Two-Dimensional Fourier Transformation in NMR€¦ · Two-Dimensional Fourier Transformation in NMR...

REVIEWS

Two-Dimensional Fourier Transformation in NMR

Ray Freeman and Gareth A. MorrisPhysical Chemistry Laboratory,

Oxford University, England

I. INTRODUCTION

The idea of two-dimensional Fourier transformationwas originated by Jeener (1) in 1971, but the very widesignificance of his proposal does not seem to havebeen generally realized until several years later, whenseveral elegant applications were described (2-6). Thebasic idea is very simple. In modern nmr spectrometers,spectra are normally obtained by Fourier transforma-tion of a transient free precession signal, which weshall call S(t2), where t2 is the usual running timevariable for signal acquisition. However, the phase andamplitude of S(t2) are dependent on the history of thenuclear magnetization prior to t2 = 0. We may definethe interval immediately preceeding t2 as an evolutionperiod f, during which the spectrometer receiver is inac-tive but the nuclear spins are forced to execute someprescribed motion that will influence the detectedsignal S(r2) in some manner. Jeener's idea was to incre-ment f, over a suitable range of values and perform asecond Fourier transformation of the signal as a func-tion of tu thus mapping out the behavior of the nuclearmagnetization during the evolution period as well asduring the detection period f2.

The process is best described as a two-dimensionalFourier transformation of a signal matrix S(tut2) to yielda spectrum in two frequency dimensions S(FUF2). Thereis no theoretical significance in the order in which thetwo successive transformations are carried out, al-though in practice it may often be convenient to trans-form with respect to t2 as soon as the free inductionsignal is acquired or as soon as time averaging iscomplete.

After the first transformation, if corresponding pointson each spectrum S(F2) axe followed as a function of tuthe result is a time-domain signal Sft) for which we

reserve the term "interferogram" in order to distinguishit from the free precession signal S(t2). Whereas S(t2)can be thought of as occurring in "real time," S(t,) mustbe built up point by point in a series of separateexperiments.

It is important to realize that the result of these twotransformations is not merely two spectra S(F,) andS(F2), but a single "spectrum" in two orthogonal fre-quency dimensions, a surface in three-dimensionalspace. A response on this diagram relates the behaviorof a given signal during f, to its behavior during t2. Thereis thus an important element of correlation involved. Atthe end of the evolution period the many components ofa high-resolution spectrum pass on their individualphase and amplitude information to the correspondingcomponents during the detection period, rather like therunners in a relay race passing on their batons to theirteammates for the next stage. Herein lies the strengthof the two-dimensional transformation technique.

Let us consider a very simple example where onlyamplitude information is transmitted to the signal S(t2).By means of a spin echo tecnhique it can be arrangedthat the nuclear spin magnetization at the end of theevolution period has decayed exponentially throughspin-spin relaxation. Echo modulation is excluded inthis simple example; this corresponds to the case ofproton-decoupled carbon-13 spectroscopy whenhomonuclear coupling is neglected. The resulting spec-trum would have a series of peaks running in the F2

dimension at frequencies determined by the carbon-13shifts and with instrumentally determined line widths.However, their profiles in the F, dimension would bepure Lorentzian with the appropriate natural line widths,all the lines being centered on F, = 0. Note that eachcomponent in the nmr spectrum transmits its own in-dividual amplitude information at the end of f,: each has

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an intrinsically different spin-spin relaxation time.The second important property of two-dimensional

Fourier transformation is the possibility of restrictingone nuclear parameter to one frequency dimension, andanother parameter to the other dimension. For example,the separation of multiplet structure from chemicalshifts, while not changing the overall information con-tent of the spectrum, can nevertheless prove extremelyuseful by preventing overlap between nearby multiplets.In solid state nmr spectroscopy, a similar separation ofchemical shift and dipolar couplings can be achieved.Separation into two frequency dimensions relies on theexistence of two independent time periods f, and f2 dur-ing which the nuclear spins are allowed to evolve underdifferent constraints.

A third general area where two-dimensional trans-formation experiments can be useful is in the indirectdetection of transitions not normally accessible in aconventional NMR experiment. It is possible to preparea coupled spin system at the beginning of the evolutionperiod f, so as to excite normally forbidden transitions.For example, multiple-quantum coherences (oscillatingoff-diagonal elements of the density matrix) are notdirectly observable in the spectrometer, but can be con-verted by a mixing pulse (at f = r,) into magnetizationthat is detectable during f2. Their oscillation frequen-cies during f, can thus be mapped out indirectly.

The purpose of this review is to describe the develop-ments that have occurred in two-dimensional spectros-copy in the three years or so since the first experimentalwork was published in 1975. The emphasis will be onapplications to high-resolution nmr of liquids ratherthan to the more specialized field of solid state nmr,and an attempt will be made to indicate which two-di-mensional experiments are most likely to be of prac-tical use to the chemist using nmr techniques. Inmid-1978 one thing is clear: the field is still developingvery fast and a definitive review is just not feasible.There are interesting parallels with the evolution of thetechniques of double resonance in the 1960s. There too,some of the early experiments were regarded as ratheresoteric, and it was a while before the practicing nmrspectroscopist was ready to apply them to realchemical problems. Yet today no nmr spectrometer iscomplete without double irradiation facilities, andcarbon-13 spectroscopy could hardly have developed atall without broad-band decoupling. One encouragingfeature of two-dimensional Fourier transform spectros-copy is that very little hardware modification is requiredto perform these experiments. Most pulse spectrome-ters are computer controlled, so the pulse sequencingand acquisition routines are relatively simple program-ming changes, and data manipulation programs arenow available from the spectrometer companies.

II. HISTORICAL PERSPECTIVE

The evolution of the concept of two-dimensionalFourier transformation has been unusual; after theoriginal idea was proposed by Jeener (1) it lay dormantfor three years before being taken up and developed.Unfortunately the original lecture and subsequent ex-perimental work by Jeener and Alewaeters were notpublished. The chronology of two-dimensional spec-troscopy is set out in references (1) to (42), covering theperiod up to 1 July 1978.

The seeds were sown at an Ampere Summer Schoolin Yugoslavia in 1971. The basic Jeener experiment hasbeen described and analyzed by Aue et al (9), relyingheavily on density matrix theory. A sequence of two 90°pulses is applied to a coupled system of protons, thepulse separation constituting the evolution period /,.Immediately after the second pulse, a free precessionsignal S(t2) is recorded and stored, and the experimentrepeated for a range of values of f, with appropriate in-tervals to allow for relaxation. Two-dimensional Fouriertransformation yields a spectrum S(FA,F2) containing in-formation about the way transitions are connected inthe energy level diagram. A full treatment of the Jeenerexperiment is quite difficult and the resulting spectraare complicated, but the experiment undoubtedly con-tains all the essential features of two-dimensionalFourier transform spectroscopy and remains the proto-type of all the investigations considered below.

Ernst was the first to appreciate the great potential ofthe method, and at a conference at Kandersteg in Switz-erland in September 1974, he presented two-dimen-sional spectra of trichloroethane and a mixture of diox-ane and tetrachloroethane obtained by the Jeenertechnique, but again the work was not published (2).The first description of two-dimensional transformationto be published in the literature was a novel method ofmapping the proton spin density within a solid object (3,4). The evolution period was employed to follow the pro-ton free precession in an imposed magnetic field gra-dient in the X-direction, and the detection periodmonitored the precession in a Z-gradient. Two-dimensional Fourier transformation produced a map ofthe proton density within the sample, projected on theXZ plane. In the same month that the full description ofthis experiment appeared (April 1975), Ernst (5) reviewedthe various possibilities for presenting nmr spectra intwo dimensions, adopting a broad definition where thesecond dimension might be frequency, time, or a dou-ble-resonance offset parameter, and Waugh et al (6)described the two-dimensional Fourier transformationof nmr signals obtained in rotating-frame double-reso-nance experiments on solids. Germination had begun.

The full potential of two-dimensional spectroscopy

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CH3-CH2-CH2-CH2-CH2-CHa b c c b a

0.0 0.2 0.6 a8kHz

Figure 1. The two-dimensional Fourier transform spectrum of carbon-13 in n-hexane obtained by the method of Miiller et al (7),reproduced by permission of the authors. During the evolution period tu the carbon-13 nuclei are coupled to protons, leading to theappearance of proton-coupled subspectra in the F, dimension, while the use of proton noise decoupling during the detectionperiod f2 restricts the F2 dimension to displaying the carbon-13 chemical shifts. The projection onto the F, axis is the full proton-coupled spectrum, shown at the top of the diagram. The absolute-value mode of display has been used in this diagram.

for high resolution nmr of liquids first became apparentwith the publication of a very simple application whichused the first "multiple trace" display (7). Carbon-13spins were allowed to precess in a proton-coupledmode during the evolution period tu but were wide-banddecoupled during the detection period t2. Two-dimensional Fourier transformation then has the effectof breaking down the carbon-13 spectrum into subspec-tra from the individual carbon sites, separated accord-ing to chemical shift in the F2 dimension, but withproton-coupled subspectra in the F, dimension. Thisturns out to be a very useful way of avoiding the overlapof spin multiplet structure in carbon-13 spectroscopy. Ithas the drawback that it requires a large number of ex-periments in the f, domain, but later developments ofthis idea use spin echo methods to eliminate thechemical shifts from F, and alleviate this problem.Already one of the principal attributes of two-dimensional Fourier transformation is apparent in thisexperiment—the separation of nmr parameters into thedifferent frequency dimensions F, and F2. This property

turns out to be particularly valuable for nmr of largemolecules, an area handicapped mainly by the prob-lems of resolving and assigning spectra with manyoverlapping lines. The two-dimensional Fouriertransform spectrum of carbon-13 in n-hexane (7),reproduced in Figure 1, provides the first opportunity tovisualize the potential of this technique in high resolu-tion nmr.

In the next year (1976), these ideas bore fruit inseveral parts of the world and in different fields of nmr.This review is principally concerned with high resolu-tion spectroscopy of liquids and consequently the solidstate work is treated in less detail. There were four suchsolid state experiments in 1976. Alia and Lippmaa (8)measured the relaxation properties of chemicallyshifted carbon-13 sites in norbornadiene, while Waughand coworkers (10,12) separated dipolar splittings andchemical shifts in a single-crystal sample of calciumformate, representing the spectrum S(FUF2) in the formof an intensity contour plot. Two other solid state ex-periments, related to two-dimensional spectroscopy but

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using only a single Fourier transformation, were de-scribed by Vega et al (13) and Stoll et al (16).

In the high-resolution spectroscopy of liquids two im-portant developments occurred during 1976. Aue et al(9) produced a detailed theoretical treatment of two-dimensional spectroscopy using the density matrix for-malism and analyzed the basic two-dimensional experi-ment of Jeener (1), illustrating the results with spectracorresponding to several possible variations of this ex-periment, including the excitation of zero- and double-quantum transitions. This was the first attempt at a for-malism and nomenclature for two-dimensional spec-troscopy, and several new concepts were addressed forthe first time, for example the question of line shapes intwo-dimensional spectra. This is clearly a key work inthe understanding of two-dimensional Fourier trans-form spectroscopy.

The second important development was the idea ofusing spin echo techniques to refocus the effect ofchemical shifts during the evolution period, therebyeliminating chemical shielding from the F, dimension.Aue et al (11) applied this principle to proton spectros-copy where (for weak coupling) the F, dimension showsonly spin-spin multiplets while the F2 dimension showsthe normal coupled spectrum. They showed that by asuitable skew projection, a proton spectrum can be ob-tained without any proton-proton splittings at all—justas if there had been complete broad-band decoupling ofthe proton-proton interactions. The application of thistechnique to the high-field proton spectra of largemolecules appears to be very promising. The spin echomethod relies on the fact that J-coupling has the effectof modulating the echoes as a function of the evolutionperiod f,. A similar J-modulation can be introduced intocarbon-13 spin echoes by a suitable gating (14) or puls-ing (15) of the proton transmitter. Since carbon-13signals may be noise-decoupled during the detectionperiod t2, this permits a complete separation of proton-carbon coupling effects from carbon-13 chemical shiftsby two-dimensional transformation of the echoes. Fur-thermore, since the echoes are relatively insensitive tofield inhomogeneity effects, significantly enhancedresolution in the F, dimension may be obtained (15).Such spectra are conveniently described by the term"two-dimensional J-spectra."

In 1977, the majority of the developments were intwo-dimensional spectroscopy of liquids, two solidstate applications (21, 22) being extensions of earliermethods for separating chemical shifts and dipolar in-teractions. Applications to high resolution work in li-quids had already raised some new problems charac-teristic of the technique itself, and many of the 1977papers addressed themselves to these technical prob-lems. For example it was becoming clear that in hetero-

nuclear J-spectroscopy (protons and carbon-13), severalpossible pulse sequences could be considered (15,17-20). No formalism existed for representing such se-quences in algebraic notation, so that each new pulsesequence had to be shown diagrammatically. Twomodes of operation that have been widely used are the"gated decoupler" experiment, where echo modulationis introduced by switching off the proton noise de-coupler at the midpoint of the evolution period (14), andthe "proton flip" method where a 180° proton pulse isapplied at the midpoint of the evolution period (15). Nor-mally the carbon-13 signal is detected under noise-de-coupled conditions. For first-order spin coupling, theprincipal difference between the two techniques is thatthe gated decoupler method halves all the CH split-tings, thus reducing the inherent resolving power of thisexperiment.

When these spin echo techniques were applied to apractical case, the carbon-13 spectrum of pyridine (15),it became evident that strong coupling among the pro-ton spins produced some interesting new features (15,18, 20, 27). Two-dimensional J-spectra of pyridine ob-tained by the gated decoupler technique were found tohave exactly the same form as the spin multiplets of theproton-coupled carbon-13 spectrum (20). However, theproton flip experiment, because of the mixing of statesby the 180° pulse, produced J-spectra of a differentform, and a new theoretical treatment was required topredict and analyze these spectra (9, 15, 27). One novelaspect of these calculations was the prediction thatcertain lines should have negative intensities even inthe absence of any spin population inversion (27).

Similar complications occur in proton spectroscopy,but become far less serious at high magnetic fields.Nagayama et al (23, 25) studied the proton J-spectra ofseveral amino acids and of bovine pancreatic trypsin in-hibitor at 360 MHz. The spin echo method allowed themto eliminate chemical shift effects in the F, dimension(although both shifts and coupling constants appear inthe F2 dimension since homonuclear noise decouplingis not feasible), thus displaying the spin multiplets freefrom the overlap problems inherent in the conventionalspectrum. This appears to be one of the more promisingapplications of two-dimensional spectroscopy, par-ticularly when combined with skew projection methods,which can eliminate the proton-proton splittings com-pletely for the weakly coupled case (11).

Another idiosyncracy of two-dimensional spectros-copy emerged when spectra were displayed in a phase-sensitive mode. The responses exhibited a novel lineshape—sections through the response parallel to onefrequency axis showed a rapid change in the mode ofthe signal, for example, negative absorption—disper-sion — positive absorption, as the second frequency

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Figure 2. A computer simulation of the "phasetwist" line shape. Cross sections parallel to the fre-quency axes through the center of the line have apure absorption-mode line shape, whereas parallelcross sections offset from the center have increas-ing amounts of dispersion mode. This line shape isa fundamental feature of two-dimensional spectraderived from phase-modulated signals whenphase-sensitive display is used (43).

parameter was varied through the resonance condition(19, 28). This so-called "phase twist" is illustrated inFigure 2.

Experimental work soon brought to light anotherproblem, the appearance of spurious signals in two-di-mensional spectra, weak replicas of the real spectrumbut displaced to one side (18,19). These artifacts turnedout to be caused by pulse imperfections and persistedwhen the pulse lengths were carefully adjusted, for thespatial inhomogeneity of the radio-frequency field en-sured that there were always some parts of the sampleexperiencing imperfect pulses. The most serious ofthese artifacts were found to be suppressed very effec-tively by a phase-cycling sequence (24).

New two-dimensional experiments tend to hinge onfinding new roles for the spins during the evolutionperiod f,. If spin-lattice or spin-spin relaxation is the onlyinfluence during f,, then two-dimensional transforma-tion allows the relaxation information to be presented inthe form of a line profile. A spectrum could thus begenerated with the usual chemical shifts in the F2

dimensions, but with natural line widths in the F,dimension. Figure 3 shows such a spectrum for car-bon-13 in butan-1,3-diol with different amounts of addedMn2+ ions, showing preferential broadening at the hy-droxyl sites in the more concentrated solution (29). Thisis a convenient visual representation for comparison ofnatural line widths.

Perhaps the most important new development during1977 arose from an extension of Jeener's experiment toheteronuclear spin systems. Maudsley and Ernst (26)were able to demonstrate that magnetization can betransferred from a nuclear species of low gyromagneticratio (the S spins) to a nuclear species of high gyromag-netic ratio (the / spins) affording an appreciable

improvement in sensitivity when mapping out the spec-trum of the S spins. The S spins are first excited by a90° pulse and then allowed to precess freely during theevolution period f,, at the end of which simultaneous90° pulses are applied to both / and S spins. Thistransfer of magnetization is most simply visualized interms of "pumping" of spin populations (31), which ex-plains why the responses always appear in antiphasepairs and why the intensity ratios within multiplets areunusual. Maudsley and Ernst (26) used the magnetiza-tion transfer technique to detect carbon-13 spectra in-directly by observation of the much stronger protonsignal, although since the method relies on couplingbetween / and S it does nothing to circumvent the lownatural abundance of carbon-13 nuclei. In fact a con-siderable sensitivity improvement is also achieved inexperiments where the roles of protons and carbon arereversed, transferring magnetization from protons tocarbon-13, since the detected carbon-13 signal strengthis then determined by proton spin populations (31). Thismagnetization transfer technique led directly to experi-ments designed to correlate proton and carbon shifts,that is to say to identify resonances of directly bondedprotons and carbon nuclei (30, 31). When CH splittingsare removed from both frequency dimensions, two-dimensional Fourier transformation can produce a shiftcorrelation map, where peaks occur at coordinatesdetermined by the proton and carbon shifts of directlybonded atoms. This important application is describedin more detail in Section V.

At the very end of 1977, an ingenious new experimentwas described that permits the detection of the nor-mally forbidden multiple-quantum transitions (32). Ex-citation of multiple-quantum "coherences" requires aselective radio-frequency pulse or a combination of two

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CH20H

- CHOH

7O Hz

"1 "1

Figure 3. Absolute-value mode two-dimensional spectra from carbon-13 in butan-1,3-diol, obtained using the evolution period f, togenerate an unmodulated spin echo. The F, dimension shows the natural line widths, and the F2 dimension the carbon-13chemical shifts. The left-hand spectrum corresponds to a solution containing 10~5 M Mn2*, and the right-hand spectrum to 10"M Mn2+. In the stronger solution there is severe preferential broadening of the hydroxyi-bearing sites.

pulses acting in a frequency-selective manner. Thesecoherences are allowed to precess during the evolutionperiod f, and are then transformed into observablesingle-quantum signals by a mixing pulse. The variousorders of multiple-quantum transitions (for example, allthe even-numbered quantum jumps) may be separatedby combining the results of experiments performed us-ing different radio-frequency phases for the excitationpulses. Multiple-quantum transitions have been utilizedin the past for the assignment of single-quantum reso-nances to the energy level diagram, and for determiningrelative signs of coupling constants.

In 1978 work continued on the application of magne-tization transfer methods to the problem of chemicalshift correlation (33). An intriguing twist to the magneti-zation transfer experiment was discovered by Maudsleyet al (37). Magnetization of the S spins, defocused byfield inhomogeneity effects during the evolution periodf,, is transferred to the / spins and refocuses during t2,forming a "coherence transfer echo." Since in generalthe gyromagnetic ratios yf and 7S are different, the

times for the "unwinding" and "rewinding" processesare unequal, and in the case of magnetization transferfrom protons to carbon-13, the echo occurs at approx-imately f2 = 4f,. The theoretical treatment of thisphenomenon (37) predicts a similar behavior when mul-tiple-quantum coherences are transferred, generating asequence of several echoes, the number and timing ofwhich reflect the number of coupled spins. A lessgeneral but perhaps simpler description can be made interms of population transfer arguments (31). It may atfirst sight seem surprising that a population transfercan pass on phase information suitable for spin echoformation. The key is that the field inhomogeneity infor-mation is coded into amplitude modulation of thecarbon-13 signal as a function of f,; this amplitudemodulation can be represented as sums of equal setsof counterrotating vectors, one set of which has the cor-rect disposition of fast and slow components to form anecho, while the other continues to defocus with time.The appearance of such "anti-echoes" as well asechoes is a basic feature of this phenomenon.

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The concept of an evolution period can be usefuleven without two-dimensional Fourier transformation.For example it is possible to exploit this idea in an ex-periment known as J-scaling (34), which scales down allthe proton-carbon splittings in a carbon-13 spectrum bya constant predetermined factor, revealing thecarbon-13 multiplicities in a manner similar to that ofcoherent off-resonance decoupling, but without someof the practical difficulties of the latter technique. Asingle sample of the nuclear magnetization is made atthe end of the evolution period, and the experiment isrepeated for several different values of f, in order tobuild up an interferogram My(f,). Since it is the signal atthe end of the evolution period that is measured, itsphase may be determined by the sum of phase anglesaccumulated as a result of two different kinds of motionduring f,. For example, proton-coupled free precessionmay be allowed during a fraction f,/ft and broad-banddecoupled precession during the remainder of f,. An ap-parent averaging process thus occurs and the interfer-ogram Mfti) transforms into a spectrum S(F,) where allthe proton-carbon splittings are reduced by a factor ft.

Although this is not strictly a two-dimensionalFourier transform experiment, it utilizes one of thebasic features, the transformation of an interferogramMjXf,), and the J-scaled spectrum may be thought of asa skew projection of a true two-dimensional spectrum.

Because of this need to build up a large number ofpoints on an interferogram, it is clear, that two-dimen-sional spectroscopy requires a large number of sepa-rate experiments and is therefore inherently time-con-suming. However, by a careful theoretical analysis, Aueet al (35) have shown that this does not entail a corres-ponding loss of sensitivity compared with a convention-al nmr experiment utilizing the same amount of time.Both experiments accumulate the same number of tran-sient nmr signals in a given time, so the overall signalenergy is of the same order in both cases. In the two-dimensional experiment this signal energy is dispersedinto two frequency dimensions, but so too is the noise,and the sensitivity remains comparable with that of theconventional experiment. The two-dimensional methodsuffers a slight loss of sensitivity attributable to signaldecay during the evolution period, and experimentsdesigned to improve the resolution in the F, dimensionare adversely affected by the necessity of sampling theinterferograms for an extended period. The generalresult is that there is a sensitivity loss by a factor whichvaries between about 2 and 6 (35). In two-dimensionalautocorrelation experiments similar to the Jeener ex-periment (9) there is a further loss attributable to thesplitting of each conventional resonance line into 2 K - 1components, where K is the number of weakly couplednonequivalent spins.

Three further papers during this period concentratedon the influence of strong coupling effects in two-dimensional J-spectroscopy (38, 39, 41). These calcula-tions confirm earlier predictions that J-spectra excitedby the "proton flip" method have exact symmetry aboutF, = 0, whereas the "gated decoupler" method pro-duces a J-spectrum that has the same form as the con-ventional carbon-13 multiplet, often markedly asym-metric. The prediction that some of the responses in thetwo-dimensional J-spectrum may have negative inten-sities (a phase property and not a population inversion)has been confirmed experimentally for AB, ABX, andABC cases (41).

This period saw the application of two-dimensionalJ-spectroscopy to the study of proton-carbon multipletstructure in the carbon-13 spectrum of cholesterol (36)showing a clear separation of shift and coupling infor-mation in a reasonably complicated spectrum, and theinvestigation of carbon-carbon spin coupling in carbon-13-enriched samples of benzyl alcohol and alanine (40).There seems to be no fundamental reason why this lat-ter technique should not be extended to natural abun-dance samples, making it generally applicable to themeasurement of these important coupling constants.

The last paper in the period covered by this reviewtreats the important subject of projections and crosssections of two-dimensional spectra (42). Cross sec-tions parallel to the frequency axis F, have been muchused to simplify the presentation of information fromtwo-dimensional Fourier transform experiments (20, 27)while projection on the F, or F2 axis has been employedin the earliest experiments (7). Skew projections andcross sections have been less common. There is an im-portant theorem (42) that states that a cross sectiondrawn through the origin across the time-domain signalS(tut2) is the Fourier transform of the projection of thefrequency domain signal S(FUF2) onto a line whichmakes the same angle with respect to the axes. A goodexample is provided by the J-scaling experiment whichis essentially the pulse sequence used by Muller et al (7)but with the signal sampled along a skew axis throughS(tut2) with tjt2 = 1/(ft-1). The result is therefore a skewprojection of the two-dimensional spectrum S(F,,F2)such that all the multiplet splittings are scaled down bya factor ft. Skew projections of J-spectra at 45 ° are thekey to generating proton spectra with spin multipletstructure suppressed (11, 23, 25, 42), while skew crosssections display just the spin multiplets one at a time(42). A theoretical analysis (42) shows that the sensitivi-ty of projections of a two-dimensional spectrum isgenerally poorer than in the spectrum itself; however, itis possible to redeem this loss if a suitable weightedprojection is calculated (42).

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III. A TYPICAL TWO-DIMENSIONAL FOURIERTRANSFORM PROGRAM

A. The Data Matrix

We may take as the starting point an existing pro-gram for conventional Fourier transformation and in-quire how this may be modified for two-dimensionalspectroscopy. The generation of a suitable pulse se-quence will not be considered here as it depends on thedetails of the particular application in mind. There re-mains the basic process:

S(tut2)~S{FuF2)

together with the details of displaying the latter. Inmany experiments Fourier transformation with respectto f2 can be accomplished as soon as the transientsignals S(f2) have been acquired or when time averagingis complete. Exponential weighting is normally em-ployed in order to optimize the signal-to-noise ratio, us-ing a time constant matched to the effective decay con-stant T2*.

The experiment is repeated several times with in-creasing evolution periods tu a sufficient time beingallowed between experiments for both spin-spin andspin-lattice relaxation and the establishment of anuclear Overhauser effect where this is appropriate.The result is a data matrix S(f1,F2), which is usually solarge that it must be stored on a magnetic disk unit. Inmost acquisition routines and in the Cooley-Tukey algo-rithm for Fourier transformation, there are severalstages of binary scaling in order to utilize the dynamicrange of the computer effectively. As a result, eachspectrum S(F2) may have a different scaling factor,a situation that must be corrected by renormalizationwith a suitable common scaling factor chosen to avoidoverflow for the largest value of the two-dimensionalarray.

At this point the information is stored sequentially asspectra S(F2) in order of increasing f,, and because thesecond transformation requires interferograms S(f,) forcorresponding points on each spectrum, a transposi-tion of the matrix is necessary, rows becoming col-umns. Since disk storage space normally exceeds dataspace in core by a large factor, transposition (whichtakes place in core) involves a large number of stages(19). The result is a set of interferograms S(f1), one foreach frequency step in the F2 dimension, stored se-quentially on the disk, ready for the second transforma-tion. The number of different r, values sampled governsthe overall duration of the two-dimensional experimentand is, therefore, kept to a minimum. The interfero-grams are therefore commonly zero-filled as well as be-

ing exponentially weighted, in order to make the bestuse of the available F, digitization.

B. Phase Adjustment

Where a conventional Fourier transform experimentproduces a sine and a cosine transform, early two-dimensional transformation programs (9, 19) generatedfour F, spectra with different phase properties, Scs(F,),Ssc(F,), Sss(F,) and S™(F,), where the superscriptsdenote the sine and cosine components of the first andsecond transformations. These four spectra are single-sided, there being no discrimination between positiveand negative frequencies in the F, dimension. The signinformation can be retrieved by calculating the twoquadrature signals:

SA =SB =

±=F

A simpler alternative would be to use a double-sidedtransformation with respect to f,.

There are several instrumental effects which can in-troduce phase errors into two-dimensional spectra; theyarise, for example, because it is not always possible tostart acquisition exactly at f, = 0 or at t2 = 0, andbecause of the effects of low-pass analogue filters usedto prevent aliasing of noise in the F2 dimension. Theusual practice is to correct these phase errors after thesecond transformation. For any one trace in the F,dimension this is accomplished in the normal way bycalculating

Srea/ = 4>

where 0 is a phase angle which includes a fixed termand a second term that depends on the frequency in theF, dimension. Furthermore </> may well prove to be afunction of F2, so the process of correcting phase overthe whole two-dimensional spectrum can be quite te-dious. It is usually accomplished by manual adjustmentof these parameters while a section of the two-dimen-sional spectrum is displayed on an oscilloscope screen.

C. The Phase Twist

In a conventional spectrum, Sjmag would representLorentzian absorption mode signals, and Srea/the dis-persion mode. The situation is far more complicated intwo-dimensional spectroscopy when phase-modulatedsignals are involved; cross sections taken through theresponse change mode as the second frequency pa-rameter is varied. Off-resonance sections might be puredispersion mode, changing to pure absorption at exact

12

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resonance and becoming dispersion in the oppositesense on the far side of resonance. The quadraturesignal would have absorption mode sections away fromresonance (in opposite senses) and pure dispersionmode at resonance. The former example is illustrated inFigure 2. This rapid change in the mode of the signal asone of the frequency parameters is changed has beencalled the "phase twist" (19). Clearly two-dimensionalspectra with several overlapping lines are likely to provedifficult to resolve or interpret with phase-twist lineshapes (41).

There are basically four different methods of circum-venting the phase-twist problem. If the signals are wellseparated in one frequency dimension (for example F2),it is quite feasible to arrange for the horizontal trace tocut a section sufficiently close to exact F2 resonancethat a pure absorption-mode profile is displayed for allsignals on that trace. Adjacent traces may be set farenough from resonance that no appreciable signals aredetected. This simple expedient has been used to ad-vantage in references (15, 18, 20, 27).

The second method can be used only when the spec-trum exhibits exact mirror symmetry about F, = 0 forthe absorptive components, and antisymmetry withrespect to the dispersive components. This is alwaystrue for carbon-13 J-spectra obtained by the "protonflip" method, and also holds for the "gated decoupler"method if the protons are only weakly coupled (20, 27).Discrimination of positive and negative F, frequenciesis no longer important and it is possible to plot SCS(F,),which presents an absorption-mode profile in both fre-quency dimensions (19, 28). This spectrum can bethought of as having been folded about F, = 0 so thatantiphase dispersive components cancel.

The third method of eliminating the phase twist ismore complicated. Two experiments are required, oneof which involves the application of a 180° pulse on the/ spins (carbon-13 for example) at the end of the evolu-tion period, so that the sense of the / spin precession isapparently reversed (28). The sum and difference of thesignals in the two experiments are taken in suitablelinear combinations to produce spectra where positiveand negative F, frequencies are discriminated and allthe signals are in the absorption mode. Importantrestrictions on this method are that it requires an exact180° pulse and cannot be applied to systems of homo-nuclear coupled spins, such as protons (28).

D. The Absolute-Value Mode

The fourth possibility of avoiding the complicationsof phase-twisted lines is to plot an absolute-value modesignal. This expedient has been very widely used in two-dimensional spectroscopy since it also sidesteps theproblem of instrumentally induced phase errors andprovides a simple solution to the problem of positiveand negative absorption signals that are encountered inmagnetization transfer experiments (26). The quantityplotted is:

S(±F,,F2) = {[Scc(FuF2) =F[Sos(FuF2) ±

The tails of such a signal are dominated by the dis-persion component; not only is this a serious line-broadening influence, but rather complicted interfer-ence effects occur in the region of overlap of two adja-cent lines. It is therefore not the ideal mode for ex-

Figure 4. A computer simulation of a two-dimen-sional line shape in the absolute-value mode (43).Cross sections parallel to the frequency axes haveprofiles identical to the absolute-value mode lineshape in conventional nmr, a curve described byW[X2 + (AF)2]1/2 where k is a constant, X the linewidth in Hz, and AF the resonance offset in Hz.Diagonal sections have profiles that fall off morequickly in the tails, and for the special case of adiagonal at an angle arctan^/X,), the cross sectionis a pure Lorentzian.

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Vol.1,No.1 13

periments aimed at exploiting the enhanced resolvingpower of spin echo experiments.

E. Characteristic Line Shapes

Even when the problem of phase twist has beenavoided by recording the absolute value mode, theshape of the resonance response is still rather morecomplicated than one might have anticipated. One wayto describe the shape is to say that four ridges rundown from the peak in the directions parallel to the F,and F2 axes. An alternative description is that the con-tours of equal intensity form neither circles nor ellipses,but a four-pointed star. This characteristic shape (43,44)is illustrated in Figure 4. Although the absolute-valuemode emphasizes the ridges, the underlying structureis also present in a double-absorption-mode signal andis in fact a fundamental property of a response derivedby two-dimensional Fourier transformation of a signalS(f,,f2) that decays exponentially in the two time dimen-sions (19, 45) since this too lacks elliptical symmetry. Incontrast, a signal S(tut2) that has Gaussian decays inthe two time dimensions, transforms to a responseS(FUF2) that has circular or elliptical symmetry in thecontour map. It has been suggested that Gaussianweighting functions would be more appropriate than ex-ponentials in two-dimensional spectroscopy. A com-parison of double-Lorentzian and double-Gaussian lineshapes is made in Figure 5.

Instrumental effects many aggravate the appearanceof ridges in two-dimensional spectroscopy. Spec-trometer instabilities during the relatively long time thatthe interferograms S(f,) are being assembled can in-troduce what is loosely referred to as "F, noise," a noisyresponse that runs out from each signal peak along aridge in the F, dimension.

F. Methods of Display

The spectrum S(FUF2) is strictly a surface in threedimensions and its accurate representation on a two-dimensional chart presents a certain challenge. Themost common method of creating a solid or three-dimensional impression is to stack a series of paralleltraces, offset from each other by small horizontal andvertical increments (46). The effect is considerablyenhanced by eliminating those sections of a given tracethat appear to lie "behind" a peak in a previous trace.The spectrometer plotting program therefore keeps arecord of the highest ordinate of all previous traces, lift-ing the pen for the appropriate sections, allowing for theinterpolation that occurs between individual datapoints. Because the earliest attempts to do thisemployed white ink, this has come to be known as the"whitewash" routine. In most situations it is possible toadjust the ratio of horizontal and vertical offsets toavoid obscuring any significant information by thewhitewash process, although where there are negative-going peaks, the program may need to be overridden.

Intensity contour plots, first employed by Waugh et al(10) can be a useful alternative to stacked traces, par-ticularly when plotting speed is an important considera-tion, since the lowest contour normally lies above thebaseline noise. Contour plots are also more convenientfor pinpointing frequency coordinates and are thereforeused in chemical shift correlation experiments, but theyare less well adapted for the study of fine structure onthe peaks. Both contour plots and stacked traces areused in the experimental examples which follow.

For certain applications a plot of the entire two-di-mensional spectrum may be unnecessary because theimportant information is carried on a small number oftraces (normally in the F, dimension). For example, incarbon-13 J-spectroscopy the key traces that occur are

Figure 5. Computer simulations of two-dimensional Lorentzian (a) and two-dimensional Gaussian (b) line shapes reproduced fromreference (19). These frequency-domain surfaces were obtained by double Fourier transformation of time-domain signals decayingwith respect to f, and f2 according to (a) exponentials and (b) Gaussians. Note that (a) has intensity contours in the form of a four-pointed star, whereas (b) has circular contours.

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the carbon-13 chemical shift frequencies in the F2 di-mension, and these are readily determined by a priornoise-decoupled experiment. It is therefore a simplematter to extract the relevant traces and ignore all therest. This has the added advantage in the phase-sen-sitive display that each trace may be separately ad-justed to pure absorption mode in spite of the phasetwist (19).

In J-spectroscopy of homonuclear spin systems (forexample, protons) the key information lies on sectionsthat make a 45° angle with the F, and F2 axes (11).Again it is more economical to compute these crosssections rather than plot out the entire two-dimensionalspectrum. In the general case the exact proton shiftswill not be known beforehand, so a trial-and-errorsearch may be necessary to determine the best loca-tions for the cross sections. For the absolute-valuemode display, a 45° section has the advantage of a nar-rower line width than sections parallel to the F, or F2

axes, and for the special case of equal line widths in theF, and F2 dimensions, the 45° section becomes a pureLorentzian (19, 42).

molecules (23, 25). Unfortunately the projection of aphase-twisted response at this angle involves a mutualcancellation of the positive absorption component withthe negative dispersion components and the projectedsignal is zero. It is therefore necessary to employprojections of the absolute-value mode signal in thiscontext; the projected lines are consequently quitebroad.

Since projections involve the summation of a greatdeal of noise, they may have significantly lower sensi-tivity than the two-dimensional spectrum from whichthey are derived. For this reason, weighted projectionsmay be preferable. This has recently been used toenhance the sensitivity of the "J-scaling" experiment,which can be thought of as a skew projection of a two-dimensional spectrum. This is a case where the traceswhich carry signals are known beforehand, so thatmatched weighting can be employed (42).

IV. TWO-DIMENSIONAL JSPECTRA

G. Projections of Two-Dimensional Spectra

In the transformation S(t)~S(F), the signal measuredat t = 0 represents the integral of the spectrum S(F);consequently in two-dimensional spectroscopy, thefirst interferogram S(tu0) and the first free precessionsignal S(0,t2) transform into the projections of the two-dimensional spectrum onto the F, and F2 axes respec-tively. In this context "projection" onto a given frequen-cy axis involves the digital summation of all the signalsthat lie on sections perpendicular to that axis.

In two-dimensional J-spectroscopy, projection ontothe F, axis superimposes all the J-multiplets onto asingle trace, all centered on F, = 0. This is the samespectrum as would be obtained by one-dimensionaltransformation of an interferogram constructed by mon-itoring the signal at the midpoint of the spin echo (47).Projection onto the F2 axis regenerates the con-ventional nmr spectrum. Although the two-dimensionalspectrum has responses with a phase twist, the disper-sion components cancel for these orthogonal projec-tions and the lines have absorption mode shapes (41).

Skew projection can also be a very useful operation.In two-dimensional proton J-spectroscopy, projection ina direction at 45° with respect to the frequency axeshas the effect of removing the spin multiplet structureleaving a spectrum with only chemical shift informa-tion, as if the protons have been completely decoupledfrom each other (11). This is expected to be of con-siderable help in simplifying the proton spectra of large

A. Heteronuclear Systems

There have been many attempts to investigate pro-ton-carbon spin coupling untrammelled by the com-plicated overlap that occurs in a conventional proton-coupled carbon-13 spectrum. These include coherentoff-resonance decoupling (48), selective excitation ofthe carbon-13 resonances from a single site (49), scalingdown the multiplet splittings by some predeterminedfactor (34), and two-dimensional Fourier transformation(7). This last method can be considerably improved bythe introduction of spin echo refocusing methods, theresult being known as a J-spectrum (47). A two-dimen-sional J-spectrum provides a convenient method ofstudying either the gross structure of the multiplets(quartets from methyl groups, triplets from methylene,etc.) or the fine structure due to long-range proton-car-bon coupling, leading to a more detailed assignment.

B. Echo Modulation by Scalar Coupling

Hahn and Maxwell (50) first showed that spin echoesfrom a homonuclear system of coupled spins are modu-lated as a function of the time 2T at which spin echoesoccur. In a simple 90°-TD-18QO-TR- echo sequence, the180° pulse refocuses chemical shift and field inhomo-geneity effects, but fails to refocus the J-splitting sinceit also has the effect of interchanging the spin statelabels of the multiplet components so that they con-

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Vol. 1, No. 1 15

tinue to diverge during the second period TR. These twodistinct effects of the 180° pulse are clearly apparent inheteronuclear experiments, since if only the / spins ex-perience the 180° pulse there is no J-modulation of theechoes, whereas if 180° pulses are applied simultane-ously to both the / and S spins, the echoes aremodulated at a frequency VzJis-

This can be most readily visualized from a "phase-time diagram" (51). Consider a spin echo experiment ona two-spin system IS where the refocusing pulse isphase-shifted by 90° as suggested by Meiboom and Gill(52). The two components of the / spin spectrum arerepresented by vectors in the rotating frame ofreference, and their phases are followed as a functionof time during the defocusing interval TD and therefocusing interval TR. Figure 6A illustrates the motion ifno 180° pulse is applied to the S spins. The two vectorsdiverge during TD and are then flipped by the refocusingpulse, changing the signs of their accumulated phaseangles. During TR they converge and form an echo withphase angle zero so there is no phase modulation as afunction of f,. This is the same behavior as for the casewhere J/s is zero, for then a single vector at the

chemical shift frequency (dotted lines) is exactly refo-cused. Figure 6B illustrates how this motion is changedwhen an additional 180° pulse is applied to the S spinsin synchronism with the refocusing pulse on the / spins.In addition to the phase jumps shown in Figure 6A,there is an interchange of the two vectors caused by the180° pulse on the S spins, resulting in a continueddivergence of the vectors during TR building up phasedeviations of ± VfeJ/sfi- In homonuclear systems both/and S automatically experience the 180° pulse (unlessthe pulse is selective) so the motion corresponds to thatof Figure 6B.

Echo modulation can also be introduced by a dif-ferent method usually known as the "gated decoupler"technique (14). Figure 6C shows the phase-timediagram. The / spins are completely decoupled from theS spins during TD, but are free to precess in a coupledmode during TR, SO they accumulate phase deviationsof ± Vt, JisU; an exactly analogous result is obtained ifthe roles of TD and TR are reversed. This explains whythe gated decoupler experiment produces J-spectrawith all the splittings halved. For systems where thereis strong coupling between several nonequivalent

Figure 6. Phase-time diagrams for the formation of modulated spin echoes. In (A) the two components of the / spin multiplet are ex-actly refocused, <p = 0 at the end of the evolution period, and there is no modulation as a function of f,. A similar result is obtainedif J/s = 0 (dotted lines). In (B) the effect of the 180° pulse on the S spins is to interchange the two / vectors, and they continue todiverge during TR, leading to phase modulation as a function of f,. In (C) the two / vectors move as one during TD when the S spinsare decoupled, but diverge during TR when the / and S spins are coupled, resulting in phase modulation as a function of f, at onehalf the frequency of case (B).

B o

C o

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a Noise Noise Protons

Noise

180°

Noise Protons

Carbon-13

90° 180 Acquisition (t2)Figure 7. Pulse sequences used to generate carbon-13 spin echoes that are modulated by proton-carbon spin coupling. There aretwo modes of operation, the "gated decoupler" (a) and the "proton flip" (b), and in both modes the same carbon-13 spin-echo pulsesequence is employed. In the "gated decoupler" mode, proton-coupled precession is allowed during one half of the evolutionperiod f, and decoupled precession during the other half (the relative order is not important). In the "proton flip" mode the 180 ° pro-ton pulse at the midpoint of the evolution period interchanges carbon spin state labels, which causes the two carbon-13magnetization components to continue to diverge in the second half of the evolution period. Compare the phase-time diagrams inFigures 6C and 6B. In both modes proton noise irradiation is employed prior to carbon excitation and throughout the acquisitionperiod f2.

-200

200

Figure 8. Two-dimensional J-spectrum ofcarbon-13 in sucrose (43), obtained by the"gated decoupler" method. Carbon-13chemical shifts are displayed in the F2 di-mension and proton-carbon multipletstructure in the F, dimension. The spec-trum is shown in the absolute-valuemode; no attempt has been made toachieve high resolution in the F, dimen-sion.

7200 Hz

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0 Hz

600-100 OHz +100

' I

Figure 9. Absolute-value mode two-dimensional J-spectrum of carbon-13 in 2(1-methylcyclohexyl>4,6-dimethylphenol, obtained bythe "gated decoupler" method. Note the fine structure on two of the methyl group quartets, which identifies them as the aromatic4-methyl (triplet structure) and 6-methyl (doublet structure).

S spins, this method has the great advantage ofgenerating a J-spectrum that has exactly the same formas the / spin multiplets of the conventional nmr spec-trum. Pulse sequences for the proton flip and gateddecoupler experiments are shown in Figure 7.

C. Carbon-13 Spectra

Consider the case where the / spins are carbon-13nuclei and the S spins are protons. Since the carbonchemical shift has no influence during tu and theproton-carbon coupling no influence during f2, a two-dimensional Fourier transformation experiment givescomplete separation of these parameters. The F,dimension shows only spin multiplet structure (so thespectral width can be quite narrow), and very highresolution can be achieved, if necessary, since the prin-cipal effects of field inhomogeneity are refocused. The

F2 dimension exhibits only carbon-13 shifts and has thesame resolution as a conventional decoupled spec-trum. Provided that all resonances in the decoupledspectrum are sufficiently well resolved, the problem ofoverlap of adjacent multiplets is eliminated.

The application of the gated decoupler method to thestudy of the gross structure of multiplets is illustrated inFigure 8 which shows the two-dimensional J-spectrumof carbon-13 in sucrose (43). The multiplicity of eachresonance is quite clear, even where there are two dou-blets poorly resolved (near F2 = 580 Hz). The absolute-value display mode has been employed, and no attempthas been made to achieve high resolution in the F,dimension.

Quite often more information is required for a com-plete assignment, and the fine structure due to long-range proton-carbon coupling must be investigated.This is the case for the three methyl groups of 2(1-meth-ylcyclohexyl)-4,6-dimethylphenol. The two<limensional

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J-spectrum of the aliphatic carbon-13 region is shown inFigure 9. The fine structure on the three methyl quartetsimmediately identifies the 6-methyl carbon because ofthe doublet splitting, the 4-methyl carbon because oftriplet fine structure and the cyclohexyl methyl becauseit is broadened by several long-range couplings (53).

D. Proton Spectra

Two-dimensional J-spectroscopy is expected toprove even more useful in helping to disentangle theproton spectra of large molecules. In these homonu-clear systems the spins necessarily remain coupledduring the detection period t2, with the result that thespin multiplets lie on 45° diagonals on the two-dimen-sional spectrum. A projection in this direction producesa spectrum with all the spin multiplets collapsed, leav-ing only chemical shift effects. For this simple result itis necessary that all spins be weakly coupled (25, 39,41), but in the very high fields of superconductingspectrometers this is usually the case. The phase-twistline shape causes a serious problem here because the45° projection leads to mutual cancellation of the ab-sorptive and dispersive components of the signal; thiscan be circumvented by projecting the absolute-valuemode signal, at the expense of a significant loss inresolution (42).

Aue et al (11) demonstrated the first such 45° projec-tion of a two-dimensional proton J-spectrum for a mix-ture of ethyl chloride, bromide and iodide, whileNagayama et al (23, 25) showed the first J-spectra ofprotons in a mixture of amino acids and in bovine pan-

creatic trypsin inhibitor (BPTI). In a later paper (42) thesesame authors used 45° projections to obtain thechemical shifts of the protons in BPTI and also usedskew projections to obtain spectra with scaled-downspin-spin splittings.

Figure 10 shows another example of proton J-spec-troscopy at high magnetic field, reproduced by kind per-mission of Dr. J. Delayre. This is a 360 MHz spectrum ofa dimethyl sulphoxide solution of gramicidin-A, a linearpeptide of 15 residues with important antibiotic prop-erties. A 2 ppm section of the proton spectrum isshown, centred on the NH region. A small coupling ofapproximately 0.8 Hz is discernible near 8.14 ppm in thetwo-dimensional J-spectrum, but is not apparent in theconventional spectrum.

V. CHEMICAL SHIFTCORRELATION MAPS

One of the most promising chemical applications oftwo-dimensional Fourier transformation is the possibil-ity of relating the chemical shifts of two different nu-clear species, for example protons and carbon-13. Thisprovides a powerful assignment technique, an alterna-tive to the rather cumbersome use of off-resonance pro-ton decoupling experiments in carbon-13 spectroscopy(48). In its simplest form the experiment identifies pairsof proton and carbon resonances that arise from direct-ly bonded atoms, although longer range spin-spin inter-actions may also be utilized where desirable.

Figure 10. Two-dimensional proton J-spectrum of the NH region of gramicidin-A at 360 MHz, reproduced by kind permission ofJ. Delayre, F. Heitz, and C. Crane-Robinson (unpublished work). This spectrum brings to light small splittings not apparent in theconventional spectrum, in particular a 0.8 Hz splitting near 8.14 ppm. The horizontal frequency axis represents the conventionalproton spectrum.

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Initial 90°proton pulse

Second 90°proton pulse

Protonpopulations

Change in I3Cmagnetization

PROTON FREE PRECESSION ( t x )

Inverted Saturated Boltzmann Saturated Inverted

Figure 11. Magnetization transfer from protons to carbon-13 explained in terms of spin state populations. Free precession of theproton magnetizations during the f, period leaves the proton vectors in various possible positions in the XV plane, so that the sec-ond proton 90° pulse rotates the V components of these magnetizations into the Z direction. The range of possible orientations inthe XY plane at time f, leads to proton spin state populations that may be inverted, saturated, or at Boltzmann equilibrium. This re-sults in changes to the population differences across carbon-13 transitions, modulating the longitudinal magnetization associatedwith these transitions as a function of f,. This modulation may be mapped out as a function of f, by applying a 90° carbon pulseand measuring the resultant signal, "reading" the information coded into the f, dependence of the carbon magnetization.

The end result of this experiment is a graph or "map"of carbon-13 chemical shifts (ordinates) plotted againstthe corresponding proton shifts as abscissae, the twofrequency axes spanning scales of about 200 ppm and10 ppm respectively. The factors that affect proton andcarbon-13 shifts are similar but by no means identical,so that most of the points on this graph lie close to adiagonal of slope about 20. The interesting cases arethose with large deviations from this diagonal, repre-senting a specific influence on the proton or carbon-13shift, not matched by a corresponding effect on theother species. Such a "shift correlation map" may bedisplayed in several possible ways. One form that hasproved convenient is the intensity contour plot; it hasthe advantage that the amount of detail presented canbe readily controlled by specifying the contour intervals.

A. Magnetization Transfer

The experiment is based on an indirect detectiontechnique pioneered by Maudsley and Ernst (26) andrelated to the Jeener experiment (9). The nmr spectrom-eter is tuned to detect only one nuclear species (the /spins) while the second species (the S spins) acts indi-rectly through the scalar spin-spin coupling J/g. The

idea is to use the evolution period f, for free precessionof the S spins, monitoring the extent of this precessionby measuring the magnetization transferred to the/ spins at the end of this period. Fourier transformationwith respect to f, then provides a spectrum of theS spins indirectly. When the technique is used purelyfor sensitivity enhancement, the / spins are naturallychosen to be those of higher gyromagnetic ratio, for ex-ample protons, while the inherently weaker resonancesof the second species are detected indirectly. Thischoice may be reversed for shift correlation studies.

One of the easiest ways to visualize the magnetiza-tion transfer experiment is in terms of spin populations(Figure 11). A 90° pulse applied to protons createstransverse magnetization which precesses for a periodf, seconds, and then a second 90° proton pulse is ap-plied. If f, = 0, then no precession occurs, the twopulses have the effect of a 180° pulse, and the protonspin populations are inverted. If f, = 1/(4AF) where AF isthe offset of a given proton resonance from the protontransmitter, then this magnetization vector precessesthrough 90° during the evolution period, so that the sec-ond 90° pulse has no effect, and the proton spins aresaturated (zero Z magnetization). If f, = 1/(2AF), giving180° precession, the second 90° pulse returns the pro-ton magnetization to the + Z axis, corresponding to a

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Boltzmann equilibrium population. Clearly as we followthe transverse proton magnetization through thevarious possible precession angles, we observe acosine modulation of the proton populations after thesecond 90° pulse. Since the two carbon-13 transitionsboth share an energy level in common with this protontransition, the Z components of carbon magnetizationare necessarily affected. Because of the progressiveand regressive arrangement of these connected transi-tions, one carbon transition must lose while the othergains an equal amount. It is an important general rulethat no net magnetization is transferred; there is alwaysan exact balance between positive and negative contri-butions. A 90° carbon pulse is then applied to "read"these population changes. It need not be synchronouswith the second proton 90° pulse.

Fourier transformation with respect to f, and f2 pro-duces a two-dimensional spectrum. Figure 12 shows anexample for methyl iodide (51). The simplest feature ofthis diagram is the 1:3:3:1 quartet, which runs in the F2

dimension along the line F, = 0 and representscarbon-13 magnetization excited by the carbon 90°pulse but not affected by the proton pulses and there-fore not modulated as a function of f,. The modulatedsignals, which carry the interesting proton information,are all to the left of this conventional quartet. In the F,dimension they are doublets and have the frequencies6 ± Vz J, identical with the two carbon-13 satellite linesof the conventional proton spectrum. However they

have opposite signs for their intensities, since as oneproton transition pumps population into the upper levelof a given carbon transition, the other proton transitionpumps it into the lower level. This spectrum is repeatedfour times, corresponding to the four carbon-13 frequen-cies in the F2 dimension. However the relative inten-sities are unusual: two are positive and two negative,but more surprisingly the conventional 1:3:3:1 ratio islost and each transition has unit intensity. The reasonfor this can be appreciated by a detailed considerationof the spin population in the appropriate energy leveldiagram, shown in Figure 13. The intensities ap-propriate to a triplet turn out to be - 1 : 0 : +1 as seenalso in Figure 13; the center transition is degenerate,one component of which is not affected by S spin popu-lations since it has no common energy levels, the othercomponent has spin population pumped in and out atidentical rates. These intensity rules for modulatedsignals were first worked out by Ernst (54).

It is important to note that the modulated signals inFigure 12 arise from proton spin population differences,which are four times larger than the carbon-13 popula-tion differences at the same magnetic field. There isthus a significant gain in sensitivity in this mode ofoperation. Moreover it is the proton spin-lattice relax-ation that determines how fast this experiment can berepeated (only the unmodulated signals lose intensitybecause of slow carbon relaxation), so this also favorsthe sensitivity of the method.

O Hz

A S

200

PROTON SPECTRUM

O Hz

Figure 12. Magnetization transferfrom protons to carbon-13 in methyliodide (51). Trie unmodulatedcarbon-13 signal is the 1:3:3:1 quar-tet running along the axis F, = 0.The four horizontal traces a, b, c,and d each correspond to thecarbon-13 satellite resonances inthe proton spectrum, except thatthe two component lines are in an-tiphase. The carbon-13 quartetswhich run in the F2 dimension havethe unusual intensities ± 1 : ± 1 :=F1 : =F1, which may be explainedby reference to Figure 13.

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Vol.1,No.1 21

(Multiplicity 2)

- 5

AX3 System

Figure 13. Spin state population transfer in AX2 and AX3 spin systems, Here A is taken to represent the carbon-13 spin system, andX the protons; the ratio JXIJA n a s been assumed to be equal to 4, so that spin state populations can be represented by simple in-tegers. The population inversion caused by a 180° pulse is indicated on one of the proton transitions by arrows. As a result the cen-tral line of the AX2 triplet experiences no change in intensity, the outer lines having equal and opposite intensity changes. In theAX3 quartet, all four lines have equal intensity changes, in opposite senses for the low-field and high-field pairs of lines; this ex-plains the intensity ratios observed in Figure 12.

B. Shift Correlation

If shift correlation maps are to be prepared for morecomplicated molecules, the multiplet splitting in twodimensions seen in Figure 12 is an embarrassment, andit is necessary to consider some kind of decoupling inboth frequency dimensions. Unfortunately any experi-ment that causes all the lines to coalesce at a commonfrequency in either dimension results in exact mutualcancellation of the antiphase components. A partialsolution may be achieved by using coherent off-reso-nance proton decoupling and incomplete refocusing bymeans of a 180° carbon pulse not quite at the midpointof the evolution period (31), so that the antiphase linesare not quite coincident in frequency. Figure 14 shows agood example of this method applied to the methyl res-onances of j3-ionone. A better solution has been de-scribed by Maudsley et al (30). Two short fixed delays A,and A2 are introduced into the pulse sequence (Figure15). They are calculated so as to allow 180° relativephase rotation between these components (which differin frequency by JQH). thus preventing mutual cancel-lation. This method relies on the fact that the directlybonded couplings are all of the same order of magni-tude, and it uses a compromise setting for doublets,

triplets, and quartets. Modulated signals transmittedthrough long-range couplings still cancel becausethese couplings are much weaker and no significantphase angles are built up during A, and A2. This greatlysimplifies the shift correlation map. Proton noise irradi-ation provides the decoupling in the F2 dimension, anda 180° carbon pulse at the midpoint of f, provides thedecoupling in the F, dimension (Figure 15). In this wayeach pair of directly bonded carbon and hydrogenatoms generates a single peak on the correlation map,apart from proton-proton splittings which are often de-liberately obscured by poor digital resolution in the F,dimension.

This simplification of the structure of the correlationmap opens the way to the use of intensity contour plots,which are particularly well suited to the determinationof frequency coordinates. The chemical shifts of pro-tons and carbon in 2(1-methylcyclohexyl)-4,6-dimethyl-phenol (55) have been displayed in this manner in Fig-ure 16. A small amount of tetramethylsilane was addedto this sample, and it is important to note that the TMSpeak then serves as a frequency reference for both fre-quency dimensions. (The frequency origins of Figures12 and 14 simply reflect the arbitrary choice of the pro-ton and carbon transmitter frequencies.) These re-

22



400 0 Hz

Figure 14. Correlation between thechemical shifts of carbon-13 andprotons for the methyl groups of 13-ionone (51). Proton-carbon split-tings in both frequency dimensionswere rendered too small to beresolved, by the use of asymmetricrefocusing during f, and coherentoff-resonance decoupling during t2.Each methyl group then gives riseto a single correlation signal in thetwo-dimensional spectrum, sites dand e having degenerate shifts.

Proton shifts

sponses along the dotted line on the right of thisdiagram are the unmodulated carbon-13 signals. Site ggives an unmodulated signal but no correspondingmodulated component because this is a quaternary sitefor which the transferred magnetization is cancelledunder decoupled conditions. Note that, site / shows adoublet structure in the F, dimension, attributed to theshift difference between axial and equatorial protons.

The power of this method is brought out even moreclearly in the shift correlation map of cholesteryl

acetate shown in Figure 17. Steroid proton spectra arenotorious for having a large number of nearly degener-ate shifts, but the shift correlation map allows them tobe separated and assigned because of the introductionof a second dimension. The numbering scheme isshown in Figure 17; the olefinic carbon resonances 5and 6 lie outside the F2 range, while the substituted site3 gives a resonance line which is folded from high F2

frequencies and appears in the trace plot of Figure 18near 46 ppm. The carbon-13 assignment is that of Reich

Figure 15. Pulse sequence used for obtaining two-dimensional chemical shift correlation spectra with proton-carbon splittingsremoved. The basic magnetization transfer is accomplished by the 90° pulses, while proton-carbon splittings are suppressed bythe carbon-13 180° pulse in the f, dimension, and by the use of proton noise decoupling in the t2 dimension. The delays A1 and A2prevent the cancellation of antiphase signals.

90°

Protons

Carbon-13

I8OC

90°

90c

Acquisition (tt)

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Vol.1, No. 1 23

- 0 ppm

- I - 2 ppm

Proton ShiftsFigure 16. Chemical shift correlation map for protons andcarbon-13 in 2(1-methylcyclohexyl)-4>6-dimethylphenol. The un-modulated carbon-13 signals lie on the broken line on the right,with shift correlation peaks to the left. Note the doubling of theresponse from site f, attributable to the different chemicalshifts of the axial and equatorial protons. Reproduced fromreference (55).

et al (48) but with the sites 12 and 16 interchanged in ac-cordance with the later work of ApSimon et al (56) andof Smith et al (57). In the contour plot of Figure 17 all theunmodulated carbon-13 signals lie on the vertical dash-ed line on the right-hand side of the diagram. Quater-nary carbon sites such as 10 and 13 generate no modu-lated signals, but all the other sites give shift correlationpeaks. When there are several close lines in the F2

dimension the contour plot gives the clearer picture, butthe trace plot of Figure 18 gives a better indication ofthe structure on the lines. The most important featureappears to be the doublet structure in the F, dimensionobserved for sites 11,12,15,16, and 22, all of which aremethylene groups with nonequivalent axial and equa-torial protons. The weak satellite lines on each side ofthe methyl group resonances appear to be an instru-mental artifact.

Since the modulated intensities of the signals insuch a spectrum depend on proton spin population dif-ferences and on proton rather than carbon-13 relaxationtimes, the sensitivity of this method is considerably bet-ter than that of most other two-dimensional experi-ments. Figure 17 and 18 represent data acquired over-

night, but usable correlation maps have been obtainedwith data gathered for as little as one minute. Only 64experiments in the f, dimension were used, the poor F,digitization simplifying the line profiles in this dimen-sion. Under these conditions there is one correlationpeak for each carbon-proton pair.

VI. DISCUSSION

We have seen that two-dimensional Fourier trans-formation experiments in high-resolution nmr havealready been used successfully for the separation of

Figure 17. Chemical shift correlation contour map of protonsand carbon-13 in cholesteryl acetate. The carbon-13 chemicalshift scale has been referenced approximately to tetramethyl-silane. Unmodulated carbon-13 signals lie on the right of thediagram with shift correlation peaks on the left. The quaternarysites (10 and 13) show no correlation peaks.

IO ppm

CARB0N-I3

SHIFTS

PROTON SHIFTS (ppm)

24

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Oppm

20

30

inro

Orr

40 <

50

60

PROTON SHIFTS (ppm)

Figure 18. Multiple trace plot corresponding to Figure 17. Line3 has been folded from high F2 frequencies. Note the doublingof peaks from sites 11,12,15,16, and 22, attributed to the shiftdifference between axial and equatorial protons.

molecular parameters, displaying for example multipletstructure and chemical shift effects in the two frequen-cy dimensions. A second important application is thatof correlation, and the most promising area here ap-pears to be the correlation of the chemical shifts of twoheteronuclei, exploiting the fact that directly boundatoms tend to have large spin-spin coupling constants.Bringing to light normally forbidden transitions is thethird major application, and an interesting series of mul-tiple-quantum experiments have been performed utiliz-ing this principle. The two-dimensional transformationtechnique also lends itself quite naturally to experi-ments incorporating spin echo methods to enhance theresolution of the spectrum. It is interesting to note thatmany of these applications have previously been at-tacked by various double-resonance techniques suchas selective decoupling and spin tickling.

There are three main barriers to the widespread useof two-dimensional nmr methods: the need for exten-sive data storage, the complexity of some of the ex-periments, and the excessively long time needed torecord the two-dimensional spectrum in durable form.The first problem has been met by the use of hard orfloppy disk units or extended core storage, and sincemany modern Fourier spectrometers are being equip-ped with disk units, this is unlikely to prove a fundamen-tal limitation. The complexity of some of the pulsesequencing and data manipulation programs repre-sents no serious activation barrier now that commercialtwo-dimensional transformation program packages arebecoming available. The time factor for plotting two-dimensional spectra is greatly reduced with new incre-mental and XVplotters (which write very fast) or by high-resolution video display terminals. The increased use ofmultiprocessing and microprocessor systems alsomeans that plotting time need not represent lost experi-mental time on the modern spectrometer. Time mayalso be saved by recording cross sections or projec-tions of two-dimensional spectra, since these can oftensummarize all the useful information.

Now that these technical barriers are being lifted,there remains the challenge of devising new ordeals forthe nuclear spins during the evolution period, leading tonew applications of the principle of the two-dimen-sional transformation. This idea, like double resonance,opens up a door to the invention of all kinds of new ap-plications, and it seems likely that the ones describedhere will soon be followed by more interesting andexciting experiments. It is hoped that this review mayencourage and stimulate such investigations.

ACKNOWLEDGEMENTS

The authors are indebted to Professor Richard Ernst forproviding several manuscripts prior to publication, and

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Vol.1,No.1 25

for permission to reproduce Figure 1 from his article inthe Journal of Chemical Physics. Dr. Jean Delayre verykindly provided Figure 10 prior to publication; thisdiagram was presented at the Fourth InternationalMeeting on NMR Spectroscopy, York, England, July1978. Some diagrams have been taken from theses atOxford University by Geoffrey Bodenhausen, MalcolmLevitt, Gareth Morris, Reinhard Niedermeyer and DavidTurner. It is a pleasure to acknowledge illuminatingdiscussions with Dr. Howard Hill.

References and Notes1J. Jeener, Ampere International Summer School, Basko Polje,

Yugoslavia, 1971, unpublished.2R. R. Ernst, Vlth International Conference on Magnetic Resonance

in Biological Systems, Kandersteg, Switzerland, 1974, unpublished.3A. Kumar, D. Welti, and R. R. Ernst, Naturwissenschaften 62, 34

(1975).4A. Kumar, D. Welti, and R. R. Ernst, J. Magn, Reson. 18, 69 (1975).5R. R. Ernst, Chimia, 29, 179 (1975).eR. K. Hester, J. L. Ackerman, V. R. Cross, and J. S. Waugh, Phys.

Rev. Lett. 34, 993 (1975).7 L MUller, A. Kumar, and R. R. Ernst, J. Chem. Phys. 63,5490 (1975).8M. Alia and E. Lippmaa, Chem. Phys. Lett. 37, 260 (1976).9W. P. Aue, E. Bartholdi, and R. R. Ernst, J. Chem. Phys. 64, 2229

(1976).10R. K. Hester, J. L. Ackerman, B. L. Neff, and J. S. Waugh, Phys. Rev.

Lett. 36, 1081 (1976). '11W. P. Aue, J. Karhan, and R. R. Ernst, J. Chem. Phys. 64, 4226

(1976).12J. S. Waugh, Proc. Natl. Acad. Sci. (U.S.A.) 73, 1394 (1976).13S. Vega, T. W. Shattuck, and A. Pines, Phys. Rev. Lett. 37,43 (1976).14G. Bodenhausen, R. Freeman, and D. L. Turner, J. Chem. Phys. 65,

839 (1976).15G. Bodenhausen, R. Freeman, R. Niedermeyer, and D. L Turner, J.

Magn. Reson. 24, 291 (1976).16M. E. Stoll, A. J. Vega, and R. W. Vaughan, J. Chem. Phys. 65, 4093

(1976).17A. Kumar, W. P. Aue, P. Bachmann, J. Karhan, L. MUller, and R. R.

Ernst, Proc. XlXth Congres Ampere, Heidelberg, 473 (1976).1*L MUller, A. Kumar, and R. R. Ernst, J. Magn. Reson. 25,383 (1977).19G. Bodenhausen, R. Freeman, R. Niedermeyer, and D. L. Turner, J.

Magn. Reson. 26, 133 (1977).20R. Freeman, G. A. Morris, and D. L. Turner, J. Magn. Reson. 26, 373

(1977).21S. J. Opella and J. S. Waugh, J. Chem. Phys. 65, 4919 (1977).22E. F. Rybaczewski, B. L. Neff, J. S. Waugh, and J. S. Sherfinski, J.

Chem. Phys. 67, 1231 (1977).

23K. Nagayama, K. Wiithrick, P. Bachmann, and R. R. Ernst, Natur-wissenschaften 64, 581 (1977).

24G. Bodenhausen, R. Freeman, and D. L. Turner, J. Magn. Reson. 27,511 (1977).

25K. Nagayama, K. Wiithrich, P. Bachmann, and R. R. Ernst,Biochem. Biophys. Res. Commun. 78, 99 (1977).

28A. A. Maudsley and R. R. Ernst, Chem. Phys. Lett. 50, 368 (1977).27G. Bodenhausen, R. Freeman, G. A. Morris, and D. L. Turner, J.

Magn. Reson. 28, 17 (1977).28P. Bachmann, W. P. Aue, L Muller, and R. R. Ernst, J. Magn. Reson.

28, 29 (1977).29G. Bodenhausen and R. Freeman, J. Magn. Reson. 28, 303 (1977).30A. A. Maudsley, A. Kumar, and R. R. Ernst, J. Magn. Reson. 28,463

(1977).31G. Bodenhausen and R. Freeman, J. Magn. Reson. 28, 471 (1977).32A. Wokaun and R. R. Ernst, Chem. Phys. Lett. 52, 407 (1977).M G. Bodenhausen and R. Freeman, J. Am. Chem. Soc. 100, 320

(1978).MR. Freeman and G. A. Morris, J. Magn. Reson. 29, 173 (1978).35W. P. Aue, P. Bachmann, A. Wokaun, and R. R. Ernst, J. Magn.

Reson. 29, 523 (1978).M D. L. Turner and R. Freeman, J. Magn. Reson. 29, 587 (1978).37A. A. Maudsley, A. Wokaun, and R. R. Ernst, Chem. Phys. Lett. 55,9

(1978).38A. Kumar and C. L. Khetrapal, J. Magn. Reson. 30,137 (1978).39A. Kumar, J. Magn. Reson. 30, 227 (1978).40R. Niedermeyer and R. Freeman. J. Magn. Reson. 30, 617 (1978).41G. Bodenhausen, R. Freeman, G. A. Morris, and D. L Turner, J.

Magn. Reson. 31, 75 (1978).42K. Nagayama, P. Bachmann, K. Wiithrich, and R. R. Ernst, J. Magn.

Reson. 31,133 (1978)." M . H. Levitt, unpublished work.**D. L. Turner, D. Phil. Thesis, Oxford University, 1977.^D. E. Pearson, Transmission and Display of Pictorial Information,

Pentech Press, London (1975)."R . R. Ernst, R. Freeman, and W. A. Anderson, J. Chem. Phys. 46,

1125(1967).47R. Freeman and H. D. W. Hill, J. Chem. Phys. 54, 301 (1971)." H . J. Reich, M. Jautelat, M. T. Messe, F. J. Weigert, and J. D.

Roberts, J. Am. Chem. Soc. 91, 7445 (1969).49G. Bodenhausen, R. Freeman, and G. A. Morris, J. Magn. Reson. 23,

171 (1976).ME. L. Hahn and D. E. Maxwell, Phys. Rev. 88, 1070 (1952).51G. Bodenhausen, D. Phil. Thesis, Oxford University, 1977.52S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958).MG. A. Morris, D. Phil. Thesis, Oxford University, 1978.MR. R. Ernst, Eighteenth Experimental nmr Conference, Asilomar,

California, 1977; Sixth International Symposium on MagneticResonance, Banff, Canada, 1977.

MR. Freeman and G. A. Morris, J. Chem. Soc. Chem. Commun. 1978,684.

M J . W. ApSimon, H. Beierbeck, and J. K. Saunders, Can. J. Chem. 51,3874 (1973).

57W. B. Smith, D. L. Deavenport, J. A. Swanzy, and G. A. Pate, J.Magn. Reson. 12, 15 (1973).


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