A SIMPLE DESCRIPTION OF TWO-DIMENSIONAL NMR SPECTROSCOPY*

AdBax

Laboratory of Chemical PhysicsNational Institute of Arthritis, Diabetes

and Digestive and Kidney DiseasesNational Institutes of Health

Bethesda, Maryland 20205, USA

I. IntroductionA. Sampling Frequency and SensitivityB. Lineshapes and Transform TechniquesC. Two-dimensional Quadrature and Absorption ModeD. Coherence Transfer by Means of Radiofrequency Pulses

II. Homonuclear Shift Correlation through Scalar Coupling

III. Homonuclear Chemical Shift Correlation through Cross Relaxation

IV. Indirect Detection of *s N through Multiple Quantum Coherence

V. Discussion

References

I. INTRODUCTION

The concept of two-dimensional (2D) Fouriertransformation in NMR was first introduced byJeener (1) in 1971. Since that time, several hun-dred different 2D pulse schemes have been pro-posed in the literature. The selection of experi-ments that will be described is based on theinsight they provide into the basis and generalityof 2D NMR, and also on the importance for solv-ing the most common types of problems.

The experiments that will be described hereare (a) 2D homonuclear shift correlation throughscalar coupling (COSY), (b) 2D cross relaxationspectroscopy based on the nuclear Overhausereffect (NOESY) and (c) heteronuclear chemicalshift correlation of 1H and x 5 N chemical shifts

*Part of this manuscript has been published inthe proceedings of the NATO Advanced StudyInstitute on NMR in Living Systems, Sicily, Italy,1984.

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through multiple quantum coherence.Before addressing those experiments, an

introduction into the principles and generalaspects of two-dimensional NMR will be pre-sented. The original 2D experiment, proposed byJeener (1), will be discussed for the simple caseof non-coupled nuclear spins, for example, theprotons in chloroform. This will illustrate how a2D Fourier transformation is accomplished andalso which type of data presentation (absorptive/ dispersive / absolute value) is generally prefer-rable. Jeener's original sequence is depicted inFigure 1. At the end of the preparation period,which is sufficiently long to establish a (close to)thermal equilibrium situation, a 90° pulseapplied along the -y axis of the rotating frame(90° _y pulse) rotates the magnetization to aposition parallel to the x axis (Figure 2a). Thesystem then evolves (usually in an unperturbedfashion) for a time t , , which is generally referredto as the evolution period. During this period themagnetization rotates through an angle Q, t , ,where £2, is the angular rotation frequency of themagnetization considered (Figure 2b). At the end

Vol. 7, No. 4 167

EVOLUTION | MIXING ' DETECTION

Figure 1. (a) General subdivision of the timeaxis of a two-dimensional NMR experiment, (b)Pulse scheme of the Jeener experiment.

a) e)

Figure 2. Evolution of the magnetization of a setof isolated spins during the pulse scheme of Fig-ure lb. (a) The 90° „ pulse rotates the z magne-tization to the x axis, (b) During the evolutionperiod, t , , the magnetization rotates through anangle a = 12, t , ; (c) The second 90° pulse rotatesthe transverse magnetization into the xz plane,leaving a transverse component proportional tocos(a) parallel to the x axis.

of the evolution period, a 90x pulse rotates themagnetization into the xz plane (Figure 2c). Atthis point in time a transverse component pro-portional to cos<fl, t , ) is left parallel to the x axisof the rotating frame. During the detectionperiod, t2 , a signal sCt^tj) will be detected thatis proportional in amplitude to c o s ^ t , ) . Ifquadrature detection is employed during thedetection period, the signal is described by:

= Mocos(fi1t1)X

12 )exp(-tt /T2 (»> )exp(-t2 /T2 (1)

where £2, and S22 denote the angular frequenciesof the magnetization during the times t, and t2,respectively. T2

( * } and T2 ( 2 ' are the decay

constants of the magnetization during the timestt and t2. Of course, in this simple experiment,ft, =Q2 and T2

( * } =T2 ( 2 } . However in many

other experiments this is not necessarily thecase, and therefore different labels are used.

Fourier transformation with respect to t2 ofthe FID's acquired for the various t, valuesgives a set of spectra with amplitude propor-tional to ^

(2)

s(t, ,«2) = Mocoslp, t, )[A2 («

£D2(«2)]exp(-t,/T2(l))

where A2(o>2) and D2(«2) are the absorptive anddispersive parts of the resonance, given by

= T2 (* >/{ 1 + [T2

2 v 2 ;

} (3a)

(3b)

The absorptive parts of a set of those spectra,obtained for different values of t , , are sketchedin Figure 3a. Cross sections parallel to the t,axis through the data matrix of Figure 3a showa cosine modulation with angular frequency 12,,if taken near o>2 =S22, and consist of near zerointensity for other o>2 values. Usually the datamatrix of Figure 3a is transposed in the com-puter memory, in order to facilitate a Fouriertransformation with respect to t, (Figure 3b).Fourier transformation of those t, sections, gen-erally referred to as interferograms (2), gives thefinal 2D spectrum, shown in Figure 3c. Usuallysuch a spectrum is presented as a contour plot(Figure 3d), rather than a stacked line represen-tation, since this simplifies measuring coordi-nates of the resonance and avoids the possibilityof low intensity resonances to be hidden behindintense resonances.

A. Sampling Frequency and Sensitivity

In conventional one-dimensional Fouriertransform NMR experiments, the spectral widthafter Fourier transformation equals l/(2At), or±l/(2At) in the case of complex Fourier trans-formation, where At is the time between sam-pling points. Analogously, if consecutive t,

168 Bulletin of Magnetic Resonance

values are separated by an amount At1, this willgive a spectral width equal to l/(2At1) afterFourier transformation with respect to

In a regular one-dimensional Fourier trans-form experiment, the signal-to-noise ratio for asingle time-domain data point can be very poor;

Figure 3. (a) A set of spectra obtained with the sequence of Figure lb for increasing durations of theevolution period, t , . (b) Transposed data matrix of Figure 3a. (c) Fourier transform of the interfero-grams of Figure 3b resulting in a two-dimensional stacked line spectrum, (d) The spectrum of Figure3c, represented as a contour plot.

Figure 4. (a) A set of phase-modulated spectra. The phase of the resonance is a linear function of theduration of the evolution period, t , . Both the real and imaginary halves of the spectra are shown, (b)Complex interferogram taken at co2 = &2. The first half of the interferogram represents a t, sectionthrough the real halves of the spectra in Figure 4a, and the second half is the interferogram through theimaginary halves. A quadrature Fourier transform of this complex interferogram determines the signand the magnitude of the modulation frequency.

Vol. 7, No. 4 169

however, the Fourier transformation takes thesignal energy of all data points and puts it allinto one (or several) narrow resonance line(s).Similarly, if for each t, value in a 2D experi-ment a spectrum with poor signal-to-noise ratiois obtained, the second Fourier transformationwith respect to tt combines the signal energy ofa particular resonance from all spectra obtainedfor different tt values, and concentrates it intoone narrow line in the 2D spectrum. Therefore,the sensitivity of 2D NMR is not necessarilylower than for the one-dimensional experiment.

In certain types of experiments one tries totransfer magnetization from one resonance toanother by means of a coherent process or bymeans of cross relaxation or chemical exchange(to be discussed later). In this case the sensitivityof 2D NMR can become rather poor if the trans-fer process is not very efficient (as is the case incoherent transfer through non-resolved couplingsor in the case of slow cross relaxation or chemi-cal exchange). However, the analogous one-di-mensional experiment also suffers in sensitivityin this case.

If sensitivity is a crucial problem, the tworules that should always be obeyed are:

1. The acquisition time in the t2 dimension,t 2 m a x , should be at least equal to 1.5 T2.2. The acquisition time in the tt dimension,t, m a x , should be chosen as short as possi-ble, i.e. not longer than absolutely neces-sary to obtain enough resolution in the «,dimension after Fourier transformation.Since the signal decays as a function of thetime t , , spectra taken for short t,- valuescontribute more to the 2D resonance inten-sity (and sensitivity) than spectra obtainedfor t, values on the order of T 2 ' ' ' orlonger.

More discussions on sensitivity of 2D NMRcan be found in the literature (3-6).

B. Lineshapes and Transform Techniques

The spectrum in Figure 3c represents thecosine Fourier transform with respect to t, ofthe real part of eqn. 2. If interferograms throughthe real and imaginary halves of the S(t, p2) areFourier transformed separately, this gives:

(4)

where " j " has the same meaning as "i", butrefers to the imaginary data in the F,

170

dimension. A, («,) and D , ^ ) are defined analo-gous to eqn. 3. The four terms at the right handside of eqn. 4 are also known as Scc, Sc s , Ss c

and S s s in the older literature (2,7). Clearly, onlyone of the four parts (Scc), shows the desirable2D absorption mode lineshape.

The signal of eqn. 2 is modulated in ampli-tude by a cosine function. It is impossible to tellfrom the second Fourier transformation whetherthe modulation frequency, ft, , is positive or neg-ative. Of course, for this simple case one knowsthat ft, =ft2, and if quadrature detection duringt2 is employed, the sign of the modulation fre-quency is determined. However, in many experi-ments this is not the case and in order to avoidconfusion, one has to make sure that all modula-tion frequencies have the same sign by position-ing the transmitter frequency at either the lowor high field side of the spectrum. Because ofsensitivity considerations, quadrature detectionin the t2 dimension is always necessary, anddata storage space is wasted if the transmitterfrequency is not set at the center of the spec-trum. Also, radiofrequency offset effects maybecome prohibitively severe in certain types ofexperiments.

A simple way around this problem is theintroduction of artificial phase modulation, to bedescribed below. If a second experiment,"90°.y—t,— 90° y—acquire(t2)" is performed,the signal detected in this second experiment willstart out along the y axis (at t2 = 0) and is mod-ulated in amplitude by sin(ft, t , ) :

s'(tt ,t2) = Mosin(fi11, )exp[i<n2t2 +7r/2)J><

exp(-t1/T2<l))exp(-t2/T2<2))

= Mosin<ft, t, )exp(ifi212 )X

exp(-t , /T2( l ) )exp(-t2 /T2

( l ) ) (5)

Adding the results of the two experiments, eqns.5 and 1, directly together and omitting therelaxation terms gives:

s*(t, ,t2) = 1, )exp(iO2t2) (6)

Eqn. 6 represents a signal of which the phaseat time t2 = 0 is a linear function of the durationof evolution period, t , . Fourier transformationwith respect to t2 will yield a resonance at

P n a s e ^i ^i:

S+(t,,a>2) =

, t1)A2(o)2)—sinfft, t, )D2(CD2)] +

Bulletin of Magnetic Resonance

l , t, )D2 1, )A2 (7)

A set of spectra obtained this way for a series oft, values, is sketched in Figure 4. Both the realand imaginary halves of those spectra areshown. An interferogram taken at « 2 =Q2 doesnot contain any of the dispersive componentssince D2(fl2)=0, and this interferogram isdescribed by:

where the real part represents the interferogramtaken through the real part of eqn. 6, and theimaginary part of eqn. 8 is the interferogramtaken through the imaginary parts of thes* (t, ,ft2) spectra. This complex interferogram issketched in Figure 4b. A complex Fourier trans-formation of eqn. 8 can be made, and the sign ofthe modulation freguency, fl1, is automaticallydetermined. Any interferogram taken at an o>2value different from £22, will have non-zero con-tributions from D2(w2), and Fourier transforma-tion with respect to t, of this D2 («2) contributionwill yield a resonance in the F , dimension that is90° out of phase relative to the interferogramtaken a t « 2 =ft2. The full 2D Fourier transformof eqn. 6 is given by:

jM0[A, («, )D2 >, )A2 (9)

Clearly, both the real and imaginary part of thisfunction are mixtures of absorption and disper-sion. The real part of eqn. 9 is sketched in Fig-ure 5. The resonance shows a so-called "phasetwist" lineshape (2), and cannot be phased tothe pure absorption mode. In practice, an abso-lute value mode calculation is usually madebefore display:

Absolute value = [Re2 +Im2f5 (10)

Since the tails of an absolute value mode reso-nance decrease proportional to l/ju>— Q|, thisresonance shows undesirable tails, decreasingresolution.

Nevertheless, this artificially induced phasemodulation is currently the most common way ofoperation for most 2D experiments. The softwareof most commercial spectrometers assume datain a 2D experiment to be phase-modulated, andthe data matrix transposition routine

Figure 5. phase-twisted line shape obtained byFourier transformation with respect to t, of thematrix of Figure 4a..

automatically places the imaginary half of theinterferogram behind the real half (as depicted inFigure 4b), ready for a second complex Fouriertransformation, this time with respect to t t .

The strong tailing of the absolute value modelineshape can be suppressed by the use of appro-priate digital filtering. All those filters have incommon that they reshape the envelope ampli-tude of the time domain data to decay in a nearsymmetrical fashion from the midpoint of theFID (8). Filters commonly used for this purposeare the sine bell function (9), the "pseudo-echowindow" (10) and the convolution difference filter(11). Figure 6 compares the contour plots for aregular absolute value mode line-shape, and onethat has been obtained after the time domaindata were multiplied by a sine bell in bothdimensions. Unfortunately, the sensitivity gen-erally suffers severely from the use of such reso-lution enhancement filtering functions (the COSYexperiment, to be discussed later, can under cer-tain conditions be an exception to this rule).

Although phase modulated experiments areexperimentally very convenient, sensitivity andresolution suffer. Another disadvantage of artifi-cial phase modulation, not discussed above normentioned explicitly in the literature, is that onlyhalf of the available signal is used: if the

Vol. 7, No. 4 171

s" (t, ,t2) = Moexp(-iJ2, t, )exp(fK212) (11)

The noise in eqns. 6 and 11 is in principle inde-pendent. If only the sum or the difference istaken, half of the available signal is wasted!

C. Two-dimensional Quadrature andAbsorption Mode

As explained above, absorption mode 2Dspectra can only be recorded if the transmitter isplaced at either the low or high field side of thespectrum. Since data acquisition during t2 has tobe done in quadrature (for sensitivity reasons),this type of experiment is very inefficient as faras data storage is concerned. Also, pulse imper-fections due to resonance offset can becomesevere. A more efficient way to obtain a 2Dabsorption mode spectrum in 2D quadrature,was first introduced by Muller and Ernst (12)and by Freeman et al.(13) and also by States etal.(14). This so-called hypercomplex Fouriertransformation approach will be described below.

Consider again the pulse sequence of Figure1 and the signal of eqn. 1. A second experiment,90°_x—tt— 90°x—acquire(t2), is performed. Thesignal in the second experiment is initially(t = 0) also along the x axis, but is modulated inamplitude by s i n ^ t , ) . The FID's obtained inthe two experiments, s(t,,t2) and s'(t,,t2), arenot co-added this time, but stored in separatelocations in memory. Hence, for every t, value,two spectra are recorded, one modulated by acosine and the other one by a sine function.Fourier transformation of the two sets of spectragives:

Figure 6. Comparison of the contour plots ofabsolute value mode line shapes, (a) Line shapeobtained after an absolute value mode calculationof a signal that has been multiplied by a negativeexponential with a time constant T2 = AT/3,where AT is the duration of the acquisition time,(b) Line shape obtained if the time domain signalis multiplied by a sine bell function. The lowestcontour level is taken at 1/12 th of the peakheight.

difference of the two experiments (eqn. 1 andeqn. 5) were taken, this gives a signal of theshape:

S(t ,^2) = Mocos(fi1t1)[A2(a.2)+£D2(a.2)] (12a)

S'(t,jw2) = Mosin(fi1t1)[A2(«2)+iD2(a)2)] (12b)

The simple but crucial trick is to replace theimaginary part of S(t^^o2) by the real part ofS'(t,^2), yielding:

tfQ, t, )A2 («2 (13)

A set of those spectra is shown in Figure 7.Complex Fourier transformation with respect tot1 of eqn. 13 gives for the real part:

S t ( t 1^2) = M0A1(«1)A2^2) (14)

which represents a 2D absorption mode reso-nance. Undoubtedly this will become the acceptedway of data processing for most 2D NMR

172 Bulletin of Magnetic Resonance

Figure 7. (a) The left halves of the spectra arethe absorption parts of the spectra obtained withan experiment where the resonance is modulatedin amplitude by a cosine function. The dispersiveparts of these spectra have been replaced by theabsorptive part of the spectra obtained with asecond experiment, where the resonances aremodulated in amplitude by a sinusoidal function,(b) Complex interferogram taken a t « 2 =fi2.

experiments in the near future.A slightly different approach for the same

problem of two-dimensional quadrature andabsorption mode has recently been proposed byBodenhausen et al. (15) and is based on an ideafirst introduced by the Pines group (16). In thisso-called TPPI method (Time Proportional PhaseIncrement), the rf phases of all preparationpulses are incremented by 90 for consecutive t,values. This causes the apparent modulation fre-quency to be increased (or decreased) by(2At1)"1, and makes it appear that all modula-tion frequencies are positive (or negative), andtherefore allows a real (cosine) Fourier transfor-mation with respect to t , . Although, at firstsight, the hypercomplex and the TPPI methodsseem totally different, a closer analysis showsthat the methods are almost identical. Also from

a practical point of view there is little difference:identical size data matrices are required andidentical data acquisition times are needed forthe two methods, giving indistinguishable resolu-tion and sensitivity in the final spectrum. Com-paring the two methods is similar to comparingthe different ways in which most commercialspectrometers versus most of the Bruker spec-trometers acquire quadrature information in aone-dimensional spectrum. In most spectrome-ters two data points, containing the x and yinformation, are sampled simultaneously,whereas in most Bruker spectrometers quadra-ture information is obtained by sampling at twicethe rate, and incrementing the receiver referencephase by 90° for consecutive sampling points,taking only one data point at a time (17).

Lines in an absorption mode spectrum can beso narrow that digitization of the resonancesbecomes a problem. In this case, the highestpoint of a resonance in a 2D spectrum can bereduced by as much as 70%, apparentlydecreasing the intensity of a resonance in a con-tour plot dramatically. This effect has beenobserved experimentally. A simple solution tothis problem is to zero fill by a number of datapoints equal to the number of signal-containingdata points, prior to Fourier transformation. Inabsolute value mode spectra, lines are broadenedsufficiently by the absolute value mode calcula-tion and use of this zero-filling procedure istherefore usually not essential.

D. Coherence Transfer by Means ofRadiofrequency Pulses

Classical vector pictures have been used forover three decades to explain the mechanisms onwhich many NMR experiments rely (18). How-ever, for a description of most of the newer typepulse sequences, applied to coupled spin systems,straightforward application of those vector pic-tures leads to erroneous results. Ernst and co-workers (19) and other workers (20,21) haveintroduced a formalism that does allow the use ofvector pictures to analyze the behavior of acoupled spin system. This operator formalismbased vector picture is more complicated than isthe use of the classical picture based on theBloch equations, but is much less tedious thanuse of the density matrix formalism. The newapproach will be briefly discussed for a homonu-clear weakly coupled two-spin system. Twoweakly coupled spins, A and B, will be consid-ered.

The total net magnetization of spin A equalsthe vector sum of the magnetizations of the two

Vol. 7, No. 4 173

doublet components (Figure 8). As in the classi-cal vector picture, the vector sum of the two

Figure 8. (a) Arbitrary orientation of the twomagnetization vectors of the doublet componentsof nuclei, A. (b) The vector sum of the twodoublet components, (c) The difference betweenthe two doublet components, representing anti-phase magnetization.

doublet components is represented by a vector I Athat can be decomposed in the standard fashioninto its components along the principal axes ofthe rotating frame (Figure 8b):

IA =(15)

This vector sum is the sum of the in-phase com-ponents of the two doublet magnetization vec-tors. The antiphase components of the two doub-let magnetization vectors do not contribute to I Ain this picture. Since those components are oppo-site to each other, they do not contribute to mac-roscopic observable magnetization. Of the anti-phase components of spin A, one componentcorresponds to spin B in the m=l /2 spin state,and the other to the m=-l/2 state. This anti-phase magnetization can then formally be writ-ten as 2IAIB z . The factor "2" in this product isneeded for normalization. The new formalismtreats the individual terms in such a product asnormal magnetization, in the classical way.

In order to make the description moreexplicit: consider the effect of a 90° x pulseapplied to the two-spin system that is initially inthermal equilibrium. The magnetization ( 1 ^ +IBz) gets rotated to a position parallel to the -yaxis*, and thus creates-(IAy+IBy). The magne-tization of the two nuclei can be considered sepa-rately, and we will concentrate on the fate of the

A spin magnetization. This will rotate withangular frequency, flA> about the z axis and thetwo doublet components will periodically (withperiod 1/J) get in antiphase. These two processescan be considered separately:

- I A x s in%t )

XAy

i A x

(16a)

IAycos(irJt) — 2IAxIBzsin(7r Jt)(16b)

IAxcos(n-Jt) + 2IAyIBzsin(7rJt)(16c)

Substitution of eqns. 16b and 16c in 16a thengives the complete evolution of the magnetiza-tion. The interesting terms in eqn. 16 are theproducts IAxlBz'an(* *Ay*Bz- Consider a 90°pulse applied to such a term:

902

90°X

(17a)

(17b)

eqn. 17a shows how antiphase A spin magneti-zation gets converted into antiphase B spinmagnetization by a 90° pulse applied perpendic-ular to the doublet magnetization vectors. Thisimportant result is the basis of Jeener's originalexperiment when applied to a homonuclearcoupled spin system. Eqn. 17b contains theproduct IAxlBy and is harder to visualize; thisterm represents so-called two-spin coherence, acombination of zero- and double quantum coher-ence. The time evolution of the product equalsthe product of the time evolution of the individualterms; i.e. the time evolution of I A x I B y is givenby

[cos(HAt)IAx + sin<8At)IAy] X

- sin(PBt)IBx] (18)

The operator formalism is easily extended tomore spins by considering product terms of allspins involved. For a more complete descriptionof this extremely useful formalism, the reader is

*In order to be consistent with conventions usedin the paper by Sorensen et al.(19), describingthe operator formalism, we adopt here the con-vention that a 90° x rotates the magnetizationanti-clockwise around the x axis, i.e. from the zto the -y axis.

174 Bulletin of Magnetic Resonance

referred to the literature (19-21).

II. HOMONUCLEAR SHIFT CORRELATIONTHROUGH SCALAR COUPLING

Jeener's original 2D pulse scheme can beconsidered as a convenient alternative for theconventional one-dimensional double resonanceexperiments, and is often referred to as theCOSY (Correlated SpectroscopY) experiment. Acomplete quantum mechanical description of theexperiment was presented by Aue et al.(22) butdid not contribute much to the popularity of theexperiment. Only in the early 1980's the wide-spread applicability of this technique was real-ized (23-27). The basic pulse scheme has alreadybeen discussed in the previous sections. Here a90°x—t, — 90°^—acquire(t2) pulse scheme isused, where the phase <t> of the final pulse andthe mode of data acquisition are selected as indi-cated in Table 1. This means that a minimum offour experiments is performed for each t1 valuewith the phase of the final 90° pulse incre-

Table 1. Cycling of the phase of theobserve pulse, <£, and of the receiverin the various steps of the COSYexperiment in order to obtain phasemodulation and to detect the coherencetransfer echo.

Step Receiver

123

xy

-x-y

mented by 90° each time and data alternatelyadded to and subtracted from memory. This wayof phase cycling results in detection of thes" (t, ,t2) signal of eqn. 11, often referred to asthe coherence transfer echo (28). The occurrenceof the echo is easily understood by consideringthat in eqn. 11, Q^ and ft2 have approximatelythe same value (in the laboratory frame). There-fore, at time t, = t 2 , the phase of the magnetiza-i [ P i ^

independent of the magnetic field strength, i.e.independent of magnetic field inhomogeneity, andconsequently an echo will occur. The entirefour-step experiment is often repeated four timeswith the phases of all radiofrequency pulses andreceiver incremented by 90° each time. Thisadditional so-called CYCLOPS cycling (29) elimi-nates imperfections in the quadrature detectionsystem of the spectrometer which otherwise maycause small mirror image signals about the«2 =0 ando>1 =0 axes.

The COSY experiment relies on transfer ofmagnetization from spin A to spin X by the sec-ond 90° pulse in Jeener's experiment, and canonly occur if spin A and X are mutually coupled.The mechanism for this magnetization transferhas been discussed in the previous section: eqn.17a shows how spin A doublet components thatare in antiphase with respect to spin X aretransferred into X spin doublet components thatare in antiphase with respect to spin A. Theamount of antiphase A-spin magnetization pres-ent before the second 90° transfer pulse dependson sinOrJ^-jjt, )• The magnetization observedduring t2 starts out in antiphase and is propor-tional to sinfr J^x^2 )• ^ n e t m i e d ° m a m signal forthe magnetization transferred from A to X isgiven by

Mosi

exp(-i ftAt, )exp(i (19)

, 2tion, exp[ i(P2t2—

is to first order

Note that for very short t, values only very littlemagnetization is transferred from A to X.Rewriting:

=— i[exp(i7rJt)—exp(-iirJt)]/2 (20)

and substitution in eqn. 19 shows that theFourier transformed signal S"(w1jw2) will showfour peaks at (w1 ^ ^ " ^ A ^ ^ A X 1 ^X^^AX^ 'and that those resonances show antiphase rela-tionships, as schematically indicated in Figure 9.Also indicated in Figure 9 are the "diagonalresonances," due to magnetization that is nottransferred from A to X or from X to A, andwhich are 90° out of phase relative to the crosspeaks. It is therefore impossible to phase allresonances in this spectrum simultaneously tothe 2D absorption mode (unless the pulse schemeis altered (30)), and an absolute value mode cal-culation before display is therefore commonlyused. The artificial phase modulation scheme

Vol. 7, No. 4 175

1

t

t

©; © 4-©;© - ^

1 /\/+\/ ©

/

©.©

_ 6.

Figure 9. Schematic diagram of the 2D COSYspectra of an AX spin system. The four compo-nents of the AX cross multiplet are pairwise inantiphase, whereas the four diagonal multipletcomponents are 90° out of phase in both dimen-sions relative to the cross peaks.

•I

6 S 1 3 2 1 PPM

(discussed before) therefore does not degradeperformance of the experiment very much. Also,since the cross peak time domain signal has anenvelope amplitude proportional tosin(7rJAX^I )sin(7rJ^^t2), and acquisition timesin the t, and t2 dimension are typically 100-300ms, use of a sine bell digital filter is close to amatched filter for those signals and favors thesensitivity of the cross multiplets, meanwhilecutting down the intensity of redundant diagonalpeaks which have a COS(TTJ^X^ )cos('r^AX^)dependence.

Figure 10 shows the COSY spectrum for a25 mM solution of amphotericin B in DMSO-d6,recorded on a Nicolet 500 MHz spectrometer.512 t, increments of 300 Msec each were used,and 512 complex data points were acquired foreach t, value. The artificial phase modulationscheme (Table 1) was employed and the totalmeasuring time was 4 h. Note that in the regu-lar one-dimensional spectrum many of thecouplings are not or poorly resolved. However, awealth of cross peaks can be observed in the 2D

Figure 10. 500 MHz absolute value mode COSYspectrum of a sample of amphotericin B inDMSO-de. 16 experiments were performed foreach t, value, and the total duration of theexperiment was 4 h. A sine bell has been usedin both dimensions prior to Fourier transforma-tion.

spectrum. For example, it is seen that proton 1is coupled to methyl group 38 (broken line inFigure 10).

Many users of the COSY experiment areoften puzzled by the absence or low intensity ofcertain cross peaks. For example, the crosspeaks between protons 1 and 2 in Figure 10have too low an intensity to be observed in thecontour plot. In general, it is hard to calculateexactly how intense a cross multiplet will be, buttwo factors that are of major importance will bementioned below.

1. A proton that is coupled to a large num-ber of other protons will show rather low

176 Bulletin of Magnetic Resonance

intensity for its individual ID multipletcomponents. In the COSY experiment, thislow intensity will now be redistributedamong all protons to which it is coupled bythe mixing pulse, yielding very low inten-sity for the cross multiplets. The effect isparticularly severe for cross peaks betweentwo multiplets that both have a complicatedmultiplet structure.2. If transverse relaxation times, T2, areshort compared with JAX~ *» t n e transfer-red magnetization in eqn. 19 will neverassume a large value and therefore only(vanishingly) low intensity cross peaks canbe observed.

From point 1, it is clear that a large couplingdoes not necessarily give rise to an intense crosspeak. However, if acquisition times t i m a x and2̂ max a r e c n o s e n short (<100 ms) this will rela-

tively emphasize cross peaks due to large scalarcouplings. Recently, it has been shown that if thesize of certain couplings can be estimated, a"designer" filter can be used to pick up thosemissing cross peaks (31).

III. HOMONUCLEAR CHEMICAL SHIFTCORRELATION THROUGH CROSS

RELAXATION

In the COSY experiment, magnetization istransferred from one proton to another throughthe scalar coupling mechanism. This experimentrelies on the existence of (partially) resolved sca-lar couplings. Another way to transfer magneti-zation between nuclei is the cross relaxationmechanism, which relies on the internucleardipolar interaction. This mechanism is generallyreferred to as the nuclear Overhauser effect(NOE). The 2D pulse scheme that is based onthe NOE effect is generally known as theNOESY experiment. This experiment was firstproposed by Ernst, Jeener et al.(32,33). It ismentioned here that the same experiment canalso be used for the investigation of chemicalexchange processes. Obviously, no resolvedcouplings are needed for this experiment; theexperiment tends to work very well for macro-molecules which have a slow tumbling rate andtherefore less ideal averaging of the dipolarcoupling mechanism. This is the reason forstrong cross relaxation and causes line broaden-ing in the conventional ID spectrum and also inthe 2D spectrum.

The pulse scheme is sketched in Figure 11and the mechanism on which this 2D experimentrelies will be briefly discussed below. Assumefor reasons of simplicity that all pulses in the

scheme are applied along the x axis of the rotat-

Figure 11. Pulse scheme of the 2D experiment todetect homonuclear cross relaxation and chemicalexchange. The phases of the rf pulses , <£, and4>2, and of the receiver are cycled according toTable 2, and the results of the odd and evensteps are stored in separate locations in memoryand processed as described in the text in order toobtain an absorption mode spectrum.

ing frame, and consider a molecule with twospins A and B that have no mutual scalarcoupling. The longitudinal magnetization of spinA, just after the second 90° pulse, is given by:

During the mixing period of duration, A, crossrelaxation with spin B takes place, changing thelongitudinal B spin magnetization by an amountCtM^ft,)—Mzg(t,)] and the A spin magnetiza-tion by the opposite amount, where C is a con-stant depending on the cross relaxation rate, thelongitudinal relaxation rates, and the duration, A.Just before the final pulse, the longitudinal Bspin magnetization is thus given by

Mz B( t i ) = fIMzB(t,)] + CM^ft,) (22)

where fTM^gft.,)] is a function depending on therelaxation of proton B during the delay, A, andon the modulation of the B spin magnetization bythe first two pulses in the sequence. It is thesecond term at the right hand side of eqn. 22that is the term of interest, since this term is dueto cross relaxation from nucleus A to 8. Thefinal 90° pulse converts this term into tran-sverse B magnetization which follows from eqns.21 and 22:

= CMoAcos^At,)exp(£flBt2)(23)

Vol. 7, No. 4 177

Table 2. Phases #1 and #3 of the pulsesin Fig. 11 in the various steps of theexperiment. Phase #2 is x for a l l 16steps. Data of odd and even steps arestored separately in the computermemory.

Step

1

3

5

7

911

13

15

2

k

6

8

10

12

U

16

•.

X

y-x- y

X

y-x- y

X

y-x-y

X

y-x-y

<f>3

XXXX

yyyy

-x-x-x-x-y-y-y-y

Receiver

XX

-x-x

yy

-y-y-x-x

XX

- y- y

yy

This signal is of the same shape as eqn. 1, and2D Fourier transformation will therefore give aresonance at (w, (W2) = (fl^,flg). In contrastwith the COSY experiment, diagonal and crosspeaks in the 2D spectrum will all have the samephase, or exactly opposite phase, depending onthe motional correlation time, r c . It is thereforestrongly desirable to record the spectrum in theabsorption mode, using the procedure outlined inSection IC. In order to eliminate transferthrough scalar coupling (the COSY mechanism)further phase cycling is necessary. In practice a16-step sequence (Table 2) is often used andadditionally, this 16-step sequence is repeatedfour times in the CYCLOPS mode (29) in orderto eliminate quadrature artifacts. Remainingcoherent transfer through zero and triple quan-tum coherence, that is not cycled out this waycan largely be eliminated by random fluctuationof A by a small amount (5%), provided that thefrequency of this zero or triple quantum coher-ence is larger than 20/A Hz.

If the cross relaxation is slow compared to

the longitudinal relaxation, maximum transferfrom spin A to B, and vice versa, occurs for amixing time on the order of the shortest T1 ofthe two spins involved. Hence, from a sensitivitypoint of view, a mixing time on the order of T, isoptimum for the detection of cross peaks. How-ever, if one wants to obtain quantitative infor-mation about the relaxation rates, one has toconsider that the cross peak buildup rate (as afunction of A) is non-linear (32-36). The simplestway around this problem is to consider onlyshort mixing times, for which the NOE buildup isstill in the linear region (35,36), but thisapproach necessarily leads to smaller crosspeaks, i.e. lower sensitivity.

As an example, Figure 12 shows a contourplot of the 2D NOE spectrum of amphotericin B,showing a large number of cross peaks, obtainedfor a mixing time of 200 ms. Analysis of thecross peak network does not only provide assign-ment for all *H resonances in this compound,additional to that obtained from the COSY spec-trum of Figure 10, but also gives informationregarding the three-dimensional structure insolution.

IV. INDIRECT DETECTION OF 1SNTHROUGH MULTIPLE QUANTUM

COHERENCE

Detection of * 5 N shifts is limited by its lowNMR sensitivity, which is due to the small andnegative value of the magnetogyric ratio and tothe low natural abundance (0.37%). The fact thatthe magnetogyric ratio, y, is negative can causesignal cancellation in the case of incompletenuclear Overhauser enhancement. NMR sensi-tivity is approximately proportional to y 5 ' 2 (37)and is therefore about a factor of 300 lower for1 5 N than for protons, which implies a factor of100000 difference in measuring time. It has beendemonstrated that the INEPT experiment(38,39) can alleviate this problem to a certainextent, but nevertheless detection remains diffi-cult. Bodenhausen and Ruben (40) demonstratedthat the 1 SN frequency could be measured indi-rectly via the protons directly coupled to the *s Nnucleus. Their rather complicated sequencetransfers proton magnetization to the l 5 N,which then evolves during the evolution period ofthe 2D experiment, before being transferred backto the protons which are detected during theacquisition time in this experiment. A much sim-pler alternative, that relies on the same principleas this "Overbodenhausen experiment" has beenproposed rather recently (41,42). Experimentalresults indicate that with this new experiment

178 Bulletin of Magnetic Resonance

1 PPM

Figure 12. Two-dimensional NOE spectrum ofamphotericin B, recorded at 500 MHz, using amixing time of 200 ms. The top trace displaysthe cross-section taken through the 2D spectrumat the chemical shift of HI, showing spacialproximity to Me39, Me40, H2, H4, and H5. Thecross-peak with H3 (3.12 ppm) is much lower inintensity suggesting a larger distance. The mix-ing time (200 ms) is not short enough to excluderelayed NOE effects, and quantitative interpre-tation of cross peak intensities is not feasiblefrom this single spectrum.

the theoretical enhancement factor of 300 really

Vol. 7, No. 4

can be obtained (43). The experiment works onlyfor protonated 1 SN nuclei. Consider the energylevel diagram of an isolated 1H-15 N pair (Figure13). The broken lines represent the two (weak)

1

a H q N A

Figure 13. Energy level diagram and wave func-tions for a 1SN-XH spin pair. The insensitive1 s N resonances (broken lines) are measuredindirectly by measuring the zero and doublequantum resonances (dotted lines) via the protonresonances (solid lines).

1 5 N transitions. The solid lines represent the(intense) proton transitions. The basic concept ofthe new experiment is to generate the dottedtransitions (zero and double quantum), thatresonate with frequencies ftjj±ftjvj. If those fre-quencies are measured indirectly, with a 2Dexperiment, the *5 N frequency can be calculatedsince the proton frequency, fljj, is known.

The pulse scheme of the experiment is shownin Figure 14. The x H 180° pulse at the center ofthe evolution period and the *5 N decoupling dur-ing data acquisition are optional and will, atfirst, not be taken into account. The theory of the

179

(a) 60*

MIL

(b) 90*

E15N

9o;

SLA t, i2

n

fffA I, A

Table 3. Phase of the first 1 SN pulsein the pulse scheme of Fig. ]h and themode of data collection for detectionof the double (DQ) and of the zeroquantum (ZQ) component.

Step DQ

123it

X

y-x-y

X

-y-x

y

X

y-x-y

Figure 14. Pulse scheme of the experiment forindirect detection of * 5 N through multiple quan-tum coherence. The first pulse applied to theprotons can be of the Redfield type or of the1-3-3-1 type and does not require additionalphase cycling, (b) The 180° *H pulse applied atthe center of the evolution period interchangeszero and double quantum coherence and causeselimination of the l H frequency contribution inthe F , dimension, leading to a regular hetero-nuclear shift correlation spectrum. The 1 s Ndecoupling during acquisition removes heteronu-clear coupling from the F2 dimension and, inprinciple, doubles the sensitivity of the experi-ment. In both (a) and (b), the phase of the first90° ! N pulse is cycled according to Table 3.

experiment is most easily described with theoperator formalism, treated in section ID. Theproton 90° pulse creates magnetization along the-y axis of the rotating frame, which then evolvesfor a time A:

(24)(IjjyCos^jjA)—IHxsin^jjA)} cos

2{— Ijjxcos(fljj^)—lHysm<^H^ s m ^

If the delay, A, is set to 1/(2JJ^JJ), only the sec-ond term at the right hand side of expression 24survives, and a 90° x *5 N pulse applied at thistime will generate product terms IjJx^Nv ant*Ijjyljvjy. The sine and cosine coefficients will be

omitted to simplify the expressions. During theevolution period, those product terms evolveaccording to:

, ) I N y —

, )IH y] X

, )INx] (25)

The final 90° *s N pulse converts those two-spincoherences back into observable XH magnetiza-tion. If all terms are taken into account, a 90° x1 s N pulse generates an observable signal:

s(t,,t2) =

(26a)

and if the first 90° *5 N pulse were applied alongthe -y axis, the second 90° J s N pulse generates

s(t,,t2) =

M )]s in%t, ) (26b)

If either of those two signals is used separatelyto calculate a 2D spectrum, two resonances in

ing the zero and double quantum frequencies,respectively. The results of eqns. 26a and 26bcan be combined in the usual fashion (eqns. 6and 11) to give either the double or the zeroquantum frequency in the 2D spectrum. Becausethe zero quantum resonance in the 2D spectrum

180 Bulletin of Magnetic Resonance

is centered at (w,,^) = ( ^ N + ^ H ^ H ^ * e c o m "puter can subtract a frequency &>2 from all <*>,coordinates, to yield a pure chemical shift corre-lation map, with resonances centered at (ftj^ftjj).Broad-band 15 N decoupling during the detectionperiod can be used to eliminate the effect of het-eronuclear coupling in this experiment.

If the first *s N 90° pulse is applied alongthe -x axis, the signal of eqn. 26a will bedetected with opposite sign. Since signals fromprotons not coupled to *5 N will not know aboutthis phase shift and their shape will beunchanged in the two experiments, subtraction ofthe two experiments will give cancellation of thesignals of protons that are not coupled to *5 N.Hence, the 1 5 N satellites in the 1 H spectrumcan therefore be detected, not hampered by the600 times stronger signal from protons coupledto * * N. In total four experiments will be per-formed for every t, value, with the phase of thefirstx 5 N pulse cycled along all four axes.

As an example, Figure 15 shows the zeroand double quantum spectra of a 30 mM solutionof a-thymosin, dissolved in 90% H2 O/10% D2 O.Spectra were recorded on a modified Nicolet 300MHz instrument, and the total measuring timewas approximately 12 h. The signals of eqns.26a and 26b were stored in separate locationsand spectrum (a) was computated from thedouble quantum signals and (b) was computatedfrom the zero quantum signals, using the sametwo sets of acquired data.

In practice, application of the sequence ofFigure 14b can sometimes be more difficult fornatural abundance samples than the sequence ofFigure 14a. This is due to the high power XSNdecoupling during data acquisition, which cancause a slight perturbation of the 2 H lock signal.A very stable lock signal is required to allowcomplete elimination of the 600 times strongersignal from protons coupled to * * N.

As pointed out in Section I, it is desirable torecord spectra in the absorption mode, both froma viewpoint of sensitivity as well as resolution.However, in this experiment the data are truelyphase modulated (not artificially induced). Con-version to amplitude modulation is possible byinsertion of a 180° pulse at the center of theevolution period (44). However, in the casewhere the proton also has homonuclear coupling,phase modulation cannot be avoided and pureabsorption mode spectra cannot be recorded. Anice feature of the simplest form of the experi-ment is that it utilizes only one pulse for theobserved protons. This pulse can be of arbitraryflip angle; a small flip angle allows a faster rep-etition rate and consequently will yield somewhat

Figure 15. Zero and double quantum naturalabundance x H-15 N spectra of a 30 mM solutionof a-thymosin in H2O. Spectra were recorded ona modified Nicolet 300 MHz instrument, andboth spectra are obtained from the same set ofacquired data. The total measuring time wasapproximately 12 h. The spectra were kindlyprovided by Dr. David Live and Dr. DonaldDavis.

higher sensitivity. More important is the factthat any type of pulse can be used for this protonexcitation. For example, a Redfield water sup-pression or a 1-3-3-1 water suppression pulse

Vol. 7, No. 4 181

can be used, making it feasible to use this exper-iment in H2 0 solution. Of course, the experimentis not limited to the indirect detection of 1 SNresonances. The technique has also great prom-ise for metabolic NMR studies in vivo, by using1 3 C labeled starting material.

V. DISCUSSION

Although only a small number out of thelarge collection of two-dimensional NMR experi-ments has been treated here, it will be clear thatthe 2D approach extends the capabilities of NMRexperiments enormously. Many applications of2D NMR in organic and biochemistry haveappeared in the recent literature. On modernNMR spectrometers, utilization of the varioustwo-dimensional experiments has become rathersimple by the introduction of suitable supportsoftware and spectrometer documentation. Themain problem the inexperienced spectroscopisthas to face is which experiment to choose andwhich results to expect. A comprehensive guideon this subject has not yet appeared, but in gen-eral it is usually a good idea to apply the stan-dard experiments before attempting to use amore sophisticated version of those experiments.It is expected that the number of experimentalpulse schemes still will expand significantly, butonce the spectroscopist realizes that all thosemethods are small variations on the same theme,the collection of 2D experiments will appear lessbewildering and practical application of thosetechniques becomes much simpler.

ACKNOWLEDGMENT

I wish to thank Ingrid Pufahl for typing andediting most of the manuscript. I am also indeb-ted to Dr. David Live and to Dr. Donald Davisfor providing the spectra of Figure 15 and to Dr.A. Aszalos for providing the sample of Ampho-tericin B and for interpretation of the corre-sponding spectra that resulted in completeassignment of the * H spectrum.

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