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Observation of 2izsz order in NMR relaxation studies for measuring cross-correlation of chemical shift anisotropy and dipolar interactions
Volume 138, number
6
CHEMICAL
PHYSICS
LETTERS
7 August
1987
OBSERVATION OF 21,& ORDER IN NMR RELAXATION STUDIES FOR MEASURING CROSSCORRELATION OF CHEMICAL SHIFT ANISOTROPY AND DIPOLAR INTERACTIONS Guy JACCARD,
Stephen WIMPERIS
and Geoffrey BODENHAUSEN
Institut de Chimie Organique, UniversitP de Lausanne, Rue de la Barre 2. CH-1005 Lausanne, Switzerland Received
5 May 1987
The cross-correlation spectral density of chemical shift anisotropy and dipolar interactions in isotropic solution of methyl formate is measured using a novel NMR technique designed to monitor the appearance of longitudinal two-spin order 2I,S, during mversion recovery. The result is tentatively interpreted in terms of the orientation of the ‘%-‘H internuclear vector with respect to the principal axes of the “C chemical shift anisotropy tensor of the carboxyl group.
1. Introduction The advances made over the past few years in the techniques of pulsed NMR spectroscopy [ l-31 have great potential for the observation of subtle relaxation phenomena that were previously inaccessible. For instance, time-domain multiple-quantum spectroscopy has greatly simplified the determination of transverse relaxation rates of forbidden transitions, making it possible to determine the cross-correlation of random fields [4] and of quadrupolar interactions [ 51. Recently developed methods based on coherence transfer have made it possible to detect deviations from a simple exponential decay, both for transverse [ 6-101 and longitudinal relaxation [ 10,111. In longitudinal relaxation studies, new techniques have made it easier to distinguish different forms of non-equilibrium population distributions. Thus, in spins with S=3/2 for example, one may separate Zeeman, quadrupolar and octupolar orders experimentally [ 10,111, whilst in systems with scalar-coupled spin-112 nuclei one can monitor the dynamics of longitudinal q-spin order 2 (q- I) n IZ,, [ 12-141. The introduction of two-dimensional exchange spectroscopy [ 15,161 has proven to be useful for studying the migration of Zeeman order from one spin to another (e.g. Zk=+Zl_). The basic twodimensional exchange experiment can easily be modified to monitor the migration of longitudinal q0 009-2614/87/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
spin order (e.g. 2Zk,Z,+2Z,,Z,;) [ 13,141. If the transition probabilities W,, across parallel transitions are equal, there can be no “cross-talk” between longitudinal spin orders of different q. This behaviour not only breaks down in the presence of strong coupling, but also in weakly coupled systems where several relaxation mechanisms of the same rank are acting simultaneously. In this case, so-called “interference” terms due to cross-correlation provide a pathway for the interconversion of longitudinal spin orders of different q. One example is discussed by Fagerness et al. [ 17 ] : in a weakly coupled system of three spinl/2 nuclei, Zeeman polarization Zk_ may be partly converted into three-spin order 4Zk,II;I,, due to crosscorrelation of the k-l and k-m dipolar interactions. We intend to return to such homonuclear threespin cases in a later study; in this Letter we focus attention on an analogous situation that arises in heteronuclear two-spin systems, where one can selectively observe the conversion S,+2Z& due to interference of chemical shift anisotropy (CSA) and dipolar interactions. It will be demonstrated that the resulting cross-correlation spectral density can be measured with a simple modification of the wellknown inversion-recovery technique.
B.V.
601
Volume
138, number
6
CHEMICAL
PHYSICS
2. Chemical shift anisotropy and dipolar interactions The use of high magnetic fields has recently lead to renewed interest in chemical shift anisotropy as a mechanism for spin relaxation in isotropic liquids, the rate of relaxation being proportional to the square of the magnetic field B,, [ 181. The CSA interaction can interfere with other second-rank mechanisms, such as dipolar or quadrupolar interactions, resulting in a non-vanishing cross-correlation spectral density [ 191. The phenomenon of cross-correlation of CSA with the dipolar interaction has long been the subject of theoretical studies [ 20-221, but is difficult to observe experimentally at low fields except if the CSA interaction is large, as is often the case for 19F[ 221. Most experimental reports concerning pairs of nuclei such as lSN-‘H [23], 3’P-‘9F [24,25], “C-‘H [ 261, 19F-‘H [ 27’1 and ‘H-‘H [ 281, have made use of high magnetic fields and are consequently quite recent. In ESR, on the other hand, the analogous cross-correlation between the anisotropic hyperfine coupling and the anisotropic electronic g factor has been amenable to study for many years [291. The occurrence of cross-correlation between CSA and dipolar interactions can be observed either for transverse (“T,“) or for longitudinal (“T,“) relaxation. For a system with two weakly coupled spinl/2 nuclei, cross-correlation leads to a linewidth difference of the two doublet components of the spin which is relaxed by both mechanisms. Measurement of this “differential line broadening” [ 24,26,30], however, is fraught with difficulties since inhomogeneous decay must be refocused, whilst ill-resolved scalar couplings to further spins make echo experiments hazardous. Consequently, it appears more attractive to measure cross-correlation by observing its effects on longitudinal relaxation, since neither inhomogeneous broadening nor scalar couplings will be detrimental, although one must pay attention to extraneous mechanisms such as external random fields.
LETTERS
7 August
of the three operators I,, S= and 2Z,S, [ 12,17,21], since the sum of the populations is always conserved. The equations of motion for these longitudinal terms may be written: $($%J=E
;
E)(%;,)
>
(1)
where (Ly__)=(Z,)-(I”), (AS,)=(X)(ST ), and where (1” ) and (ST ) are the expectation values of the operators 1= and S, at thermal equilibrium. Considering CSA interactions for both nuclei in addition to the dipolar interaction, but neglecting contributions from external random fields, we obtain: A = - 9 Jm,s - 451.1 , B=
-‘pJws-4Js.s
>
C= - 2Jws - 4Js.s - 4J,, ,
E= - 4J1s.s , F= -4J.w
>
(2)
where J,s,~~, JI., and Js,s are the auto-correlation
spectral densities for the dipole-dipole and the two CSA relaxation mechanisms, and J1s,s and Jls,, are the cross-correlation spectral densities. If I= ‘H and S= 13C,the terms JI.I and Jls.[ which depend on the (very small) proton chemical shift anisotropy are negligible. It is this situation that will concern us. With the assumptions of fast molecular motion (“extreme narrowing”) and isotropic motion, the remaining spectral densities simplify to the following expressions: J ,s,,s=~t~~4x)2Y:YBfi2r-6r,,
(3a)
Js.s= 3b~3W44~
t3b)
,
Jls.s= - h W4~ )YIYifir-3& x WC, ~.a., ~YZ~hC
,
3. Theory
where
The four populations of a two spin-112 system IS can be expressed in terms of the expectation values
L(b)=~::+a:+a~-a,aY-a*az--yrQZ
602
1987
and
(3c)
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138, number
CHEMICAL
6
PHYSICS
The values ox, u y and gz refer to the frame OXYZ of principal axes of the shielding tensor. The IS internuclear vector lies along the OZ’ axis. The angles 8,,. and eyz. define the relative orientation of the IS internuclear vector with respect to the principle axes X and Y of the CSA interaction. These expressions are consistent with those given by Goldman
[211. The solution of eq. (1) will show multi-exponential behaviour. However, if the experiment is commenced by inverting or saturating only the S spin magnetization and if the linear initial rate approximation is invoked, we obtain: (I_) = (I”)
+Dt(AS:)
(22) = (S!?) +BT(AP)
(21:&S,) =El(ASL?) )
)
(da) )
(4b)
(4c)
where (Asp)=-(SF), and (SF) is the expectation value of S, immediately after the perturbation that commences the experiment, for example, a 180” inversion pulse applied uniformly to both S spin doublet components. Thus the initial change in the Z, magnetization is linear (a transient nuclear Overhauser effect), while S, initially recovers linearly towards its equilibrium value. Most importantly in the present context, there is a linear growth of longitudinal two-spin order 21-S,. Note that the initial expectation value of this term is zero provided the perturbation affects both S spin transitions equally. The appearance of a non-zero expectation value for the operator 21-S: is due to unequal longitudinal relaxation rates across the two parallel S spin transitions. This manifests itself as a differential recovery of the two components of the S spin doublet. The difference in the intensity of the two lines is directly proportional to ( 215). If the internuclear distance r and the principal values of the shielding tensor are known from other sources, the correlation time 7c can be determined from the recovery of the S, magnetization, which is proportional to the sum of the amplitudes of the two
LETTERS
7 August
1987
doublet components. According to eq. (4b) the S, recovery occurs with the initial rate B= -‘pJ,.sls - 4Js.s, provided external random field relaxation is negligible. Similarly, if the growth of (2Z&) can be measured then, from eq. (4c), the initial slope will give the constant E= - 4Jls.+ To determine the sign of the initial rate E it is necessary to identify the sign of the term (2Z&). If the absolute sign of the heteronuclear scalar coupling constant is positive, ( 2Z&) is positive if the low-frequency (high-field) component of the S spin doublet relaxes faster than the high-frequency component [ 12,2 1,221. Knowing the sign and magnitude of E, and having measured the value of ‘so it is possible to determine K( cr, eXz,, 8yz ) (see eq. ( 3~)). Under certain simplifying assumptions, one may then determine the relative orientation of the internuclear 13C-‘H vector with respect to the principal axis system of the 13Cshielding tensor - a quantity normally obtained only from solid-state NMR.
4. Experimental methods and results The molecule chosen for study was methyl formate, where there is cross-correlation between the carboxylic “C CSA interaction and the 13C-‘H dipolar interaction. Asymmetric longitudinal relaxation was observed in the enriched compound H’3COOCD3 (the methyl group was deuterated to prevent further dipolar interactions from interfering). The material was prepared by alkylation of formic acid (90% 13C enriched) with excess methanol-d, in the presence of traces of acid. A 10 vol”!‘o solution in methanol-d, was sealed in vacua in a 10 mm tube after three freeze-pump-thaw cycles. NMR spectra were recorded at 275 K using a Bruker AM 400 spectrometer (B,,= 9.4 T). Conventional 13C inversion-recovery measurements on this sample showed only a very slight asymmetry in the recovery of the two carboxyl 13C lines, indicating that the cross-correlation effect is quite weak compared to the overall relaxation rate. Thus, during the first second of the recovery, where the growth of 2ZS= magnetization is expected to be linear, it was barely possible to detect an asymmetry. Much as in conventional steady-state Overhauser 603
Volume 138, number 6
CHEMICAL PHYSICS LETTERS
Intensity .. : 0.010
. . I
.
.
I
: . . . .
I
”
10
”
I
”
20 Delay
”
I
30
”
,’
1,
40
”
I
50
(s)
Fig. 1. Plot of the intensity of - (2I,S,)l(AS9) as a function of the recovery delay r for the experiment on methyl formate described in the text. The inset shows the difference spectrum observed after a relaxation delay of 6 s. A plot (not shown) of the recovery of SZmagnetization (defined as (&)I( AS! ) ) was extracted from the same data. Analysis of these plots yielded an estimate of the orientation of the “C-‘H internuclear vector with respect to the principal axes ofthe “C CSA tensor ofthe carboxyl group.
studies, we therefore found it necessary to use a form of difference spectroscopy to separate the weak antiphase component from the dominant in-phase component. Two spectra with v,= _+x were recorded and stored separately for each value of the delay r in the sequence: Z S
90X90,, 180~r-90,-acquire.
The sequence applied to the S ( 13C) spins is a conventional inversion-recovery experiment. Longitudinal relaxation gives rise to S, and 2Z_Szterms which are transformed into observable S, and ? 2ZSXoperators, where the sign of the antiphase term depends on the relative phases of the two pulses applied to the Z spins. The sum of the spectra gives the coefficient of S,; the difference gives the (much smaller) coefftcient of 2Z_S,.This experiment can be seen as a heteronuclear form of multiple-quantum filtered inversion-recovery [ IO]. Fig. 1 shows the coefficient of the 2Z_&term as a function of the recovery delay r. A typical difference spectrum is shown in the inset. Nine measurements of the S, and 21-S, terms were taken from the lirst 604
7 August 1987
second after the inversion pulse where linear rates were observed. Linear regression gave the rates B=-(6.19+0.19)x10-* s-r and E=-(0.484f 0.086) x lo-* s-l. The proton-carbon internuclear distance in methyl formate r= 1.101 x 10 -lo m is known from microwave rotational spectroscopy [ 3 11. The principal values of the carboxyl shielding tensor in methyl formate are known from solid state powder spectra [ 321: 6,r = -90.5 ppm, a,,=29.1 ppm and cr33= 6 1.5 ppm with respect to the isotropic shift. With these values the correlation time is calculated to be r =(2.91 kO.09) x lo-‘* s. InseAion of this value of the correlation time into the expression for the initial rate E of the 2ZSz term (see eqs. (2), (3)) gives K(a,8,,6Yz)= -(9.25*1.99)x10-5. Assigningo,, to o,and 033 to cry and assuming that 033= oy is perpendicular to the OCO plane (i.e. eyz, = 90’) in accordance with solid-state studies of carboxyl shielding tensors [ 331, we find that eXz, the angle between the least shielded axis ( CJ, , ) of the carboxyl CSA tensor and the 13C-‘H internuclear vector, can take one of two values: 45 ’ ? 3 oor 135 ’ k 3 ‘. This ambiguity, which follows from the fact that the cosine function occurs as a square in the Legendre polynomial, is common to many NMR measurements. The single-crystal NMR study of Cornell [ 341 for dimethyl oxalate indicates that err , lies in the OCO plane, perpendicular to the C=O bond. If this is also true for methyl fox-mate, the angle OX=.would be 145”. Hence of our two possible results the angle 135” + 3’ would seem most plausible. The agreement of this result with the twodimensional powder spectra study of Linder et al. [ 321 is also satisfactory. The errors quoted in the above results refer only to errors that have propagated from our experimental uncertainties. We have made use of a number of simplifying assumptions that might lead to additional errors in the final estimate of the angle 8,,. The assumption of extreme narrowing appears to be quite safe, but the hypothesis of isotropic motion implicit in eq. (3) could be challenged. We have also assumed that external random field mechanisms did not contribute to the rate B, which in this case seems to be a good approximation [ 351.
Volume 138, number 6
CHEMICAL PHYSICS LETTERS
7 August 1987
5. Prospects for further study
Acknowledgement
The pulse sequence used in this Letter for selective detection of heteronuclear 2Z& order can be considered as a prototype of a family of novel experiments. Various means can be devised for the detection of longitudinal q-spin order arising from relaxation. For instance, in a homonuclear two-spinl/2 system, where both S spins could be 13C, the 2&S,= term arising from cross-correlation between the CSA and dipolar interactions can be selectively detected using a double-quantum filter. Likewise, in a homonuclear system with three spin-l/2 nuclei, where all nuclei could be ‘H, the conversion of Zkr into 41,,1,,1,,,, terms due to cross-correlation between pairs of dipolar interactions [ 171 can be monitored using a triple-quantum filter. A variation of twodimensional exchange spectroscopy could be devised for selective observation of processes which involve interconversion of different classes of longitudinal qspin order. At the beginning of the mixing period, a composite 90” pulse would be used to convert Zkxto 4, without allowing antiphase terms 4Z&ZjzZ,,,,to be converted into three-spin order 4Zk,Z,,Z,= [ 131. At the end of the mixing period, a triple-quantum filter could be used to convert only three-spin order terms 4Z,,Z,=Z,,, into observable transverse magnetization. Such an experiment would allow measurement of cross-correlation spectral densities between pairs of dipolar interactions, which holds the promise of determining the angle subtended by two internuclear vectors.
The authors are grateful to Dr. J. Keeler for a preprint of ref. [ 271. This research was supported in part by the Swiss National Science Foundation (Grant 2.925-0.85).
6. Conclusions In this Letter we have estimated the orientation of the CSA tensor principal axis system of the carboxyl 13C of methyl formate in an isotropic liquid phase. The fact that this seems to be possible with a fair degree of accuracy, without resort to solid-state NMR of single crystals [ 33,341, or two-dimensional NMR of polycrystalline solids in static [ 32,331 or spinning samples [36,37], holds much promise for further study.
References [ 1] R.R. Ernst, G. Bodenhausen and A. Wokaun. Principles of nuclear magnetic resonance in one and two dimensions (Clarendon Press. Oxford, 1987). [ 21 K. Wiithrich, NMR of proteins and nucleic acids (Wiley, New York, 1986). [3 ] N. Chandrakumar and S. Subramanian, Modem techniques in high-resolution FT-NMR (Springer, Berlin, 1987). [4] A. Wokaun and R.R. Ernst, Mol. Phys. 36 (1978) 3 17. [ 51 R.L. Vold, R.R. Void, R. Poupko and G. Bodenhausen, J. Magn. Reson. 38 (1980) 141. [6] N. Miiller, G. Bodenhausen, K. Wiithrich and R.R. Ernst, J. Magn. Reson. 65 (1985) 531. [ 7 ] N. Miiller, G. Bodenhausen and R.R. Ernst, J. Magn. Reson., to be published. [ 81 M. Rance and P.E. Wright, Chem. Phys. Letters 124 (1986) 572. [9] J. Pekar and J.S. Leigh, J. Magn. Reson. 69 (1986) 582. [IO] G. Jaccard, S. Wimperis and G. Bodenhausen, J. Chem. Phys. 85 (1986) 6282. [ II] N. Miiller, Chem. Phys. Letters 131 (1986) 2 18. [ 121 O.W. Sorensen, G.W. Eich, M.H. Levitt, G. Bodenhausen and R.R. Ernst, Progr. NMR Spectry. 16 (1983) 163. [ 131 G. Bodenhausen, G. Wagner, M. Rance, O.W. Sorensen, K. Wiithrich and R.R. Ernst, J. Magn. Reson. 59 (1984) 542. [ 141 G. Wagner, G. Bodenhausen, N. Miiller, M. Rance, O.W. Sorensen, R.R. Ernst and K. Wiithrich, J. Am. Chem. Sot. 107 (1985) 6440. [ 151 J. Jeener, B.H. Meier, P. Bachmann and R.R. Ernst, J. Chem. Phys. 71 (1979) 4546. [ 161 A. Kumar, R.R. Ernst and K. Wiithrich, Blochem. Biophys. Res. Commun. 95 (1980) 1. [ 171 P.E. Fagemess, D.M. Grant, K.F. Kuhlmann, C.L. Mayne and R.B. Parry, J. Chem. Phys. 63 (1975) 2524. [ 181 A. Abragam, The principles of nuclear magnetism (Oxford Univ. Press, Oxford, 196 I ) . [ 191 R.L. Vold and R.R. Vold, Progr. NMR Spectry. 12 (1978) 79. [ 201 H. Shimizu, J. Chem. Phys. 40 (1964) 3357. [ 2 11 M. Goldman, J. Magn. Reson. 60 (1984) 437. [22] E.L. Mackor and C. MacLean, J. Chem. Phys. 44 (1966) 64. [ 231 M. GuCron. J.L. Leroy and R.H. Griffey, J. Am. Chem. Sot. 105 (1983) 7262. [24] T.C. Farrar and R.A. Quintero-Arcaya, Chem. Phys. Letters 122 (1985) 41.
605
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[25] S.G. Withers, N.B. Madsenand B.D. Sykes, J. Magn. Reson. 61 (1985) 545. (261 T.C. Farrar, B.R. Adams, G.C. Grey, R.A. Quintero-Arcaya and Q. Zuo, J. Am. Chem. Sot. 108 (1986) 8 190. [27] J. Keeler and F. Sanchez-Ferrando, J. Magn. Reson., to be published. [28] F.A.L.Anet, J. Am.Chem. Sot. 108 (1986) 7102. [ 291 H.M. McConnell, J. Chem. Phys. 25 (1956) 709. [ 301 L.G. Werbelow, J. Magn. Reson. 7 1 (1987) 151. [31] R.F.Curl Jr., J.Chem. Phys. 30 (1959) 1529. [ 321 M. Lindert, A. Hohener and R.R. Ernst, J. Chem. Phys. 73 (1980) 4959.
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[ 331 M. Mehring, NMR - Basic principles and progress, Vol. 11. High resolution NMR spectrosppy in solids (Springer, Berlin, 1976). [34] B.A. Cornell, J. Chem. Phys. 85 (1986) 4199. [ 351 T.D. Alger, D.M. Grant and J.R. Lyerla Jr., J. Phys. Chem. 75 (1971) 2539. [ 361 M.G. Munowitz, R.G. Grifftn, G. Bodenhausen and T.H. Huang, J. Am. Cheni. Sot. 103 (1981) 2529. [ 371 M.G. Munowitz and R.G. Grit%, J. Chem. Phys. 76 (1982) 2848.
6
CHEMICAL
PHYSICS
LETTERS
7 August
1987
OBSERVATION OF 21,& ORDER IN NMR RELAXATION STUDIES FOR MEASURING CROSSCORRELATION OF CHEMICAL SHIFT ANISOTROPY AND DIPOLAR INTERACTIONS Guy JACCARD,
Stephen WIMPERIS
and Geoffrey BODENHAUSEN
Institut de Chimie Organique, UniversitP de Lausanne, Rue de la Barre 2. CH-1005 Lausanne, Switzerland Received
5 May 1987
The cross-correlation spectral density of chemical shift anisotropy and dipolar interactions in isotropic solution of methyl formate is measured using a novel NMR technique designed to monitor the appearance of longitudinal two-spin order 2I,S, during mversion recovery. The result is tentatively interpreted in terms of the orientation of the ‘%-‘H internuclear vector with respect to the principal axes of the “C chemical shift anisotropy tensor of the carboxyl group.
1. Introduction The advances made over the past few years in the techniques of pulsed NMR spectroscopy [ l-31 have great potential for the observation of subtle relaxation phenomena that were previously inaccessible. For instance, time-domain multiple-quantum spectroscopy has greatly simplified the determination of transverse relaxation rates of forbidden transitions, making it possible to determine the cross-correlation of random fields [4] and of quadrupolar interactions [ 51. Recently developed methods based on coherence transfer have made it possible to detect deviations from a simple exponential decay, both for transverse [ 6-101 and longitudinal relaxation [ 10,111. In longitudinal relaxation studies, new techniques have made it easier to distinguish different forms of non-equilibrium population distributions. Thus, in spins with S=3/2 for example, one may separate Zeeman, quadrupolar and octupolar orders experimentally [ 10,111, whilst in systems with scalar-coupled spin-112 nuclei one can monitor the dynamics of longitudinal q-spin order 2 (q- I) n IZ,, [ 12-141. The introduction of two-dimensional exchange spectroscopy [ 15,161 has proven to be useful for studying the migration of Zeeman order from one spin to another (e.g. Zk=+Zl_). The basic twodimensional exchange experiment can easily be modified to monitor the migration of longitudinal q0 009-2614/87/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
spin order (e.g. 2Zk,Z,+2Z,,Z,;) [ 13,141. If the transition probabilities W,, across parallel transitions are equal, there can be no “cross-talk” between longitudinal spin orders of different q. This behaviour not only breaks down in the presence of strong coupling, but also in weakly coupled systems where several relaxation mechanisms of the same rank are acting simultaneously. In this case, so-called “interference” terms due to cross-correlation provide a pathway for the interconversion of longitudinal spin orders of different q. One example is discussed by Fagerness et al. [ 17 ] : in a weakly coupled system of three spinl/2 nuclei, Zeeman polarization Zk_ may be partly converted into three-spin order 4Zk,II;I,, due to crosscorrelation of the k-l and k-m dipolar interactions. We intend to return to such homonuclear threespin cases in a later study; in this Letter we focus attention on an analogous situation that arises in heteronuclear two-spin systems, where one can selectively observe the conversion S,+2Z& due to interference of chemical shift anisotropy (CSA) and dipolar interactions. It will be demonstrated that the resulting cross-correlation spectral density can be measured with a simple modification of the wellknown inversion-recovery technique.
B.V.
601
Volume
138, number
6
CHEMICAL
PHYSICS
2. Chemical shift anisotropy and dipolar interactions The use of high magnetic fields has recently lead to renewed interest in chemical shift anisotropy as a mechanism for spin relaxation in isotropic liquids, the rate of relaxation being proportional to the square of the magnetic field B,, [ 181. The CSA interaction can interfere with other second-rank mechanisms, such as dipolar or quadrupolar interactions, resulting in a non-vanishing cross-correlation spectral density [ 191. The phenomenon of cross-correlation of CSA with the dipolar interaction has long been the subject of theoretical studies [ 20-221, but is difficult to observe experimentally at low fields except if the CSA interaction is large, as is often the case for 19F[ 221. Most experimental reports concerning pairs of nuclei such as lSN-‘H [23], 3’P-‘9F [24,25], “C-‘H [ 261, 19F-‘H [ 27’1 and ‘H-‘H [ 281, have made use of high magnetic fields and are consequently quite recent. In ESR, on the other hand, the analogous cross-correlation between the anisotropic hyperfine coupling and the anisotropic electronic g factor has been amenable to study for many years [291. The occurrence of cross-correlation between CSA and dipolar interactions can be observed either for transverse (“T,“) or for longitudinal (“T,“) relaxation. For a system with two weakly coupled spinl/2 nuclei, cross-correlation leads to a linewidth difference of the two doublet components of the spin which is relaxed by both mechanisms. Measurement of this “differential line broadening” [ 24,26,30], however, is fraught with difficulties since inhomogeneous decay must be refocused, whilst ill-resolved scalar couplings to further spins make echo experiments hazardous. Consequently, it appears more attractive to measure cross-correlation by observing its effects on longitudinal relaxation, since neither inhomogeneous broadening nor scalar couplings will be detrimental, although one must pay attention to extraneous mechanisms such as external random fields.
LETTERS
7 August
of the three operators I,, S= and 2Z,S, [ 12,17,21], since the sum of the populations is always conserved. The equations of motion for these longitudinal terms may be written: $($%J=E
;
E)(%;,)
>
(1)
where (Ly__)=(Z,)-(I”), (AS,)=(X)(ST ), and where (1” ) and (ST ) are the expectation values of the operators 1= and S, at thermal equilibrium. Considering CSA interactions for both nuclei in addition to the dipolar interaction, but neglecting contributions from external random fields, we obtain: A = - 9 Jm,s - 451.1 , B=
-‘pJws-4Js.s
>
C= - 2Jws - 4Js.s - 4J,, ,
E= - 4J1s.s , F= -4J.w
>
(2)
where J,s,~~, JI., and Js,s are the auto-correlation
spectral densities for the dipole-dipole and the two CSA relaxation mechanisms, and J1s,s and Jls,, are the cross-correlation spectral densities. If I= ‘H and S= 13C,the terms JI.I and Jls.[ which depend on the (very small) proton chemical shift anisotropy are negligible. It is this situation that will concern us. With the assumptions of fast molecular motion (“extreme narrowing”) and isotropic motion, the remaining spectral densities simplify to the following expressions: J ,s,,s=~t~~4x)2Y:YBfi2r-6r,,
(3a)
Js.s= 3b~3W44~
t3b)
,
Jls.s= - h W4~ )YIYifir-3& x WC, ~.a., ~YZ~hC
,
3. Theory
where
The four populations of a two spin-112 system IS can be expressed in terms of the expectation values
L(b)=~::+a:+a~-a,aY-a*az--yrQZ
602
1987
and
(3c)
Volume
138, number
CHEMICAL
6
PHYSICS
The values ox, u y and gz refer to the frame OXYZ of principal axes of the shielding tensor. The IS internuclear vector lies along the OZ’ axis. The angles 8,,. and eyz. define the relative orientation of the IS internuclear vector with respect to the principle axes X and Y of the CSA interaction. These expressions are consistent with those given by Goldman
[211. The solution of eq. (1) will show multi-exponential behaviour. However, if the experiment is commenced by inverting or saturating only the S spin magnetization and if the linear initial rate approximation is invoked, we obtain: (I_) = (I”)
+Dt(AS:)
(22) = (S!?) +BT(AP)
(21:&S,) =El(ASL?) )
)
(da) )
(4b)
(4c)
where (Asp)=-(SF), and (SF) is the expectation value of S, immediately after the perturbation that commences the experiment, for example, a 180” inversion pulse applied uniformly to both S spin doublet components. Thus the initial change in the Z, magnetization is linear (a transient nuclear Overhauser effect), while S, initially recovers linearly towards its equilibrium value. Most importantly in the present context, there is a linear growth of longitudinal two-spin order 21-S,. Note that the initial expectation value of this term is zero provided the perturbation affects both S spin transitions equally. The appearance of a non-zero expectation value for the operator 21-S: is due to unequal longitudinal relaxation rates across the two parallel S spin transitions. This manifests itself as a differential recovery of the two components of the S spin doublet. The difference in the intensity of the two lines is directly proportional to ( 215). If the internuclear distance r and the principal values of the shielding tensor are known from other sources, the correlation time 7c can be determined from the recovery of the S, magnetization, which is proportional to the sum of the amplitudes of the two
LETTERS
7 August
1987
doublet components. According to eq. (4b) the S, recovery occurs with the initial rate B= -‘pJ,.sls - 4Js.s, provided external random field relaxation is negligible. Similarly, if the growth of (2Z&) can be measured then, from eq. (4c), the initial slope will give the constant E= - 4Jls.+ To determine the sign of the initial rate E it is necessary to identify the sign of the term (2Z&). If the absolute sign of the heteronuclear scalar coupling constant is positive, ( 2Z&) is positive if the low-frequency (high-field) component of the S spin doublet relaxes faster than the high-frequency component [ 12,2 1,221. Knowing the sign and magnitude of E, and having measured the value of ‘so it is possible to determine K( cr, eXz,, 8yz ) (see eq. ( 3~)). Under certain simplifying assumptions, one may then determine the relative orientation of the internuclear 13C-‘H vector with respect to the principal axis system of the 13Cshielding tensor - a quantity normally obtained only from solid-state NMR.
4. Experimental methods and results The molecule chosen for study was methyl formate, where there is cross-correlation between the carboxylic “C CSA interaction and the 13C-‘H dipolar interaction. Asymmetric longitudinal relaxation was observed in the enriched compound H’3COOCD3 (the methyl group was deuterated to prevent further dipolar interactions from interfering). The material was prepared by alkylation of formic acid (90% 13C enriched) with excess methanol-d, in the presence of traces of acid. A 10 vol”!‘o solution in methanol-d, was sealed in vacua in a 10 mm tube after three freeze-pump-thaw cycles. NMR spectra were recorded at 275 K using a Bruker AM 400 spectrometer (B,,= 9.4 T). Conventional 13C inversion-recovery measurements on this sample showed only a very slight asymmetry in the recovery of the two carboxyl 13C lines, indicating that the cross-correlation effect is quite weak compared to the overall relaxation rate. Thus, during the first second of the recovery, where the growth of 2ZS= magnetization is expected to be linear, it was barely possible to detect an asymmetry. Much as in conventional steady-state Overhauser 603
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Intensity .. : 0.010
. . I
.
.
I
: . . . .
I
”
10
”
I
”
20 Delay
”
I
30
”
,’
1,
40
”
I
50
(s)
Fig. 1. Plot of the intensity of - (2I,S,)l(AS9) as a function of the recovery delay r for the experiment on methyl formate described in the text. The inset shows the difference spectrum observed after a relaxation delay of 6 s. A plot (not shown) of the recovery of SZmagnetization (defined as (&)I( AS! ) ) was extracted from the same data. Analysis of these plots yielded an estimate of the orientation of the “C-‘H internuclear vector with respect to the principal axes ofthe “C CSA tensor ofthe carboxyl group.
studies, we therefore found it necessary to use a form of difference spectroscopy to separate the weak antiphase component from the dominant in-phase component. Two spectra with v,= _+x were recorded and stored separately for each value of the delay r in the sequence: Z S
90X90,, 180~r-90,-acquire.
The sequence applied to the S ( 13C) spins is a conventional inversion-recovery experiment. Longitudinal relaxation gives rise to S, and 2Z_Szterms which are transformed into observable S, and ? 2ZSXoperators, where the sign of the antiphase term depends on the relative phases of the two pulses applied to the Z spins. The sum of the spectra gives the coefficient of S,; the difference gives the (much smaller) coefftcient of 2Z_S,.This experiment can be seen as a heteronuclear form of multiple-quantum filtered inversion-recovery [ IO]. Fig. 1 shows the coefficient of the 2Z_&term as a function of the recovery delay r. A typical difference spectrum is shown in the inset. Nine measurements of the S, and 21-S, terms were taken from the lirst 604
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second after the inversion pulse where linear rates were observed. Linear regression gave the rates B=-(6.19+0.19)x10-* s-r and E=-(0.484f 0.086) x lo-* s-l. The proton-carbon internuclear distance in methyl formate r= 1.101 x 10 -lo m is known from microwave rotational spectroscopy [ 3 11. The principal values of the carboxyl shielding tensor in methyl formate are known from solid state powder spectra [ 321: 6,r = -90.5 ppm, a,,=29.1 ppm and cr33= 6 1.5 ppm with respect to the isotropic shift. With these values the correlation time is calculated to be r =(2.91 kO.09) x lo-‘* s. InseAion of this value of the correlation time into the expression for the initial rate E of the 2ZSz term (see eqs. (2), (3)) gives K(a,8,,6Yz)= -(9.25*1.99)x10-5. Assigningo,, to o,and 033 to cry and assuming that 033= oy is perpendicular to the OCO plane (i.e. eyz, = 90’) in accordance with solid-state studies of carboxyl shielding tensors [ 331, we find that eXz, the angle between the least shielded axis ( CJ, , ) of the carboxyl CSA tensor and the 13C-‘H internuclear vector, can take one of two values: 45 ’ ? 3 oor 135 ’ k 3 ‘. This ambiguity, which follows from the fact that the cosine function occurs as a square in the Legendre polynomial, is common to many NMR measurements. The single-crystal NMR study of Cornell [ 341 for dimethyl oxalate indicates that err , lies in the OCO plane, perpendicular to the C=O bond. If this is also true for methyl fox-mate, the angle OX=.would be 145”. Hence of our two possible results the angle 135” + 3’ would seem most plausible. The agreement of this result with the twodimensional powder spectra study of Linder et al. [ 321 is also satisfactory. The errors quoted in the above results refer only to errors that have propagated from our experimental uncertainties. We have made use of a number of simplifying assumptions that might lead to additional errors in the final estimate of the angle 8,,. The assumption of extreme narrowing appears to be quite safe, but the hypothesis of isotropic motion implicit in eq. (3) could be challenged. We have also assumed that external random field mechanisms did not contribute to the rate B, which in this case seems to be a good approximation [ 351.
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5. Prospects for further study
Acknowledgement
The pulse sequence used in this Letter for selective detection of heteronuclear 2Z& order can be considered as a prototype of a family of novel experiments. Various means can be devised for the detection of longitudinal q-spin order arising from relaxation. For instance, in a homonuclear two-spinl/2 system, where both S spins could be 13C, the 2&S,= term arising from cross-correlation between the CSA and dipolar interactions can be selectively detected using a double-quantum filter. Likewise, in a homonuclear system with three spin-l/2 nuclei, where all nuclei could be ‘H, the conversion of Zkr into 41,,1,,1,,,, terms due to cross-correlation between pairs of dipolar interactions [ 171 can be monitored using a triple-quantum filter. A variation of twodimensional exchange spectroscopy could be devised for selective observation of processes which involve interconversion of different classes of longitudinal qspin order. At the beginning of the mixing period, a composite 90” pulse would be used to convert Zkxto 4, without allowing antiphase terms 4Z&ZjzZ,,,,to be converted into three-spin order 4Zk,Z,,Z,= [ 131. At the end of the mixing period, a triple-quantum filter could be used to convert only three-spin order terms 4Z,,Z,=Z,,, into observable transverse magnetization. Such an experiment would allow measurement of cross-correlation spectral densities between pairs of dipolar interactions, which holds the promise of determining the angle subtended by two internuclear vectors.
The authors are grateful to Dr. J. Keeler for a preprint of ref. [ 271. This research was supported in part by the Swiss National Science Foundation (Grant 2.925-0.85).
6. Conclusions In this Letter we have estimated the orientation of the CSA tensor principal axis system of the carboxyl 13C of methyl formate in an isotropic liquid phase. The fact that this seems to be possible with a fair degree of accuracy, without resort to solid-state NMR of single crystals [ 33,341, or two-dimensional NMR of polycrystalline solids in static [ 32,331 or spinning samples [36,37], holds much promise for further study.
References [ 1] R.R. Ernst, G. Bodenhausen and A. Wokaun. Principles of nuclear magnetic resonance in one and two dimensions (Clarendon Press. Oxford, 1987). [ 21 K. Wiithrich, NMR of proteins and nucleic acids (Wiley, New York, 1986). [3 ] N. Chandrakumar and S. Subramanian, Modem techniques in high-resolution FT-NMR (Springer, Berlin, 1987). [4] A. Wokaun and R.R. Ernst, Mol. Phys. 36 (1978) 3 17. [ 51 R.L. Vold, R.R. Void, R. Poupko and G. Bodenhausen, J. Magn. Reson. 38 (1980) 141. [6] N. Miiller, G. Bodenhausen, K. Wiithrich and R.R. Ernst, J. Magn. Reson. 65 (1985) 531. [ 7 ] N. Miiller, G. Bodenhausen and R.R. Ernst, J. Magn. Reson., to be published. [ 81 M. Rance and P.E. Wright, Chem. Phys. Letters 124 (1986) 572. [9] J. Pekar and J.S. Leigh, J. Magn. Reson. 69 (1986) 582. [IO] G. Jaccard, S. Wimperis and G. Bodenhausen, J. Chem. Phys. 85 (1986) 6282. [ II] N. Miiller, Chem. Phys. Letters 131 (1986) 2 18. [ 121 O.W. Sorensen, G.W. Eich, M.H. Levitt, G. Bodenhausen and R.R. Ernst, Progr. NMR Spectry. 16 (1983) 163. [ 131 G. Bodenhausen, G. Wagner, M. Rance, O.W. Sorensen, K. Wiithrich and R.R. Ernst, J. Magn. Reson. 59 (1984) 542. [ 141 G. Wagner, G. Bodenhausen, N. Miiller, M. Rance, O.W. Sorensen, R.R. Ernst and K. Wiithrich, J. Am. Chem. Sot. 107 (1985) 6440. [ 151 J. Jeener, B.H. Meier, P. Bachmann and R.R. Ernst, J. Chem. Phys. 71 (1979) 4546. [ 161 A. Kumar, R.R. Ernst and K. Wiithrich, Blochem. Biophys. Res. Commun. 95 (1980) 1. [ 171 P.E. Fagemess, D.M. Grant, K.F. Kuhlmann, C.L. Mayne and R.B. Parry, J. Chem. Phys. 63 (1975) 2524. [ 181 A. Abragam, The principles of nuclear magnetism (Oxford Univ. Press, Oxford, 196 I ) . [ 191 R.L. Vold and R.R. Vold, Progr. NMR Spectry. 12 (1978) 79. [ 201 H. Shimizu, J. Chem. Phys. 40 (1964) 3357. [ 2 11 M. Goldman, J. Magn. Reson. 60 (1984) 437. [22] E.L. Mackor and C. MacLean, J. Chem. Phys. 44 (1966) 64. [ 231 M. GuCron. J.L. Leroy and R.H. Griffey, J. Am. Chem. Sot. 105 (1983) 7262. [24] T.C. Farrar and R.A. Quintero-Arcaya, Chem. Phys. Letters 122 (1985) 41.
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[25] S.G. Withers, N.B. Madsenand B.D. Sykes, J. Magn. Reson. 61 (1985) 545. (261 T.C. Farrar, B.R. Adams, G.C. Grey, R.A. Quintero-Arcaya and Q. Zuo, J. Am. Chem. Sot. 108 (1986) 8 190. [27] J. Keeler and F. Sanchez-Ferrando, J. Magn. Reson., to be published. [28] F.A.L.Anet, J. Am.Chem. Sot. 108 (1986) 7102. [ 291 H.M. McConnell, J. Chem. Phys. 25 (1956) 709. [ 301 L.G. Werbelow, J. Magn. Reson. 7 1 (1987) 151. [31] R.F.Curl Jr., J.Chem. Phys. 30 (1959) 1529. [ 321 M. Lindert, A. Hohener and R.R. Ernst, J. Chem. Phys. 73 (1980) 4959.
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[ 331 M. Mehring, NMR - Basic principles and progress, Vol. 11. High resolution NMR spectrosppy in solids (Springer, Berlin, 1976). [34] B.A. Cornell, J. Chem. Phys. 85 (1986) 4199. [ 351 T.D. Alger, D.M. Grant and J.R. Lyerla Jr., J. Phys. Chem. 75 (1971) 2539. [ 361 M.G. Munowitz, R.G. Grifftn, G. Bodenhausen and T.H. Huang, J. Am. Cheni. Sot. 103 (1981) 2529. [ 371 M.G. Munowitz and R.G. Grit%, J. Chem. Phys. 76 (1982) 2848.