MAGNETIC RESONANCE IN CHEMISTRY, VOL. 26, 90s910 (1988)

An Alternative Approach to Multi-Dimensional NMR Spectroscopy

Ray Freeman,* Jan Friedrich and Simon Daviest Department of Physical Chemistry, Cambridge University, Cambridge, UK

Multi-dimensional NMR spectroscopy must satisfy the Nyquist sampling condition in all frequency dimensions. This entails very long experiments and very large data arrays, and may mean that fine structure of cross-peaks cannot be adequately digitized. This sampling restriction can be sidestepped in an alternative approach which explores the new frequency dimensions by scanning selective radiofrequency pulses in small steps over narrow frequency ranges. A pulse envelope shaped according to the first half of a Gaussian curve is particularly well suited to this task. Thus a form of double resonance experiment can be used to generate twedimensional correlation spectra that have all the features of the well known COSY experiment, with an additional zoom capability that reveals the detailed fine structure information. For systems of three coupled spins the corresponding triple reson- ance experiment generates a three-dimensional correlation spectrum, the frequency scans being restricted to the region of interest (the 3D cross-peak) based on information from the conventional NMR spectrum. For correlation spectroscopy of higher dimensionality no frequency search is employed, the selective pulses simply being set at predetermined chemical shift frequencies. A four-dimensional correlation experiment is described which uses popu- lation transfer to establish that four non-equivalent protons are coupled in a chain I-SR-P. It employs an initial 22-pulse applied at the I-spin frequency to excite longitudinal two-spin order (2Z2Sz) which is then propagated along the chain by the application of selective ZZZ-pulses to the intermediate spins S and R, creating an anti- phase intensity perturbation (2R2P2) on the P multiplet. The procedure is recursive and can in principle be extended to N spins coupled pairwise in a linear chain. This is a powerful diagnostic tool for structure determi- nation and can be adapted to recognize other topological features such as chain branching and ring closure.

KEY WORDS Multi-dimensional spectroscopy Nuclear magnetic resonance NMR

~~ ~

INTRODUCTION

Two-dimensional NMR spectroscopy has proved so universally successful that it may now seem strange to suggest alternative techniques for accomplishing the same end-result. Yet we can regard the two-dimensional Fourier transform technique simply as an indirect method for exploring a new frequency dimension (F, ) . In principle the same information can be extracted from a double resonance experiment. The essential require- ment is a second radiofrequency source ( F , ) that is frequency-selective and which can be scanned in small increments across a suitable frequency range. In the present experiments F , is applied in the form of a soft radiofrequency pulse, suitably shaped so as to avoid sinc function side-lobes in the excitation spectrum.2

There is, of course, a disadvantage in the double- resonance approach ; it does not share the high sensi- tivity of two-dimensional Fourier transform spectroscopy because the multiplex advantage is lost. A two-dimensional Fourier transform experiment gathers signal at every increment of t,, whereas the double- resonance experiment only collects signals when F , happens to fall on a resonance line. However, this differ-

* Author to whom correspondence should be addressed. t Physical Chemistry Laboratory, Oxford University, Oxford, UK.

ence should not be over-emphasized because we seldom embark on a double resonance experiment without prior knowledge of the conventional one-dimensional spectrum, so it is usually feasible to limit the F , scan to interesting regions of the spectrum. Indeed, the F , irra- diation may be restricted to exact resonance for a small number of transitions.

Step-by-step exploration of frequency space has an important intrinsic advantage over the indirect Fourier transform method. It is not subject to the restrictions of the Nyquist sampling theorem. In two-dimensional Fourier transform spectroscopy, the sampling rate in the t , dimension must be high enough to avoid aliasing of any part of the spectrum. There is no method for filtering out high-frequency components of the interferogram S(t , ) before the sampling operation. Therefore, if we require fine digital resolution in the F , dimension and if the spectrum covers a wide frequency range, the two-dimensional experiment must employ a large number oft, increments, which requires a protrac- ted experiment and a large amount of data storage. By contrast, the analogous double-resonance experiment can be set up to examine a very narrow region of the F , frequency dimension at very high digital resolution- the equivalent of the zoom operation of a camera.

A very simple example is provided by the nuclear Overhauser experiment, usually performed either as a one-dimensional double-irradiation experiment, or by two-dimensional cross-relaxation spectroscopy

0749-1 58 1/88/100903-08 $05.00 0 1988 by John Wiley & Sons, Ltd.

Received 22 June 1988

904 R. FREEMAN, J. FRIEDRICH AND S. DAVIES

(NOESY). Between these two extremes, we could con- sider exploring a limited range of chemical shifts with a selective radiofrequency pulse which is stepped in fre- quency through the region of interest. The resulting intensity changes in the spectrum could be displayed (in the difference mode) as an intensity contour map, using the standard two-dimensional spectroscopy software. We might call this experiment pseudo-NOESY, since the spectra would have a similar appearance to two- dimensional NOESY spectra. However, for applications where only part of the total chemical shift range is of interest, the results could be obtained in a shorter time than the full two-dimensional experiment and with finer digital resolution.

The restriction on the minimum sampling rate in a Fourier transform experiment reveals itself as the 'Archilles heel' if we try to extend the experiment into three or more dimensions, for each new evolution domain must be adequately sampled. Data matrices become enormous, transformation times become tediously long and the experimental time is unrealisti- cally protracted. Nevertheless, some successful three- dimensional experiments have recently been reported3 that require several days of data acquisition. Alterna- tively, semi-selective excitation"8 may be used to restrict the spectral width, either to a narrow range of chemical shifts or to a single well-separated spin multi- plet.

For these reasons we have examined multiple reson- ance methods as alternatives to multi-dimensional Fourier transformation. They are not intended as replacements of the Fourier transform methods, but as complementary experiments to improve digital resolution or to facilitate the extension into the third or fourth frequency dimension. Extensive software already exists to facilitate the display of two-dimensional spectra in various forms, for example intensity contour maps or stacked trace plots. Usually this can be adapted for the multiple resonance experiments so the spectra may bear a strong resemblance to conventional two-dimensional spectra. We shall call these double res- onance experiments pseudo-two-dimensional spectros- COPY (*-2D).

IS system. The more general case is complicated by the introduction of several 'passive' couplings J, , , J,, , etc., but otherwise follows the same behaviour. In the for- malism of product operator theoryx6 the soft pulse applied at exact resonance for one of the I transitions generates a term of the form 2 I , S , which is converted by the hard pulse into - 21, S , , an antiphase S doublet in the pure absorption mode. When the soft pulse is at resonance for the other I transition the sense of the coherence transfer signal is reversed. Consequently, the 'cross-peak' consists of the familiar square pattern of four lines with intensities ; !. There is a symmetrically related cross-peak generated when S transitions are irradiated by the soft pulse and the I spectrum is observed. Diagonal peaks arise from I-spin coherence that remains at the I-spin frequency or S-spin coherence that remains at the S-spin frequency. They are gener- ated when the soft pulse is slightly offset from the exact resonance condition. They take the form of a square pattern of four lines in the dispersion mode and all in the same sense. All the signals of the conventional NMR spectrum are cancelled by the phase alternation of the hard 90" pulse (which has no effect on the sense of the coherence transfer signal).

Consequently, a $-COSY spectrum has all the fea- tures of the corresponding COSY spectrum-dispersion mode lines along the principal diagonal representing the conventional one-dimensional spectrum, and antiphase absorption-mode cross-peaks symmetrically disposed on either side of the diagonal indicating resolved scalar couplings. From the contour map it would be virtually impossible to decide whether the spectrum had been obtained by the COSY or the $-COSY technique. However, the $-COSY experiment can be set up to examine the detailed fine structure of a cross-peak under high digital resolution. This is illustrated in Fig. 1 for one of the cross-peaks of the two-dimensional cor- relation spectrum of the protons of reserpine recorded at 400 MHz.17 The conventional COSY spectrum is necessarily poorly digitized because of the limitation on data storage for two-dimensional matrices; the $-COSY spectrum is digitized six times more finely in both frequency dimensions. More experimental details of this experiment can be found in Ref. 15.

~~ ~ ~~~ ~~~

CORRELATION SPECTROSCOPY

Of all the two-dimensions1 experiments, homonuclear shift correlation spectroscopy (COSY) has been the most suc~essful~-'~ and has been the subject of many refinements in recent years. The double-resonance ana- logue ($-COSY) is a simple experiment:"

Soft 90" (+ X) Hard 90" (+ X) Acquire (+)

Soft 90" (+ X) Hard 90" (- X) Acquire (+)

There are no intervals of free precession equivalent to the evolution period t,. The soft pulse can be made highly frequency selective. It has an envelope shaped according to the first half of a Gaussian curve; this ensures a narrow excitation range for the Y component of magnetization.' The process of coherence transfer is most simply visualized in terms of a coupled two-spin

THREE-DIMENSIONAL SPECTROSCOPY

When two-dimensional correlation spectra become overcrowded, some simplification is to be expected by spreading the information out into a third frequency dimension, lifting any accidental degeneracies in chemi- cal shifts. Unfortunately, the data matrix then becomes very large and the three-dimensional experiment would require a long time to complete. For example, if we allowed only lo3 data points to cover the spectrum, the data storage requirement is lo9 words, and a lo6 steps would be required as the two evolution periods were explored. While advances in computer technology will surely allow us to employ large data tables of this kind, it seems unrealistic to contemplate experiments that require several days of data acquisition.

AN ALTERNATIVE APPROACH TO MULTI-DIMENSIONAL NMR SPECTROSCOPY 905

a I b

Figure 1. A cross-peak between protons H-18 and H-20 of reserpine observed at 400 MHz. (a) The conventional COSY experiment. (b) The yr-COSY experiment with digital resolution increased six-fold; note the better definition of the 32 components. Reproduced with permission from Ref. 17.

The three-dimensional version of the $-COSY e~periment '~ . '~ circumvents these problems. By using information from the conventional NMR spectrum and the two-dimensional COSY spectrum, values for chemi- cal shifts and most of the correlations are already known. Any remaining ambiguity may be resolved by searching for the appropriate three-dimensional cross- peak centred at the chemical shift coordinates ( a I , as, aR). Only a very limited number of frequency increments is needed to search this small volume of fre- quency space, defined approximately by I J I s + J I R I Hz by I J I R + J R s I Hz by I J I s + J R s l Hz. This three- dimensional experiment is surprisingly simple. The two frequency dimensions F , and F , are explored with simultaneous selective radiofrequency pulses; an entire F , scan must be made for each step of the F , scan. The third dimension (F, ) is examined by calculating the Fourier transform of the free induction signal S(t3) excited by a hard read pulse. There are no stringent practical restrictions on the fineness of digitization in

The simplest spin system that can be studied by this technique has three non-equivalent spins (ISR) coupled by three resolvable coupling constants JIs , J I R and J S R . Without loss of generality we may assume that we explore the I-spin response with F , and S-spin response with F , and display the R-spin response in the F , dimension. The two selective pulses ( F , and F,) are applied simultaneously. In practice this is achieved by using the centre-band response of a DANTE sequence" to provide the F , pulse while the first side-band response (360" precession between pulses) provides the F , selective pulse. Thus F , is swept by varying the transmitter frequency while F , is swept by varying the pulse repetition rate (allowing for the variations in the transmitter frequency). In this way the two selective pulses are phase coherent, and phase cycling is readily implemented. A more sophisticated version of this experiment employs a double-DANTE sequence where there are two independent side-band conditions."

We are interested in detecting coherence that has

F3 -

been transferred from I to S to R (or from S to I to R) by the action of the two selective radiofrequency pulses followed by a hard 90" read pulse affecting all the spins. It is therefore very useful to incorporate a phase cycle which suppresses all signals not affected by both selec- tive radiofrequency pulses. A four-step cycle is used:

Soft 90" ( I ) Soft 90" ( S ) Hard 90" Acquisition

+ X + X + X + + X - X + X - X + X + X - X - X + X +

In this way the conventional NMR spectrum is elimi- nated, along with other signal components that would be undesirable in the final three-dimensional spectrum.

If, for simplicity, we consider the case where both F , and F , are at exact resonance for an I and an S tran- sition, respectively, then it is found that coherence is transferred to R only when the F , and F , pulses excite pure (antiphase) I S zero-quantum coherence, repre- sented by the product operator expression [21, S ,

occur if F , and F , irradiate I and S transitions that are regressively connected in the energy level diagram. Of the sixteen possible combinations of I and S transitions, only four combinations have this regressive configu- ration. The final hard 90" pulse generates only one observable product operator term 41, S, Rr . This is an absorption-mode signal at the R-spin frequency consist- ing of four lines with the characteristic intensity pattern (- + + -). The three-dimensional cross-peak consists of four such four-line patterns, two of them inverted

We can appreciate these rules most easily by con- sidering a typical three-spin system, the protons of acrylic acid. A schematic diagram of the conventional NMR spectrum is presented in Fig. 2; note that each proton exhibits a four-line multiplet pattern. The three- dimensional cross-peak" is illustrated in Fig. 3. Actually the experiment has been simplified by record-

- -

+ 41xSx R, + 2IrSr + 4IrSy Rz]/4. This will only

(+ - - +).

906 R. FREEMAN, J. FRIEDRICH AND S. DAVIES

I R S

Figure 2. Schematic diagram of the spectrum of protons in acrylic acid, an IRS spin system with all couplings positive.

ing responses in the four (F, F3) planes corresponding to the (known) frequencies of the I-spin transitions. Intensity contour maps are shown, with filled contours representing negative intensity. We can recognize the four-line patterns running in the F , dimension, having intensities (+ - - +) or (- + + -). The dis- position of these in the (F, F,) plane reflects the fact that J I R and J,, have the same sign in this compound.

It is easy to see now why the cross-peak would disap- pear if any of the three coupling constants were van- ishingly small. If J,, -+ 0 then the two four-line patterns at the top right merge and cancel, as do the two pat- terns at the bottom left. If either J I R + O or J S R + 0 , positive and negative lines merge in each four-line pattern and all signals disappear. A three-dimensional cross-peak only appears if all three coupling constants are large enough to be resolved. (An equivalent require-

72

I3

Figure 3. A cross-peak from the three-dimensional correlation spectrum of acrylic acid. Filled contours represent negative inten- sity. Note the doubly-antiphase intensity patterns in the f , dimen- sion. Reproduced with permission from Ref. 19.

ment applies to cross-peaks in a conventional two- dimensional COSY experiment-the active coupling must be resolvable.)

This type of three-dimensional correlation spectro- scopy allows us to detect groupings of three coupled spins and to examine the fine structure of the cross- peaks under very high digital resolution. In cases where the classic two-dimensional COSY spectrum is too crowded for a straightforward analysis, the three- dimensional $-COSY experiment provides a new fre- quency dimension to spread out the signals. The experiment can be short in duration and the data storage requirements are modest.

MULTI-DIMENSIONAL SPECTROSCOPY

It is natural to enquire whether there might be any application for four-dimensional correlation spectro- scopy or even experiments of higher dimensionality. We believe there is. Not all assignment problems can be solved by establishing pairwise correlations of the type obtained from COSY spectra. We may, for example, see IS cross-peaks and SR cross-peaks in a COSY spectrum and conclude that I is coupled to S, which is in turn coupled to R. However, it may have been the case that what appeared to be the S response was, in fact, the superposition of two different responses S and S with similar chemical shifts. Attempts to resolve this kind of ambiguity have led to the introduction of the RELAY experiment2122 and total correlation spectro~copy,~~ often called the HOHAHA technique24 since it involves a homonuclear variant of the classic Hartmann-Hahn e~periment.~ In this technique, as the mixing time z is increased, coherence is transferred through more than one stage, eventually identifying all the spins of a coupled spin system.

The structural chemist might prefer a more specific test. A particularly useful feature would be the ability to pin down a linear chain of coupled spins. Here linear is used in a special sense. Suppose we have four coupled spins, I-S-R-P (Fig. 4). We might consider this as a linear chain if all the coupling constants J I , , J , R and J R p are relatively large compared with the line widths, whereas the other couplings, J , , , J I p and J,, , are con- siderably smaller, perhaps not resolved. In other words, the path I-S-R-P represents the path of least resist- ance to magnetization transfer through the scalar coup- lings. We require a technique which tests this pathway,

Figure 4. A linear chain of coupled spins. The couplings J j S , J,, and JRp must be large enough to be resolved, whereas the couplings J,, , J,, and Jsp need not be resolvable.

AN ALTERNATIVE APPROACH TO MULTI-DIMENSIONAL NMR SPECTROSCOPY 907

giving a null response if any of the three scalar coup- lings is too small to be resolved.

A search of four-dimensional frequency space, either by Fourier transform methods or by direct exploration with selective pulses, appears to be out of the question, but of course it is not necessary. At this stage of the investigation, the conventional NMR spectrum and the COSY spectrum would already have been recorded and analysed. The structural chemist might already have a good idea which spins might be coupled in a chain; all that is required is an unambiguous confirmatory experi- ment.

There is a straightforward population transfer experi- ment which achieves this result. It starts with the appli- cation of a pulse to a spin 1 at one end of the chain, creating a population disturbance known as longitudi- nal two-spin order, represented by the term 21, S , in the product operator formalism. For his reason, this initial pulse is known as a ' Z Z pulse'. The tuning of this pulse is not critical, and the irradiation frequency is roughly centred on the multiplet in question. The resulting population disturbance affects both the 1 and S reson- ances and could be displayed as an 'up-down' intensity pattern if a read pulse of small flip angle is used.

A Z Z pulse is a semi-selective composite radiofre- quency pulse made up of two parts. The first part is a 90" (X) pulse shaped according to the first half of a Gaussian curve, terminating abruptly at the highest point. Applied near the mid-point of a J doublet, this half Guassian pulse converts one component into pre- dominantly + X magnetization and the other com- ponent into - X magnetization (the two resonance offsets are equal and opposite in sign). Typical magne- tization trajectories are illustrated in Fig. 5. This is a refinement of an experiment first performed by Bron- deau and Canet26 using rectangular soft pulses. The half-Gaussian pulse is reasonably effective over a range

7

X

Figure 5. Magnetization trajectories calculated for the first part of a 'ZZ pulse,' a half-Gaussian go"(+%) pulse of duration 0.2 s applied at the centre frequency of a J doublet. The two magne- tization vectors turn about tilted effective fields and end up close to the +% and -% axes.

'2 + 2 I ,S ,

- 2 I , S ,

Figure 6. Schematic representation of Z-magnetization of the I spin in an IS spin system acted on by a ZZ pulse (or a ZZ* pulse) applied at the /-spin chemical shift.

of J values and the frequency setting is not particularly critical. The second part is a 90"(Y) pulse designed to turn + X magnetization into + Z , and to turn - X magnetization into - Z . It can be less selective (shorter duration) than the first part of the Z Z composite pulse. The envelope can be shaped according to the second half of a Gaussian curve; other shapes are possible. Hence the overall effect of the Z Z pulse is to act on Boltzmann spin populations and convert them into the 'up-down' pattern characteristic of two-spin order. Figure 6 shows this schematically for a coupled two- spin system. This mode of preparation is similar to that used in the INEPT polarization transfer experiment2' except that it is more tolerant of variations in the coup- ling constant J I S .

This is where population transfer experiments have a considerable practical advantage over coherence trans- fer methods. Any pulse imperfections in the Z Z excita- tion produce transverse components of magnetization M,, . Since population disturbances persist for times of the order of the spin-lattice relaxation tines, there is plenty of time to suppress M,, by the application of field gradient pulses, by phase cycling the pulses or by the introduction of short random delays between pulses. Population transfer can therefore be made into a very 'clean' experiment with few undesirable artifacts. The net result of the pulse imperfections is merely a loss of amplitude of the population transfer signal.

The next requirement is to propagate this population disturbance along the chain to the R spin. This is achieved by applying a new kind of semiselective pulse (called a ' Z Z Z pulse') at a frequency approximately centred on the S multiplet (again the adjustment is not critical). We can represent the population differences across the S-spin transitions by the stick diagram in Fig. 7). After the initial Z Z pulse applied to the 1 multi- plet, these populations are altered as shown in Fig. 7b. The Z Z Z pulse has the effect of inverting the popu- lations of the two centre S transitions; it is in fact a semi-selective 180" pulsc calculated not to affect the outer lines significantly. The overall result is to convert a population disturbance represented by 21, S, into a population disturbance represented by 2S, R, . If required, this can be read by a hard 45" pulse (a 90"

908 R. FREEMAN. J. FRIEDRICH AND S. DAVIES

(b) # - urr Figure 7. Effect of a ZZZ(S) pulse on (a) S magnetization at Boltzmann equilibrium (b) S magnetization already perturbed by a Z(/) pulse.

pulse would create only multiple-quantum coherence). Under these conditions, the R-spin experiences the up- down perturbation of intensities 2S, R, superimposed on the natural R-spin multiplet (R,).

This operation is recursive; the population dis- turbance 2SzRz can be transmitted to the P spin by applying a ZZZ pulse to the R multiplet (Fig. 8). In principle, a chain of any length N can be pinned down by such a population transfer experiment. One could argue that this is formally an N-dimensional experi- ment, although in practice the results are presented as a one-dimensional (difference) spectrum. Two scans are made, one initiated by a ZZ pulse, the second by a ZZ* pulse (Fig. 6) that creates a population disturbance in the opposite sense:

ZZ = 90( +X) 90( + Y) ZZ* = 90( + X ) 90( - Y).

In the difference mode these population disturbance signals reinforce, but all other responses cancel.

The overall result is that the characteristic updown pattern of intensities is propagated to the last spin in the chain. The power of the method lies in the fact that the observed updown signature disappears if any one (or more) of the coupling constants is too small to be resolved. (A similar criterion governs the appearance of cross-peaks in conventional COSY spectra.) In this way a valuable piece of structural evidence is obtained, more definitive than the individual pairwise connectivity determinations of the COSY method.

Parasitic coupling

Protons are seldom coupled in a simple chain without additional parasitic coupling paths. Consider, for example, the chain of four spins I, S, R and P illustrated

2 1 , S , 2 S z R , 2 R z P ,

222 0 b serve 22 222 pulse pulse pulse

Figure 8. Population transfer along a linear chain IS-R-P.

in Fig. 4. The path of least resistance I-S-R-P is deter- mined by the fact that J I s , J S R and J R p are well re- solved couplings. In general, however, there will be other couplings J I R , Jsp and J p which, although smaller in magnitude, may still produce resolvable split- tings. They will also allow some undesirable transfer of the population disturbance created by the initial ZZ pulse applied to the I spin. The efficiency of the ZZ and ZZZ pulses will be small for these parasitic transfers, so the resulting intensity changes should be correspond- ingly weak. In most cases they induce antiphase inten- sity disturbances of the very small splittings of the observed P multiplet, not on the main splitting. Never- theless, it could be important to suppress these intensity perturbations so as to avoid drawing erroneous conclu- sions from the experiment. For example, we would wish to eliminate the effect of a population transfer I-S-P which could occur even if JSR and J R p were zero pro- vided that J s p were resolvable. This is achieved by repeating the experiment with the ZZZ(R) pulse omitted, so that only the parasitic transfer I-S-P occurs. The results of the two experiments are then sub- tracted, leaving only the response from the three-stage transfer I-S-R-P. Analogous experiments can be used to test that each and every intermediate spin in the chain is essential for the population transfer.

Spin topology

The topology of coupling within spin systems, particu- larly proton systems, is usually complex. Considerable research has been carried o ~ t ~ ~ * ~ ~ on the recognition of the detailed topological features of the various possible groups of spins. Methods have been devised for multiple-quantum filtrati~n~*~ to separate the spectra from groups of 2, 3, . . . , N coupled spins, supplemented by spin topology filtration which divides each such group into subgroups based on the details of connecti- vities. For example, four-spin systems have the charac- teristic topologies illustrated in Fig. 9. This approach employs a whole armoury of spin manipulation tech- niques and extensive data processing of the experimen- tal spectra. The method lends itself particularly well to searching spectra of biological molecules for a predeter- mined spin topology.28 Not surprisingly, some of these separations are incomplete, yielding ambiguous conclu- sions, or the spectroscopist has to make judgements regarding the relative reliabilities of a large number of structural features found by the computer p r ~ g r a m . ~ ~ . ~ ~

Square Tetrahedron Hang-glider

Chain Kite

6 Star

Figure 9. Topologies of couplings in a four-spin system.

AN ALTERNATIVE APPROACH TO MULTI-DIMENSIONAL NMR SPECTROSCOPY 909

There is an alternative approach. It engages the spec- troscopist at each stage of the process, starting with a partial connectivity diagram derived from (say) conven- tional COSY experiments, leading on to some postu- lated feature of the topology that can be directly verified by a multi-stage population transfer experiment of the t y p described above. It is based on the principle that there are only three essential elements of connectivity- chain formation, chain branching and ring closure. From these basic elements the more detailed patterns (for example those in Fig. 9) can be deduced. Chain for- mation has been described above and is the basis of the two other tests. Branching can be characterized by per- forming two separate chain experiments following first the left-hand fork and then the right-hand fork. Alterna- tively, a simple population transfer experiment involv- ing only ZZ pulses can be employed to test for the branch point. Consider, for example, the 'star' configu- ration of four coupled spins (Fig. 10) with the I spin at the centre and all three couplings J I s , J , , and J I p resolved. The I-spin spectrum would be a doublet of doublets of doublets. After the pulse sequence

ZZ(S) Hard 45" Acquire (+)

ZZ*(S) Hard 45" Acquire (-) the difference-mode I-spectrum reveals an antiphase pattern of intensities characteristic of a product oper- ator term 2ZySz (Fig. 1Oa). Population transfer from R to I (Fig. lob) and then from P to I (Fig. 1Oc) confirms the star configuration.

22 22

22

Figure 10. Testing for a branch point by population transfer from three neighbouring spins S, R and P to the central spin 1.

(a)

222

* Q2

zz

z z z

s2 R Z

Figure 11. Population transfer experiments used to establish the presence of a five-membered ring. The first experiment establishes that 0 4 4 - R - P forms a chain and the second that R-P-Q-IS forms a chain.

A ring is a special case of a chain in which the final spin is the same as the initial spin. However, the meth- odology used above for a chain must be used with some care because the initial population disturbance can be transferred to two immediate neighbours and propa- gated both clockwise and counterclockwise around the ring. To fix ideas, consider the five-membered ring illus- trated in Fig. 11. For simplicity we neglect all but the coupling between immediate neighbours in the ring. We initiate the population disturbance by applying a ZZ pulse at the centre of the I multiplet which (in this approximation) has the characteristic pattern shown in Fig. 12. For the two central transitions the Z Z pulse is sufficiently close to exact resonance that it creates mainly Y magnetization, equivalent to saturating the spin populations across these transitions. The two outer transitions generate predominantly + Z and - Z mag- netization, leaving the pattern shown in Fig. 12. In the product operator formalism this would be represented by the terms 21, S, + 21z Qz . Consequently, this pulse creates population disturbances ('up-down' patterns) on both the S and Q multiplets.

Figure 12. Schematic diagram representing the effect of a ZZ pulse on the 2-magnetization of the /-spin in the five-membered ring in Fig. 11.

910 R. FREEMAN, J. FRIEDRICH AND S. DAVIES

It might be confusing to propagate these population disturbances on to the same site from two opposite directions, since the resulting intensity pattern is the superposition of two pairs of antiphase doublets (Fig. 12). A clearer result is obtained by omitting the final link that closes the ring; two 'chain' experiments are performed :

ZZ(Z) ZZZ(S) ZZZ(R) Hard 45" Acquire(+)

ZZ*(Z) ZZZ(S) ZZZ(R) Hard 45" Acquire( -)

generates antiphase doublets on both P and Q, estab- lishing that Q-I-S-R-P form a chain (Fig. lla), whereas

1 lb). Together these two tests establish the presence of a five-membered ring.

It would appear that most examples of spin topol- ogies can be investigated by suitably conceived popu- lation transfer experiments employing Z Z and Z Z Z pulses. This would be useful in a situation where it is necessary to test for a characteristic group of spins in order to establish (for example) the presence of a partic- ular amino acid in a biochemical molecule. Neverthe- less, it seems that the detection of a coupled chain of spins is the most powerful diagnostic of all.

ZZ(Z) ZZZ(Q) ZZZ(P) Hard 45" Acquire( +) Acknowledgements

ZZ*(Z) ZZZ(Q) ZZZ(P) Hard 45" Acquire( -) The authors are indebted to Professor J. S. Rowlinson and Dr P. J. Hore for access to the Varian XL-400 spectrometer at Oxford. Per- mission to reproduce diagrams was granted by Academic Press Inc. (Fig. 1) and Taylor and Francis (Fig. 3).

generates antiphase doublets on both R and S, demon- strating that R-P-Q-I-S also constitutes a chain (Fig.

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