(O) - 10J,(oO) + 4J,(2co,) - 6J,(O) + lOJ,(w,) - 4wdi Ydt), dY&)ld~ = EJakkJ - 2Ja@%) + JC(%)lYl(~) + UP) [9Jh”) + ~J,(w,) + 1w~d~~w - c9u~d - a4 + 4~,(2~,)1 y3(t).
PI
191 DOI
A unique solution to these three coupled differential equations may be found from the initial conditions resulting from application of an rf-pulse along the y-axis in a rotating frame, causing the equilibrium magnetization to rotate by 8”, where relaxation effects during the pulse are ignored. It then follows that ~~(0) = [COSe - mmT, ~43 = W)YI(~), Y3Kn = (1/4)YI(O).
u11
In order to solve Eqs. [&-lo], it is necessary to choose a molecular system and define a model for its rotational dynamics. For the general (but still moderately tractable) system of a symmetric top molecule with the symmetry axis of a rotating methyl group
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attached at angle /3with respect to the symmetry axis of the top, Hubbard has shown that there are just two kinds of spectral densities, J,(W) and J,(o) (13) : (1/4)(3cos2~-
1)‘6DI
+ 3sin2/3cos2/?(5D1 + D,,)
(603 + (km,)’ (5D, + D ,;)” + (kcoo)2 _L (3/4) sin”P(2D, + 40 ,,) + (9/S) sin”P(GD, f 4Di”t) ’ (20, + 40 !,)’ + (key
(60, + 4Dj,J2 + (kqJ2 + (j/2) sin’P(1 + COS”~) (5D, + D !, + 4Dint) (5D,+ D 11+ 4Dint)’ + (k~,)’ + (3/S) [(I + COS’/?)’ + ~COS’~] (20, + 40 11+ 4Dint) , (20, + 40 I/ + 4Di”t)’ + (koo)’ 9 (9/8 sin4P(6D, + 4D,,,) Jc(kwo) = J’(k0o) - @ (60, + 4Di,J2 + (k~,)~
Cl21
+ (3/2 sin’/?(l + cos2~)(5DI + D ,, t 4Di,J (5D,+ D,, + 4Dint)’ + (LOO)’ + (3/X) [(I + COS2p)2 + ~COS’~] (20, + 401, + 4Dint) (20, + 40 11+ 4Dint)’ + (k~,)~ ’ In these equations, D, is the diffusion constant for rotational diffusion about an axis perpendicular to the symmetry axis of the symmetric top, D,, is the parallel diffusion constant, and Dint is the diffusion constant for internal rotation of the methyl group with respect to the molecular frame. By the usual methods, the three simultaneous, first-order, linear homogeneous differential equations [S-lo] with constant coefficients may be solved determina~tall~7 by Laplace transform methods. The form of the solution for the expectation value of I, is given by, (I,) - (Iz)T = (I,)‘(cos i3- 1) [A, e-‘.lf + A2 em+ + A, e-“3t], /I41 for an initial preparation of the system by a Q-pulse. Initial conditions dictate that Ci Ai = 1. The pre-exponential and exponential factors would be exceedingly bulky for listing in closed form, but the longitudinal magnetization is readily computed numerically and is in any case most suitably displayed in graphical form (see Figs. 3-4). For the fast-motion limit, (6DJ2 s 4w& considered previously (12), yl(t) and yz(t) are no longer coupled to y3(t), and the time-dependence for the transverse and longitudinal magnetizations become identical. Since (I,)’ = 0 (there is no equilibrium magnetization in the x-y plane), x(t) = o(t), and we seek (I,) =Tr[a(t)l,]. El151 On setting up the density matrix equations using the same basis set as for the iongitudinal calculation, it becomes evident that the appropriate parametrization is in terms of three linearly independent combinations of the matrix elements of 0.
+
&> =
(l/2)
(l/2) (056
93(f) = 92(t) +
632
k.56 + g65 + d87 + ‘?,I, + G65 + fl87 + a78)~ +
023.
81161 iI71 PSI
NONEXPONENTIAL
METHYL
RELAXATION
305
With these definitions, and using Eqs. [14-201 of (II), the equations for time rate of change of ql, q2, and q3 may be obtained. 4lW~
= l-(3Jm + 3Jc(O)) + (65J,(%) - Jc(%>> + (-2Ja(2%) + 6(Jcb.J + JcG’wJ) q&) + 6(Jc(O) - J&d) dt),
+ 2JcGh3))191(~~ f191
4?,(0/~~= -[Ja(%>- J,(Qh)19&) + Fw,(%) - Jc(aJ>> - 2(Jach) - Jc@%))192(~) + L-Va@> - J,(O))+ (J&4 - JC(wJ)l93@)? PO1 &,(W = -(J,(Q-d - Jc(%>>41(t) + [3(Ja(%) + Jc(wJ)+ ~Jcw%)192~0 + [-3(J,(O) - JdQN - (4J&%) + 2Jc(f%)) - (2~,(2~~) + 4~(2~,))1 h(t).
L-31
‘The initial condition of the spin system is again taken as the instant following a Q-pulse counterclockwise about the y-axis, so that a(f = 0) = e-isry cT e+i6r,. Now for the high-temperature
approximation,
124
Fu, 4 kT,
e-(WkT) (J‘T=
Tr[e’-hE’kT)]
-- (I - (tzE/kT))/Tr[I]
L233
where I denotes the unit matrix. Substituting Eq. [22] into [23], and making note of the fact that E = --cooI,, o(t = 0) = (l/S) + (Aoo/8kT) e-ielyIZ efi61y,
~41
but
eeiexyI, e+isry= I, sin Q + 1, cos 6.
E251
Therefore, cr(t = 0) = (l/S)
x
10000000 01000000 00100000 00010000 00001000 00000100 00000010 00000001
300
010 00-10 0 0 0 000 000 000 000
w,f3cos$ + 16kT
IO
3+ 0 3+ 0 2 0 2 0 0 0 3+
0 0 0 \O
0 0 0 0
0 0 0 0
0 0
0 0 -3 0 0 0 0
0 0 0 0 34 0 0 0 0 0 0 0100 0 1000 0 0001 0 0010
00 00 00 0 0 10 0 -1 00 00 0 0 0 0
00 00 00 0 0 00 0 0 10 0 -1 0 0 0 0
306
WERBELQW
APU’D MARSHALL
and since
/27]
Wt=o = WdO) f,l, ql(0) = sin 19(1,>~,
Lw
do) = (l/6) 41(O), q&a = (v9q1(0).
pcq c303
In these calculations, the second order Zeeman correction (which is of the order of the linewidth if the reciprocal Larmor frequency is of the order of the correlation time of the bath) is neglected. For longitudinal relaxation, this omission has no effect, whereas for tranverse relaxation, inclusion of the correction leads to a small modulation of the time decay of the magnetization envelope, since each of the three decaying exponentials will precess with slightly different frequency. Fraenkel (J. Chem. P&s. 42,4275 (1965)) describes in great detail the effect of this small Zeeman correction, and ref. 28 illustrates the complexity in both calculations and results when the correction is included. In our judgment for the present case, the trivial analytical difference to be expected between the results described here and the “exact” calculation does not warrant the additional computational difficulty required. Solution of the system of differential equations, Eqs. [19-211, subject to the initial conditions, Eqs. [28-301, yields a solution of the form, (I,)* = sin d(1,)’ [B, e+lt + B, emyzf+ B, ecYst].
:311
Again, the B’s are constrained, the sum being equal to unity. This solution is adapted to the rotating frame in which the rapid time dependence has been suppressed (i.e., the true solution of Redfield’s equations yields a result where the r.h.s. of Eq. [31] is multiplied by a factor of exp(--iwO(t + t’)) where t’ is an initial phase factor). As with the longitudinal relaxation, it is not worth listing the pre-exponential and exponential factors analytically; the graphical results are analyzed in the Discussion. It may be pointed out that in the limit of extreme narrowing, both Eqs. [14] and [31] reduce to the sum of two exponentials, and the time-dependence of both equations becomes identical, reducing to Hubbard’s result (12, 13). RESULTS
AND
DISCUSSION
Disregarding cross-correlations (i.e., setting all J,(o) = 0), it can be seen from Eqs. [g-10] and [19-211 respectively that (CL> - WT)/(cos
tJ - l)(L>T = exp[(-2J,(a)
- 8Jat2~d
and (I,)/sin
Q (I,)T = exp[(-3&(O) - 5J,(c0,,) - 2J,(200)) t].
rl,
1321 WI
Woessner et al. (1.5) have previously considered the (uncorrelated) spin-lattice relaxation for axially symmetric ellipsoids with internal motion. It may be verified that Eqs. [32-331 are identical to the result of Ref. (15). Since analytical expressions for the relaxation would be too bulky for ready presentation, we have chosen to express all results graphically in Figs. l-4. For all plots, it was assumed that the methyl group axis was coincident with the symmetry axis of the symmetric top to which the methyl was attached; it was found in general that the nonexponential character of the relaxation was most exaggerated for this case, and in any
NONEXPONENTIAL
METHYL
RELAXATION
307
case, explicit display of this angle-dependence would be unprofitably lengthy. For each plot, the magnetic field was chosen as typical for high-resolution NMR experiments : CC)*= 2n x 108sec-I. Finally, for each plot, an interproton distance of 1.8 &I was assumed, since X-ray and microwave values average close to 1.77 III 0.03 A. Figure I shows plots of log(magnetization) versus time for three values of D,, the diffusion constant for rotational diffusion about an axis perpendicular to the symmetry axis of the symmetric top: the upper plot corresponds to an extreme-narrowed case, 60,* 2c00, and in this limit the transverse and longitudinal magnetizations have the same time-dependence so only one plot is required; the middle plots show the longitudinal (left-hand plot) and transverse (right-hand plot) magnetization for a case where 6D, LZ20,; the bottom plots give the longitudinal (left-hand plot) and transverse
FIG. 1. Plots of Iog(magnetization) versus time, following a &pulse along the y-axis of a rotating frame, for a methyl group attached along the symmetry axis of a symmetric top molecule. For plots lb and Id, g(t) =
308
WERBELOW
AND
MARSHALL
(right-hand plot) magnetizations for a methyl group on a very large molecule in solution (or small molecule in very viscous solution), 6DL c 2~0,. The extreme-narrowed case (Fig. la) is identical to Hubbard’s Fig. 1 (12) except for a change in notation of the abscissa scale. From comparison of curves P-V for each graph, for a symmetric top of any size, relaxation (whether longitudinal or transverse) is most nonexponential when the top is most prolate and/or exhibits fastest internal motion. For large molecules (bottom two plots), the longitudinal magnetization decays very nearly according to a single exponential, while the transverse decay is markedly nonexponential. Two general properties proved by Runnels (34) are manifest in Fig. 1: first, that inclusion of crosscorrelations always retards the relaxation; and second, that the initial slope of each
FIG. 2. Resolution of a plot of log(longitudina1 magnetization) versus time into its (three) component exponentials. Curve denotes plot V of Fig. lb. Dotted line denotes linear result obtained by neglecting cross-correlation effects. The three solid lines (in order of increasing slope) are plots of Iog(O.125exp [-0.678t]), log((0.125 + 0.704) exp[-5.34t)), and log(exp[-12.5t]) vesus f. See Eq. [34] and accompanying discussion.
curve approaches the slope of the single exponential decay obtained when cross-correlation is neglected (see Fig. 2). Finally, it should be noted that in Fig. 1 the abscissa is in specific units of seconds rather than the more general “reduced time” variable as seems to be the common convention (12,31). This is because, unless working within the eonfines of the limit wOz, < 1, the explicit field dependence detracts from the usefulness of this form of presentation. Although the scale as shown in Fig. 1 is specific for the case ofan isolated methyl group in a 23.5 kG field (w, = 100MHz); the graphs are still rather general in their presentation. For example, if the gyromagnetic ratio were smaller (as for 19F), the shapes of all curves would remain the same, but the time scale would be contracted (slower relaxation). If r were smaller, the shapes of the curves would again remain the same, and the time scale would be expanded (faster relaxation). For diEerent values of LOO,the curves will again retain the same shapes, provided that the rotational diffusion constants are changed by the appropriate factor. In Fig. 2, curve V of Fig. lb is reproduced (curved line), and resolved into its three component exponentials (the three solid lines), and compared to the result obtained when cross-correlations are neglected (dotted line). The actual equation of the curve is, ((I,) - (I,)‘)/(cos
6 - 1) (I,)= = (0.125) e-0.678t + (0.171) e-12.5t + (0.704) e-5Q34t-~ D41
NONEXPONENTIAL
METHYL
RELAXATION
309
Note that the y-intercepts of the three solid lines are: log(O.125), log(O.125 + 0.704) and log(1.) ; this seems to be the most useful way to show the limiting behavior at long times. From this plot, it is clear that the initial slope of the curve approaches that ofthe dotted line. The most general display of the present calculations is provided by Figs. 3 and 4, which show the relative contributions and time constants of the three exponentials for the longitudinal (Fig. 3) and transverse (Fig. 4) magnetizations. The pre-exponential factors are shown in the upper three plots of each figure, and the relative time-constants in the lower two plots. This manner of presentation serves the same purpose as an ex-
7
8
9
10
11
LogiD~~Dt)
Log(D,, +Di) FIG. 3. Nonexponentiality of time-decay for longitudinal magnetization, displayed as contour plots, where each contour is a line of constant magnitude for either preexponential factors (AJ or exponential ratios Aj/?,k from Eq. [14], as a function of D, and (D,, + DJ. See text for further details.
tensive digital table or a 3-dimensional plot depicting the mutual effects of size, shape, and internal flexibility on the relaxation behavior of the spins. These two figures are simply contour plots and are interpreted as one would use a topographical map. For example, say one wishes to determine the relaxation behavior of an isolated, flexible methyl group attached to the backbone of a large molecule in solution. Furthermore, for the purpose of serving as an example, assume the large molecule is roughly spherical in shape and the system can be characterized by the three diffusion constants, D, = D,, = 10grad2/sec, and Dint = 101°rad2/sec. Referring to Fig. 3, it is found that 3b2z A, E 4i., and the preexponentials for the two distinctive terms are (AZ + AX) sz0.92 and A 1 z 0.08. Of course, the same relaxation would be expected if lOD, = D ,, = 1O1O
310
WERBELOW
AND
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rad2/sec and Dint = 109rad/%ec (this is due to the fact that for these plots, fi = 0”; therefore, (D,! + Dint) is effectively a single variable). Figures 3 and 4 are most informative when used in conjunction with Fig. 1. By referring to Fig. 1, it is now possible to conclude how small the ratio of &/& (and/or &/,I,) and also how comparable AI and A2 (and/or AS) must be in magnitude in order that relaxation be markedly nonexponential. Also, only the ratios of the exponential factors are shown in Figs. 3 and 4. Therefore, relative, not absolute behavior is depicted in these figures. Again, Fig. 1 shows some absolute plots for fixed choices of Dint, D ,,, and DL. With the aid of Figs. 3 and 4, it is
7
8
9
10
7
8
9
10
J
8
9
10
11
Log [ D,,‘Di]
FIG. 4. Nonexponentiality of time-decay for transverse magnetization, displayed as contour plots, where each contour is a line of constant magnitude for either preexponential factors (I?,) or exponential ratio yj/yr factors from Eq. [31], as a function of D, and D ,, + Of). See text for further details. (Note the trivial change in notation: D, in Figs. 3 and 4 corresponds to I&,, in the text.)
possible to make quite precise qualitative interpolations or extrapolations. In order for nonexponential behavior be manifest, it is necessary botlz that at least two of the pre-exponential factors (A’s or B’s) be of the same order of magnitude and that the ratio of the corresponding exponentials differ appreciably from unity. From Figs. 3 and 4, it is seen that nonexponential relaxation is most evident for very prolate tops and/or when internal rotation is fast compared to DI. In the extreme narrowing limit (upperright hand region of each plot), A, and B, approach zero and/or &/I,,+1 ; in either event, the result reduces to the two exponentials obtained by Hubbard (12). It should also be mentioned that the regions of the plots where D, > (a,,, + D ,,) have little physical meaning since (even for an extremely flattened ablate top) D, doesn’t differ appreciably from the value for D ,,.
NONEXPONENTIAL
METHYL
RELAXATION
311
A final intriguing (but unexplained) feature of the transverse magnetization is that for a [(D,, $ &)/DL]-ratio of 7/4, the relaxation approaches that of a single exponential, i.e., B,+ 1.OOin Eq. [31], as shown in Fig. 4. This fact can be shown analytically for the extreme narrowed case (22) and appears simply to be an accident of the algebra. Extension of the treatment to -CF, groups follows directly. The interfluorine distance is typically 2.21 & 0.04 1$, and the lgF magnetogyric ratio is (25179/267§3) that for protons; thus for the same value of w0 and identical molecular dynamics, the 19F dipole-dipole relaxation rate will be down by a factor of [2.52/2.6714 [1.77/2.2116 z (l/5) that for a proton methyl group. From Figs. 3 and 4, it is evident that the best system in which to observe nonexponential decay in magnetization should be a molecule possessing a methyl group whose internal rotation rate is fast compared to reorientation of the molecule as a whole. For small molecules in liquid state, this means that for the effect of cross-correlations to be noticeable, Dint will be restricted to cases where the methyl group rotation approaches the “free diffusion” limit where Dint is of the order of (kT,lmethvi)a z 1Oi3 rad2/sec, where Imcthy, is the moment of inertia for the methyl group. Just such an example is provided by liquid acetonitrile [the molecule for which cross-correlation was first manifested in the solid state (2)]; by comparing proton and nitrogen relaxation times in. CH,CN, it may be shown that the rotation about the C-N axis is about ten times as fast as reorientation of that axis itself (45). However, the proton relaxation was observed to be exponential. This incongruity is resolved by a study of the temperature dependence of T,, which shows that a spin-rotation interaction is present and provides the dominant intramolecular relaxation mechanism above 25°C. Moreover, it has recently been argued (46,47) and demonstrated (48) that the magnetic field due to internal rotation can also produce a spin-rotation interaction; thus even for methyl groups on Iarge molecules, the presence of free or nearly free internal rotation should introduce a large (exponential) spin-rotation relaxation which should mask any nonexponential effects due to dipole-dipole interactions. There is however one situation in which nonexponential decay might be observable in liquids. The spin-rotation interaction generally has the form (46), P51 (l/T&pin-rotation = (213) kTF2 I, C,z rSR, where I, is the moment of inertia of the internal rotor, C, is the corresponding internal rotation spin-rotation constant, and rsRis the correlation time for changes in magnitude of the spin-rotation interaction. rsR i s distinguished in two ways from the average correlation time, rDD, for changes in magnitude of the dipole-dipole interaction: first, rsRmay be thought of as the average time between “collisions” which change either the principal axis direction or rate of the internal rotation, so that rsR becomes sJzorter as the rotational motion becomes more hindered (this may be contrasted with rDD, which may be regarded as the average time it takes for the dipole-dipole axis to reorient of the order of a radian, so that rDD becomes loizgev as the rotation becomes more hindered; thus the dipole-dipole and spin-rotation interactions generate opposite temperature dependence for relaxation, which provides a means of resolving their relative contributions); and second, rsR is in general shorter than t,,,,. Thus, in order to emphasize the dipole-dipole interaction compared to the spin-rotation, one need simply locate a .molecular system with relatively hindered internal motion; subject to this constraint,
312
WERBELOW
AND
MARSHALL
one couid then reasonably expect to observe nonexponential relaxation, provided that the (hindered) internal motion were still fast compared to reorientation of the molecule as a whole. While the importance of spin-rotation contribution to relaxation is notoriously difficult to assessin advance (49), the form of the relaxation is well established for rotational diffusion of spherical (50,51) and symmetric (52,53,54) tops, and recently for a more general motional model for symmetric tops (55). In any case, there is well documented evidence that spin-rotation is an important contribution to relaxation for both -CH, groups (50,54) and -CF, groups (48, SO), on small molecules in liquids. CONCLUSIONS
It has been shown that intramolecular dipole-dipole relaxation for an isolated methyl group can be markedly nonexponential, especially for internally rotating methyl (or trifluoromethyl) groups on large molecules in solution. However, unless the internal motion itself is appreciably hindered, the spin-rotation relaxation will contribute and probably mask any nonexponential effect experimentally. Thus, to detect nonexponential decay in magnetization, the methyl group internal rotation should be fast compared to rotation of the molecule as a whole, but slow compared to a “free diffusion” model. For -CF, groups, the effect will probably never be observable in liquids, because the dipole-dipole relaxation rate is slower by a factor of five, and the spin-rotation coupling is larger than for protons. For -CF3, then, nonexponential relaxation would be expected only when the internal rotation of the -CF, group is very “hindered” However, in order that the nonexponential effect be observable, one would have to invoke a macromolecule whose rotational anisotropy differs by an order of magnitude from a sphere, which is highly improbable in practice. There is one other way in which fluorine magnetic relaxation might be nonexponential, and that is due to possible crosscorrelation effects between competing relaxation processes. This problem is formally similar to the present calculation and will be treated in a later paper. Finally, it has been shown that a plot of either the transverse or longitudinal magnetization versus time gives an initial slope which is the same as would be obtained by neglecting all cross-correlation effects, further compounding the experimental difficulty of detecting nonexponentiality. Nevertheless, an internally rotating methyl group on a large molecule in solution should show nonexponential relaxation. The fact that none has been reported for such systems should signal the likelihood that other relaxation mechanisms are operative (principally spin-rotation and intermolecular dipoledipole interactions); there is at least one example for intermolecular contributions to relaxation for a macromolecule (57). REFERENCES I. 2. 3. 4. 5. 6. 7. 8.
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12