Two-Dimensional NMR

Two-Dimensional NMR Peter L Rinaldi, The University of Akron, Akron, OH, USA Masud Monwar, Dittmer Research Lab, Florida State University, Tallahassee, FL, USA ã 2017 Elsevier Ltd. All rights reserved.

LC-NMR MEM NOE NOESY PEP PFGs TPPI

Abbreviations COSY DNP FDM FID GES GFT

correlation spectroscopy dynamic nuclear polarization filter diagonalization method free induction decay gradient enhanced spectroscopy G-matrix transform

Synopsis of Material to be Covered Although NMR has been a valuable tool for scientists who must understand the structures, reactions, and dynamics of molecules, there have been two major advances in the past 30 years, which more than any other contributions, have kept this an exciting and rapidly evolving field. The first of these was the introduction of the Fourier transform NMR (FT-NMR) technique by Ernst and Anderson in 1966. This development helped to reduce problems associated with the biggest limitation of NMR – poor sensitivity. It also set the stage for a second important development. The dispersion of NMR signals and thus the complexity of molecules that can be studied is related to the magnetic field strength of the instrument. At a time when scientists were preparing ever more complicated structures, the incremental increases in the magnetic field strengths of commercially available instruments were growing smaller. However, the proposal by Jeener in 1971 and first demonstration by Muller et al. in 1975 of multidimensional NMR spectroscopy resulted in a quantum leap in the capabilities and the prospects for NMR. By dispersing the resonances in a second frequency dimension additional spectral dispersion could be achieved. The dispersion from a 2D experiment performed on a 1980 vintage 200 MHz spectrometer can match that obtained in the 1D spectrum from a modern 800 MHz spectrometer. In 2D spectroscopy, the spectral dispersion increases as the square of the magnetic field strength. Furthermore, 2D experiments can have the unique characteristic of providing structural information based on the correlation of the frequencies at which peaks occur. This article deals with the background and practical aspects of obtaining 2D NMR data. There are quite a few variations of 2D NMR experiments in which properties such as retention time (in liquid chromatography-NMR (LC-NMR)), distances (imaging), or diffusion coefficients (diffusion ordered spectroscopy) are the variables along one or more axes in the spectra. However, discussions in this article will be restricted to experiments in which two frequencies, related to NMR parameters, are plotted along the two axes of the spectra. Other forms of 2D NMR are discussed in other parts of this work. Although this article can be read alone, it is useful to refer to other articles to learn the details of various techniques (e.g., weighting, zero filling,

Encyclopedia of Spectroscopy and Spectrometry, Third Edition

liquid chromatography-NMR maximum entropy methods nuclear Overhauser enhancement nuclear Overhauser enhancement spectroscopy preservation of equivalent pathways pulsed field gradients time-proportional phase incrementation

sampling rates, complex vs real Fourier transforms, linear prediction, sparse sampling, and other methods of fast 2D NMR) that are discussed in this article as they pertain to 2D NMR.

Fourier Transform NMR Spectroscopy Figure 1(a) shows the time domain signal, called the free induction decay (FID), obtained by measuring the response of nuclear spins to a radio frequency (rf) pulse. The FID is the sum of many exponentially decaying cosine waves, one for each resolvable signal in the spectrum. In this time domain spectrum, a single frequency is observed; by measuring its period, the frequency can be determined. A typical FID will contain the sum of many oscillating signal components, making it impossible to identify individual frequency components by visual inspection of the time domain signal. By converting the time domain signal to a frequency domain signal using a mathematical process called Fourier transformation, a readily interpretable spectrum (Figure 1(b)) with peaks at discrete frequencies (one for each cosine wave in the original FID) can be obtained. Each point in the time domain spectrum contains information about every frequency in the frequency domain spectrum. In a typical 1D spectrum up to 100k points are collected in the time domain; thus, information about each peak is measured 100k times. The laws of signal averaging tell us that the signal (S) from n measurements increases linearly (n
S), but that the noise (N) from n measurements increases as n1/2 (n1/2
N).

(a)

(b) t1

f1

Figure 1 One-dimensional FT-NMR data: (a) time domain FID detected after an rf pulse and (b) frequency domain spectrum after Fourier transformation of the signal in (a).

http://dx.doi.org/10.1016/B978-0-12-803224-4.00087-X

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476

Two-Dimensional NMR

Therefore, the signal-to-noise ratio (S/N) in the final spectrum improves as n
S/(n1/2
N)¼n1/2
S/N as long as the signal is present throughout the FID. Consequently, the S/N in the final spectrum will be (105)1/2 ffi300-fold better than that obtained in a single scanned spectrum. This improvement is known as the Felgett advantage. It is described here because it has some important consequences when n-dimensional experiments are performed. In practice, S/N gains, in 1D NMR are lower than those predicted by the Felgett advantage, because the intensities of the signals decay exponentially during the signal acquisition period. However, in multidimensional NMR, short evolution and acquisition times have typically been used to minimize the size of the data sets and the time needed to collect the data. Consequently, very little signal decay occurs and S/N improvements are close to those expected from theory.

evolution (t1), mixing, and detection (t2) times. The filled rectangular boxes represent 90 pulses, which are applied at the 1H resonance frequency in this experiment. In general, these pulses can be at a variety of flip angles and can be applied at a variety of frequencies depending on the requirements of the experiment and the information desired. The preparation period is used to put the nuclear spins into the initial state required by the experiment being performed. In this particular sequence, the preparation period is a relaxation delay to allow the spins to return to their equilibrium Boltzmann distribution among the energy levels. In some sequences, the preparation period might also contain a coherence transfer step (e.g., by an INEPT-type polarization transfer pulse sequence) to move NMR signal components from one nucleus to another in preparation for the evolution period. The evolution period is used to encode frequency information in the indirectly detected (t1) dimension. The mixing period is used to transfer magnetization from one nucleus (whose chemical shift information is encoded during t1) to a second nucleus for detection during the acquisition period, t2. The NOESY sequence contains a delay during the evolution period to encode 1H chemical shifts; however, some pulse sequences contain 180 refocusing pulses to remove chemical shift modulation or coupling to a second nucleus (if the pulse is at the frequency of that second nucleus), or combinations of pulses to remove selected signals or coupling interactions. The key to the success of 2D NMR is the collection of a series of FIDs, while progressively incrementing the value of t1. At the end of data collection a set of 100–1000 FIDs is obtained (the number of FIDs collected depends on the desired resolution and spectral window in the t1 dimension) as shown in Figure 3(a). The intensities of these FIDs are modulated based on the length of the t1 period and the precession of the coherence during t1. If each of these FIDs is Fourier transformed (with respect to t2), a series of spectra is obtained as shown in Figure 3(b). Each spectrum contains signals that correspond to those found in the normal 1D spectrum of the detected nucleus. The intensity of a signal at a specific chemical shift

Fundamentals of 2D NMR General Sequence for Collection of 2D NMR Spectra Figure 2 contains a diagram of a 2D NMR pulse sequence called the NOESY (nuclear Overhauser enhancement spectroscopy) experiment. NMR spectroscopists have been very liberal in their methods for selecting acronyms to name their experiments. This pulse sequence contains the four basic elements that are common to 2D NMR experiments: preparation,

t1

Preparation

t2

Evolution

Mixing

Acquisition

Figure 2 NOESY 2D NMR pulse sequence.

t1

t1

(b) (a) FT(t2)

a t2

b f2 a

FT(t1) (d)

t1 (c)

f2 b f1

Figure 3 Schematic illustration of the process used to produce a 2D NMR spectrum.

t1

Two-Dimensional NMR

varies from one spectrum to the next. Its intensity is modulated by the NMR interaction (J-coupling, chemical shift, multiple quantum coherence, etc.) that is in effect during t1 and by the duration of the t1 period. If one were to plot the intensities of the two peaks in Figure 3(b) as a function of t1, the curves in Figure 3(c) would be obtained. The modulation frequencies of these two curves are different because the detected signals in t2 originate from different coherences that have different precession frequencies in t1. The intensity variations in these curves are reminiscent of the 1D FIDs. The obvious (in 1971 it was not so obvious until Jeener pointed this out) thing to do with these signals is to transpose the data matrix, and at each frequency in f2, Fourier transform the data with respect to t1. The result is a spectrum with signal intensity variations as a function of two frequencies as shown in Figure 3(d). The frequencies plotted along the f2 dimension correspond to those that are detected during t2 (i.e., 1 H chemical shift and J-coupling if the sequence in Figure 2 is used). The frequency plotted along the f1 dimension corresponds to the precession properties of the coherences that are selected by the pulse sequence.

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intensity values (usually at levels xn, where x is a user selectable number and n¼0, 1, 2, 3, . . .) above a user-determined threshold in the spectrum. The contour map is generated by plotting the intersections of the peaks with these planes. The more intense peaks will intersect a larger number of planes; therefore, peaks in Figure 4(b) that are defined by a larger number of contours are more intense than those defined by a small number of contours. In this display mode, peaks are not obscured and the printout is generated fairly rapidly. Although it does take a significant amount of computer power to calculate a contour map, modern computers are capable of doing so in much less time than it takes to transmit the data to most plotters. Commercial software packages for manipulating NMR data permit the adjustment of the number and spacing of the contours so as to best display all the peaks in the spectrum. In cases where all the signal intensities are of the same order of magnitude, x can be small (between 1.1 and 1.4) so that a large number of contours accurately define the peak shapes. In cases where there is a considerable dynamic range of peak heights, x can be large (2–4) to prevent the generation of a large number of contours around intense peaks.

Presentation of 2D NMR Data Figure 4(a) shows a stacked plot of a COSY (correlation spectroscopy) spectrum from ethanol. Detailed fine structure is not resolved in this spectrum because of the greatly reduced digital resolution compared to that obtained in 1D NMR. This reduced digital resolution is not very detrimental, and is necessary to keep the 2D data files to a manageable size (vide infra). The stacked plot (Figure 4(a)) is not the most desirable way to present the data because it involves a lot of plotting time and background peaks are often hidden by those in the foreground. The preferred method of presenting 2D data is in the form of contour maps, as shown in Figure 4(b). For those unfamiliar with generation of contour maps, planes are set at a range of (a)

Classes of 2D NMR Experiments Part of the power of 2D NMR comes from its ability to provide tremendous spectral dispersion; however, the structural information present from the correlation of frequencies is equally important. Organic chemists have been doing elegant syntheses for many decades. They chose a target molecule for preparation, and based on their knowledge of chemical reactions selected the proper reagents from their stockroom to carry out the chemical transformations necessary to obtain the desired product. Since the development of multidimensional NMR similar possibilities exist for studying molecular structure, reactivity, and dynamics. The NMR spectroscopist first defines the (b)

0.8 0.9

f2 (ppm)

1.0 1.1 1.2 1.3 1.4

1.4

1.3

1.2

1.1

1.0

f1 (ppm)

Figure 4 COSY 2D NMR spectrum of ethanol: (a) stacked plot and (b) contour plot with contours plotted at intensities of 2n.

0.9

0.8

0.7

478

Two-Dimensional NMR

(b)

(a)

O CH3

CH2

CH2

CH2

C

CH2

f2 (ppm) 1.0

CH3

1.2 1.4

b

1.6 c

1.8 t1

2.0 t2

2.2 2.4 2.6

a

d 2.4

2.0

1.6

1.2

0.8

f1 (ppm)

Figure 5 (a) COSY pulse sequence and (b) COSY spectrum of 3-heptanone with its 1D 1H spectrum plotted across the top.

nature of the information needed to solve a particular problem. It is then possible to go to the ‘NMR stockroom,’ and choose from a variety of ‘NMR reagents,’ which include pulses, delays, frequencies, rf phases, rf amplitudes, magnetic field gradients, and so on. Using the right combination of these reagents, a spectrum can be produced with selected signals containing the needed information, while removing other undesired signals that interfere with observation of the interesting signals or the interpretation of the data. As an example, Figure 5 shows a COSY pulse sequence (Figure 5(a)) and the 2D COSY spectrum of 3-heptanone with the 1D 1H spectrum plotted across the top (Figure 5(b)). The 2D spectrum contains a series of peaks along the diagonal whose positions in f1 and f2 correspond to the positions of the peaks in the 1D spectrum. Off the diagonal, cross peaks exist, which correlate the frequencies of proton pairs that are coupled to each other. In COSY spectra, these correlations indicate protons that are on adjacent carbons (or nonequivalent protons on the same carbon). Separate sets of cross peaks are observed for the ethyl (A) and butyl (B, C, and D) fragments of the molecule. Because the protons on these two fragments are more than three bonds away from each other, there is no J-coupling between protons on the two different groups. Consequently, none of the resonances from protons on the butyl fragment contain correlations to the resonances of protons on the ethyl fragment. In general, the NMR parameter plotted along the f2 dimension is related to the signal detected during the t2 time period. The NMR parameter plotted along the f1 dimension is determined by the precession frequencies of the NMR coherences during the t1 period. A specific sequence of pulses can be used to place the NMR coherence on selected nuclei (e.g., 1H or 13C) to encode their NMR properties during t1; a second series of pulses and delays are then used to transfer that coherence to the detected nucleus based on an NMR interaction (e.g., by J-coupling, dipolar coupling, or internuclear relaxation). Cross

peaks in the 2D NMR spectrum identify pairs of nuclei that share this interaction. Some common 2D NMR experiments are shown in Table 1, along with the NMR parameters plotted along the f2 and f1 dimensions, the interaction that produces the correlations, the structure information obtained from the spectrum, typical experiment times, and some comments. The experiments can arbitrarily be classified into four groups: homonuclear chemical shift correlation, heteronuclear chemical shift correlation, J-resolved, and multiple quantum experiments. The first six experiments in Table 1 are homonuclear chemical shift correlation experiments. In one subset of homonuclear chemical shift correlation experiments, COSYand TOCSY-type experiments, the same chemical shifts (usually those of 1H) are plotted along the f1 and f2 axes. The 2D spectrum contains peaks along a diagonal at the intersections of the chemical shifts of each nucleus. If there is J-coupling between two nuclei, then off-diagonal cross peaks connect the diagonal peaks to form a box as in the COSY spectrum of 3-heptanone described earlier. The TOCSY experiment can produce, in addition, cross peaks for all protons involved in an unbroken chain of spin couplings. A second subset of homonuclear chemical shift correlation experiments (NOESY and ROESY) has an appearance identical to that of COSY-type experiments, but with off-diagonal cross peaks indicating proximity of two nuclei in space (usually the nuclei must be within 5 A˚ to produce NOESY cross peaks). The second group of experiments produces 2D spectra with the chemical shifts of different nuclei along the two axes (e.g., 1 H along the f1 axis and 13C along the f2 axis). A single cross peak is observed for each coupling interaction in HETCOR, COLOC, HMQC, HSQC, and HMBC experiments. The latter three experiments are sometimes put in their own classification, and are called indirect detection experiments. HETCORtype experiments, which involve detection of the 13C signal during t2, were the first commonly used experiments to provide 1H–13C correlations. Later, after the performance of NMR

Table 1

Some common 2D NMR experiments and related information

Experiment name

f2

f1

NMR interaction

Structure information

Timea/sample quantity

Comments

COSY DQ-COSY

dH dH

dH dH

2 2

JHH and 3JHH JHH and 3JHH

H–C–H and H–C–C–H H–C–H and H–C–C–H

15 min/1 mg 15 min/1 mg

Relayed COSY TOCSY NOESY ROESY HETCOR Long-range HETCOR COLOC HMQC/HSQC IMPRESS-HSQC

dH dH dH dH dC dC

dH dH dH dH dH dH

JHH and 3JHH JHH and 3JHH H–H dipole–dipole H–H dipole–dipole 1 JCH 2 JCH and 3JCH

Hs in a spin system Hs in a spin system rHH conformation rHH conformation C–H C–C–H and C–C–C–H

15 min/1 mg 15 min/1 mg 4–12 h/5 mg 4–12 h/5 mg 2–8 h/10 mg 2–8 h/10 mg

Easy Double quantum filtered COSY – same as COSY, but eliminates singlets that contain no information Easy Easy Usually 10–100 times weaker than COSY Usually 10–100 times weaker than COSY 13 C detected 13 C detected

dC dH dH

dH dC dC

2

JCH and 3JCH JCH 1 JCH (spectral folding and editing possible)

C–C–H and C–C–C–H C–H C–H

2–8 h/10 mg 1 h/5 mg

dH dH dH

dC dC dC

1

JCH 1 JCH 1 JCH

C–H C–H C–H

dH dH dH dH dC dH

dC dC dC dC dH JHH

JCH and 3JCH JCH and 3JCH 2 JCH and 3JCH 1 JCH and 3JHH C–H dipole–dipole JHH scalar coupling

C–C–H and C–C–C–H C–C–H and C–C–C–H C–C–H and C–C–C–H H–C–C–H rCH Conformation

1–4 h/5 mg 1–4 h/5 mg 1–4 h/5 mg 1–4 h/5 mg 12 h/50 mg 20 min/1 mg

1 H detected Compensates for range of J couplings Simultaneous detection of C–C–H and C–C–C–H couplings Selective detection of two-bond C–H Difficult Easy and fast

dC

JCH

2

2–8 h/10 mg

Moderate

dC

dCa þ dCb

1

Conformation and number of attached H 13 Ca–13Cb

12–16 h/100 mg

1/104 molecules, extremely difficult

a

2

1

2 2

JCH and 3JCH JCC

Typical experiment times for a molecule with MW¼500 and experiments performed on a 300–400 MHz spectrometer.

13

C detected H detected 1 H detected 1

DPFGSE sequence incorporated for selective excitation Adiabatic 180 pulses on X channel Adiabatic 180o pulses on both H and X channel Single scan collection of entire 2D spectrum

Two-Dimensional NMR

CRISIS-HSQC CRISIS2-HSQC Ultra-SOFAST HMQC HMBC CIGAR-HMBC 2J, 3J–HMBC H2BC HOESY Homonuclear 2D-J Heteronuclear 2D-J 2DINADEQUATE

2

479

480

Two-Dimensional NMR

instruments improved, the more sensitive and more challenging 1H–13C correlation experiments involving detection of the 1 H signal during t2 became popular. In these experiments, the chemical shift of the X nucleus (usually 13C) is indirectly detected in the t1 dimension. Perhaps, if HMQC type experiments were popular first, then HETCOR-type experiments would now be called indirect detection experiments. The HOESY experiment is the heteronuclear version of the NOESY experiment, and contains cross peaks between the resonances of dissimilar nuclei if there is an NOE interaction between those nuclei. The third class of experiment is the J-resolved series of pulse sequences. These produce spectra with the peaks at the frequencies along f2 corresponding to the resonances observed in the 1D spectrum of the detected nucleus. In homonuclear 2D J-spectroscopy, the peaks are dispersed into the f1 dimension based on homonuclear J-coupling. In heteronuclear 2D J-spectroscopy, the peaks are dispersed into the f1 dimension based on heteronuclear J-coupling (e.g., detection of 13C in f2 and at the shift of each 13C a multiplet, resulting from all of the resolved JCH-couplings, is observed in f1). The fourth class of experiments involves multiple quantum spectroscopy such as 2D INADEQUATE. In these experiments, homonuclear shifts (such as those of 13C) are plotted along the f2 dimension. If two or more nuclei are J-coupled to each other, then they can be made to share a common multiple quantum precession frequency during the t1 period. In the 2D INADEQUATE experiment if CA and CB are coupled to each other, then the signals from each of these components will precess at a common double quantum frequency (nAþnB) in the f1 dimension. Usually, two or more experiments are run, where the correlations in each experiment provide a set of structure fragments. The combined fragments can then be fitted together like the pieces of a puzzle, and in most instances, the right combination of multidimensional experiments can provide complete information about the structure of an unknown molecule. As an example, if HSQC and HMBC spectra were obtained from p-nitrotoluene, the HSQC spectrum would provide C–H connectivities illustrated by the highlighted bonds in structure 1; HMBC would provide information that relates the 1H shifts with 13C shifts of atoms two and three bonds away. Some of these correlations are illustrated on structure 2. The combined information from the two experiments provides a complete structure of the molecule. Although a complete description of these experiments is beyond the scope of this article, some comments on experimental characteristics are worth noting. Experiments that involve 1H detection are generally much more sensitive than those that involve 13C detection, largely due to the higher g of 1H. The use of 1H detection necessarily involves the elimination of all 1H NMR peaks of protons attached to 12C, approximately 99% of the total intensity, and this is achieved by suitable pulse sequences (vide infra). Even though HETCOR and HSQC provide similar frequency correlations and identical structure information, the former involves 13C detection and generally requires approximately 30 times more sample to produce a spectrum of the same quality.

H

H

H

H

H

CH2

H

CH2

NO2

H

NO2

H

H

H

[1]

[2]

Although many of the experiments use similar interactions to provide correlations, experiments that use smaller, longrange J-couplings, require longer delays (usually approximately 1/2J) than experiments that use large one-bond J-coupling. During these longer delays, relaxation effects reduce the intensities of the signals that are finally detected during t2, especially for large molecules having short T2 relaxation times. Consequently, experiments like HMBC produce spectra with poorer S/N than its counterpart, HSQC. Although all the entries in the second and third columns of Table 1 refer to 1H and 13C, other combinations of NMR-active nuclei can be used to perform most of these experiments. For example, the experiments in the first six rows of Table 1 are 1 H–1H homonuclear correlation experiments. These experiments will work just as well with 19F–19F homonuclear correlation experiments if there are a number of mutually coupled fluorine atoms in the structure to be studied. Likewise, 15N or 31 P, for example, could be substituted for 13C in HSQC and HMBC experiments.

Experimental Aspects of 2D NMR Acquisition Conditions Instrument requirements Most instruments that have been installed in the past 10 years are capable of performing all of the experiments shown in Table 1. Collection of 2D NMR spectra requires a stable instrument and a stable instrument environment. The exact requirements become more stringent at higher magnetic fields. For example, 600–800 MHz spectrometers generally require room temperature fluctuations less than 0.5  C, minimized air currents, and the magnet mounted on vibration isolation pads. In some instances, it might be necessary to mount other mechanical equipment near the instrument (near is not used in an absolute sense since some buildings are more efficient at transmitting vibrations throughout the structure than others) on its own vibration isolation equipment. All of the experiments shown in Table 1 can be performed on standard two-channel (i.e., 1H and X channels) spectrometers.

Spectral resolution and data size As mentioned earlier, a number of separate 1D FIDs are collected, each with a different value for t1. In 1D NMR spectroscopy, 50–100k data points are collected and Fourier transformed to provide a spectrum. The exact number of points depends on the spectral window, expected line widths, and the

Two-Dimensional NMR

desired digital resolution (usually 0.1–0.5 Hz per point) in the 1D spectrum. If this digital resolution is maintained in both dimensions of a 2D experiment, the file size could grow to many gigabytes, and would be difficult to manipulate and store. Consequently, shortcuts are used to minimize the sizes of 2D data files. The first of these shortcuts is to minimize the spectral windows to include only those regions that are expected to contain peaks of interest. For example, in a COSY experiment that contains cross peaks between the resonances of coupled protons, the spectral window is narrowed to exclude singlets and solvent resonances. It is usually worthwhile to collect 1D spectra that correspond to the windows in the two dimensions before attempting to run the 2D experiment. In some cases, however, this may not be possible (e.g., if sample quantity is limited and the f1 dimension is the 13C chemical shift in an HSQC experiment). A second shortcut is to drastically reduce the digital resolution in the 2D spectrum; typically the data is collected to provide 1–2 Hz per point digital resolution in the final spectrum. Typical t2 (acquisition) times are 0.05–0.2 s, more than an order of magnitude smaller than those used in 1D NMR spectroscopy. The use of longer acquisition times has very little effect on data collection times. If a 1 s relaxation delay is used, increasing t2 from 0.05 to 0.2 s results in 15% longer experiment time and provides a fourfold increase in digital resolution in f2 (and a fourfold increase in the size of the data file). The same is not true in the t1 dimension. Typically, 100–500 separate FIDs are collected to produce the points in the t1 dimension of a 2D spectrum. To obtain a fourfold increase in digital resolution in f1, four times as many FIDs must be collected, increasing the experiment time by more than fourfold (the t1 period for the last FID will be significantly longer than for the first increment where t1¼0). If four transients are averaged for each of the 100 FIDs, the S/N in the resulting 2D spectrum would be comparable to that obtained in the corresponding 1D version of the experiment (i.e., if only the first t1 increment is collected) obtained by averaging 400 transients. However, in many cases the additional sensitivity achieved by collecting so many FIDs is not required and the 2D experiment is longer than is required for signal detection. To mitigate this problem various shortcuts have been devised to shorten 2D data accumulation times while improving spectral resolution (see sections below on Linear prediction and Sparse sampling of 2D NMR data).

Phase cycling for artifact suppression and coherence selection There are a number of artifacts that can appear in 2D spectra, including peaks and ridges at the transmitter frequencies in f1 and f2, and mirror images of the real peaks on the opposite side of the spectrum. These can arise from a number of sources, including the fact that some spins experience imperfect pulses. Even with a properly functioning instrument, nuclei whose resonances lie near the edge of the spectral window experience significantly different flip angles compared to those nuclei whose resonances lie near the transmitter, as a consequence of resonance offset effects. Additionally, even those nuclei whose resonances fall near the transmitter experience a gradation of flip angles depending on the position of the nuclei relative to the probe’s transmitter coil (i.e., the nuclei in the portion of the

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sample near the middle of the tube experience a larger flip angle than nuclei in those portions of the sample near the top and bottom of the tube). These artifacts are reduced by using either composite pulses or adiabatic pulses in place of simple 180 pulses (CRISIS versions of pulse sequences in Table 1). To further reduce artifacts, the phases of 180 pulses are shifted by 180 in alternate transients; and the phases of 90 pulses are usually incremented by 90 in a sequence of four transients. A sequence with both a 90 and a 180 pulse ideally would require averaging of eight transients to obtain a spectrum resulting from all permutations of the two phase cycles. In experiments with many pulses, the number of transients required to cycle the phases of all pulses becomes extremely large (spectra in which the number of transients per FID is 64–256 can be found in the early literature). Usually, artifacts from imperfections in one pulse are more prominent than those arising from imperfection in the other pulses in the sequence. In those cases, the phases that remove the most severe artifacts are cycled first. When setting up an experiment, it is necessary to know the number of transients needed to complete this minimum phase cycling, and to set up the experiment to collect an integral multiple of this number of transients. Some of the 2D spectra result from cancellation experiments (i.e., coherence selection). The HMQC is an excellent example of experiments in this class. As described earlier, HMQC provides a 2D spectrum correlating the shifts of 1H and directly bound 13C nuclei. In the 1H spectrum of 3-heptanone, the peaks that are normally observed are those from 1H bound to 12C (99% of the protons, Figure 6(a)); however, if the vertical scale of the spectrum is increased 100fold, a set of satellite resonances from 1H atoms bound to 13C (1.1% of the signal, Figure 6(b)) are observed. To selectively detect the desired component from 1H atoms bound to 13C, the pulse sequence in Figure 7(a) is used. If the phase, F1, of the 90 13C pulse is applied along the þx and x axes on odd and even transients, respectively, the sign of the undesired signals from 1H bound to 12C is unaffected; however, the phases of the desired signals from 1H bound to 13C are altered in odd and even transients. If the FIDs from odd transients (Figure 6(c)) are subtracted from those in even transients (Figure 6(d)) (by altering the phase of the receiver F3), the undesired signals cancel and the desired signals add (Figure 6 (e)). Detection of the desired signals requires observation of small differences between two large signals. Minor variations in the state of the instrument or its environment will result in imperfect cancellation (as evident by the large residual center signal in Figure 6(e)) and will produce large noise ridges that will obscure the cross peaks of interest. With a 64-cycle sequence, the residual center peak can be significantly reduced; however, once adequate S/N is achieved after a single transient, the experiment must still be run 64 times longer just to complete the phase cycling necessary to remove the artifacts. Furthermore, even when limited sample quantities result in the need to perform signal averaging, the residual signal intensity varies randomly from one pair of transients to the next, and adds like noise. The result is a ridge of noise along f1 at the f2 frequency of intense signals. This noise ridge, often called t1 noise, limits the ability to detect weak cross peaks in the spectrum.

482

Two-Dimensional NMR

1

H spectrum

HMQC spectra

(a)

(c)

(e) d-c NT = odd

2.7

2.5

2.3 ppm

2.1

2.7

(b)

2.5

2.3 ppm

2.1

(d)

2.7

2.5 ppm

2.3

2.1

2.7

2.5

2.3 ppm

2.1

(f)

NT = even A x100

2.7

2.5

2.3 ppm

2.1

2.7

2.5

2.3 ppm

2.1

Figure 6 Methylene regions between 2.2 and 2.5 ppm from the proton NMR spectra of 3-heptanone. (a) Normal 1H spectrum, (b) 100
vertical amplification of (a), (c and d) spectra obtained from collecting a single HMQC transient with phase cycling for odd and even transients, respectively, (e) spectrum in (d) minus spectrum in (c), and (f) single transient from HMQC spectrum obtained with PFG coherence selection.

(a)

1

2

H

t2

Δ

Δ

3

1 13

t1

C

Decoupling

(b) 1

H

13

Δ

Δ

t1

C g1

t2

Decoupling

orthogonal axes (x, y, and z) of the sample; however, few instruments are equipped with this capability.) The spectrum in Figure 6(f) was obtained by collecting a single HMQC transient with the aid of PFG coherence selection. The residual center peak from 1H–12C fragments is completely suppressed in one transient. The first obvious advantage of PFG spectroscopy is that excellent coherence selection is obtained in a single transient. Under these circumstances, the number of transients per FID is determined by sensitivity requirements and not the need to complete a phase cycle. Additionally, in cancellation experiments, the suppression level is less sensitive to instrument instabilities; and because the large undesired signal component never reaches the receiver, instrument gain settings can be optimized for detection of the weak signals of interest.

g2 g3

Figure 7 (a) HMQC pulse sequence with phase cycling for coherence selection and (b) PFG-HMQC pulse sequence. In these diagrams, solid rectangles are 90 pulses, unfilled rectangles represent 180 pulses, and hashed rectangles represent field gradient pulses.

Pulsed field gradients for coherence selection Pulsed field gradients (PFGs), also known as gradient enhanced spectroscopy (GES), can be used to achieve coherence selection and minimize the need for extensive phase cycling. In PFG spectroscopy a large z-gradient is introduced along the sample’s vertical axis (magnitude approximately 0.1–0.5 T m1 and duration approximately 1 ms); additional PFGs are introduced later in the sequence to selectively refocus the coherence components of interest and continue to destroy undesired coherence components. (In some rare cases, it is possible and advantageous to apply gradients along the three

Quadrature detection in f1 and f2 When collecting 1D spectra on many modern instruments, two detection channels are present that independently measure the signals 90 out of phase with respect to one another (Figure 8 (a)). Two FIDs, a real component and an imaginary component (which is 90 out of phase with respect to the real component of the signal), are saved; frequency information is obtained from the real component and phase information (i.e., whether the signal is to the left or right of the transmitter) is present in the imaginary component. A complex Fourier transformation produces a spectrum showing peaks with the proper relationship with respect to the transmitter, depending on the relative phase of the imaginary component in Figure 8(a). In 2D NMR, it is not possible to use a second detector in the f1 dimension. There are alternatives that provide the equivalent of phase-sensitive detection in the f1 dimension; the commonly used methods are the States’ method, time-proportional

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Phase sensitive absolute value

483

Dispersion signals Absorption signals (b)

(a) PH mode

a × r + (1−a) × i

FT r

(c) i

AV mode

(r 2+ i 2)1/2

Figure 8 (a) Schematic illustration of real (r) and imaginary (i) components of the spectrum before FT. After FT the data can be represented in (b) phase-sensitive (PH) mode or (c) an absolute value (AV) mode and spectra can be calculated.

phase increment (TPPI), and the combined States-TPPI method. In the TPPI method a single data set with 2n t1 increments is collected. In each successive t1 increment the phase of the 90 pulse at the end of the t1 period is incremented by 90 with respect to the phase of the corresponding pulse in the previous t1 increment. (An equivalent experiment can be performed in which the phases of the pulses before the t1 period are shifted by 90 .) This is equivalent to changing the reference frame in f1 so that the transmitter in the t1 dimensions appears to be shifted to one edge of the spectrum. After performing a real Fourier transformation, all peaks will appear to be shifted to one side of the transmitter in f1. The main disadvantage of this technique is that phase distortions can appear for resonances in strongly coupled spin systems. To obtain true quadrature detection, two sets of data with real and imaginary components in t1 must be obtained. In the States method, two sets of 2D FIDs are collected and saved. Both data sets contain n FIDs (2n total FIDs) and the t1 delays in the corresponding FIDs in the two data sets are identical. Their only difference is that the second set of FIDs is collected with the phase of the 90 pulse immediately after the t1 period shifted by 90 compared to the phase of the same pulse in the first set of FIDs. The first set of FIDs contains the frequency modulation information in t1 and the second set of FIDs contains the phase information in t1, similar to quadrature detection in 1D NMR. After each of the FIDs in the two data sets are Fourier transformed, corresponding points from the two data sets are paired to form complex points in t1 and a complex Fourier transformation is performed with respect to t1. This latter method provides data sets that are identical in size (and digital resolution) to those obtained from the TPPI method with equivalent digital resolution. The States method of phase-sensitive detection is usually preferred because artifacts are less problematic. The third commonly used method of quadrature detection in t1 combines the elements of the States and TPPI methods. Two sets of n FIDs are collected, 90 out of phase with respect to one another, but with the same t1 values (as in the States method). Within each of these data sets the phase of the 90 read pulse at the end of t1 is incremented in 90 steps while incrementing t1 (as in the TPPI method). With this technique,

axial artifacts are shifted to the edges of the spectral window where they are less likely to interfere with cross peaks to be detected. Kay et al. have devised a method called preservation of equivalent pathways (PEP) as data collected with PFG coherence selection and with States phase-sensitive detection components retain the P and N components of the signal in both 2D data set components. At the processing stage the real and imaginary components in t1 are generated by the sum and difference of the collected data. In ideal cases, this method provides 21/2 increase in S/N.

Establishing a steady state Ideally, a relaxation delay of 3–5T1 should precede the cycle of pulses used to collect each transient. However, this would make experiments impractically long. Normally, 1–2 s relaxation delays are used, even though T1s might be 10 s or more. For larger molecules with shorter T1s, relaxation delays of 100–500 ms are used. Under these conditions, incomplete relaxation occurs. Consequently, the intensities of the signals in the first t1 increment are artificially enhanced relative to the same signals in later increments. This means that the first point in t1 is offset, leading to a large zero frequency offset in the baseline at slices corresponding to the resonance frequencies of all peaks in t2. To eradicate this problem, it is customary to perform 8–32 dummy scans, which are discarded, before collection of the data to be saved. During these dummy scans, ‘steady-state’ magnetization is established before data collection commences.

Linear prediction Limited access to instrument time and the volume of data that must be collected means that shortcuts must always be taken, which adversely affect the appearance of the spectrum. Digital signal processing techniques can significantly enhance the appearance of a spectrum without increasing data collection times. In some cases, when instrument time is at a very high premium, it might even be desirable to deliberately reduce the experiment time below the minimum needed for a reasonable spectrum knowing that processing techniques can be used to compensate for the missing data. Mathematically, it is possible to use the

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behavior of a function during time t, during which a measurement is made, to predict the behavior of the function if the measurement time had been extended by time t0 (Figure 9(a)). As of this writing, in multidimensional NMR, linear prediction is one of the most used and most useful of these mathematical methods. Essentially, the oscillatory behavior of the signal intensity as a function of t1 (at a specific f2) is fit to the sum of a series of cosine waves. Since the number of peaks present at a single f2 in a 2D spectrum is relatively small, the sum of a relatively small number of frequency components is sufficient to simulate the behavior of the FID in t1. The function can then be used to artificially synthesize values for the FID in t1 as if a much larger number of t1 increments had been collected. Usually the data are increased to two to four times the original size, and zero filling is applied to double the length of the data (e.g., if 2
256 t1 increments were collected, linear prediction would be used to forward extend the data to 2
1024 and zero filling could be used to further increase the size in t1 to 2
2048). This permits improvements in resolution comparable to what would be achieved from an experiment that is up to four times longer than the actual experiment time. Linear prediction can also be used to remove experimental artifacts from data. For example, the intensity of a single FID in the middle of a 2D experiment (Figure 9(b)) could be distorted if the field were perturbed for some reason. If steady-state pulses were not applied at the beginning of the experiment, (a) t

t′

(b)

the first few points in t1 might be more intense than they should be (Figure 9(c)). In the former case, with linear prediction, the behavior of the FID on either side of the distorted point could be used to approximate the correct value of the distorted point. In the latter case, if the first two points are distorted, the behavior of the function for points 3–10 could be used to back predict the proper value of the points in the beginning of the FID.

Sparse sampling of 2D NMR data As mentioned earlier, usually the limiting factor in collecting 2D data is not S/N; rather, it is the need to sample sufficient data points in the t1 dimension to provide sufficient digital resolution. In most cases, long after sufficient signal is obtained, signal averaging is continued to provide sufficient digital resolution in the indirectly detected dimension. In simple 2D experiments, the experiment time increases approximately linearly as the digital resolution (number of t1 increments) in the t1 dimension is increased. Simultaneously, as the digital resolution increases so does the S/N also, as each point sampled in t1 contributes to the S/N in the entire spectrum. The above-mentioned Felgett advantage also applies to the indirectly detected dimension. If n FIDs are collected, a factor of n1/2 increase in S/N can be expected compared to the 1D spectrum with averaging of the same number of transients. In most cases, limited availability of spectrometer time places a limit on the digital resolution in the t1 dimension. The sampled data are schematically illustrated by the subset of sampled points in the lower left corner of Figure 10(a). To improve digital resolution and minimize truncation artifacts in spectra with good S/N, linear prediction is used to fill in the data set with additional calculated data (gray data points in the central part of Figure 10(a)) before Fourier transformation to produce the final spectrum. Recently, a number of other data sampling/processing methods have been proposed to contend with this sampling problem. These include selective excitation of subregions in the f1 domain – analytical methods such as Hadamard techniques and application of numerical methods such as the filter diagonalization method (FDM) and maximum entropy methods (MEM). All these techniques make use of sparse sampling schemes, followed by mathematical reconstruction of the nD NMR spectrum.

Selective excitation (c)

Figure 9 (a) Time domain NMR signal (FID) detected (full line), and calculated (dashed line) using linear prediction. (b) An FID from the middle of a 2D experiment with a single distorted data point (c) An FID with distorted signal intensity at the beginning.

If only a few (n) peaks are present in the f1 dimension, selective excitation of each spectral region is performed and correlations to that region can be observed. After Fourier transformation with respect to t2, n separate selective 1D spectra are obtained, where correlations to the excited resonance are observed in the n respective regions. The concept of using n selective 1D spectra to replace the full 2D spectrum was first described two decades ago. The primary factor that has increased the utility of these methods in recent years has been the development of flexible instrument hardware and software that enables the reproducible and accurate generation of rf pulse waveforms that can excite any arbitrary desired spectral window or combination of windows. These methods work well when correlations to a relatively small number of peaks are desired. The drawback of this

Two-Dimensional NMR

485

(a) FT(t2) t1

FT(t1) t1

f1

t2

f2

f2

(b) FT(t2) f1

f1

t2

f2

(c) SA

FT(t2) f1

f1

SA + SB f1 S − S A B

SB

t2

f2

f2

(d) FT(t2)

t1

MEM(t1) t1

f1

t2

f2

f2

(e) FDM(t1, t2) t1

f1 f2 t2

f2

Figure 10 Methods of dealing with sparse sampling of 2D NMR data sets: (a) small number of FIDs are collected and after FT(t2) linear prediction is used to calculate the gray points before FT(t1); (b) selective excitation at the few known peak positions in f1, and FT(t2), produces a 2D spectrum with information only at the two selective excitation frequencies in f1 – no data is present between these frequencies; (c) Hadamard method of 2D NMR in which all peaks in f1 are excited simultaneously, but with shaped pulses that alter the phases (þ/) of the peaks from one trace to another – the final spectrum is produced by taking linear combinations for the raw data based on the Hadamard mask; (d) sparse sampling of the FIDs in t1 followed by MEM fitting of the data to produce the final high-resolution spectrum; and (e) FDM method of 2D NMR where only a few t1 increments are sampled, followed by numerical transformation of the data from S(t1,t2) into S(f1,f2); sampling of many points in t2 contributes to the information about peak positions in f1.

method comes when n (the number of spectral regions) becomes large. In these cases, many selective spectral windows must be collected, and the S/N values in each of these spectra are independent of one another. The experiment times can become quite long to signal average the necessary number of transients in each of the many desired spectral regions. To circumvent these problems, in recent years the selective excitation methods have been adapted to Hadamard spectroscopic techniques. The Hadamard adaptation of selective experiments involves the use of rf waveforms such that in each FID a combination of selective pulses is simultaneously applied to all of the resonance positions in f1 in such a manner as to excite (þ) or invert () the peaks in n different regions; n different FIDs are collected containing linear combinations of the spectral components based on the Hadamard mask. Linear combinations of these n FIDs are calculated based on the

Hadamard mask and these combined FIDs are Fourier transformed to produce n spectra each with one of the desired correlations. The appropriate linear combinations can be calculated on either the time or frequency domain signals in f2. In the left part of Figure 10(c), the simplest Hadamard data set, where n¼2, is used as an example. Two FIDs are collected, where the excitation waveforms are such that the two regions have signal phases þþ in the first FID and þ in the second. Fourier transformation with respect to t2 produces two spectra (SA and SB) in which the two sets of correlations have the same phase or opposite phase, respectively (center of Figure 10(c)). In this simplest case, the Hadamard mask requires taking the sum and difference of the two data sets to produce the two final spectra (right part of Figure 10(c)). The advantage of this technique is that all FIDs have information about all signals in the spectrum, producing n1/2 improvement in S/N

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compared to the experiment where n separated selective experiments are performed. Adaptations of this use of Hadamard spectroscopy have been described where the multiplicities of correlations in f1 have been regenerated from the information in f2; and where narrow spectral windows rather than a single frequency in f1 have been collected to produce a series of 2D spectra with narrow f1 windows. The Fourier transformation is an analytical relationship between the time and frequency domain spectra with specific requirements for calculating the transform between the two domains. By using numerical methods it is possible to produce a simulated spectrum, transform the trial spectrum to produce a calculated time domain signal, compare that signal with the experimental time domain spectrum, and iteratively adjust the trial spectrum so that its time domain signal matches the experimentally collected version of the data. MEM have been used for this purpose. Numerical methods like this remove the restriction imposed by the Fourier transformation technique that sequential data points be sampled. As shown in the left part of Figure 10(d), instead of sampling n separate FIDs with t1 incremented in steps of 1/sw1 from 0 to (n1)/(sw1), where sw1 is the spectral width in f1, m separate FIDs are collected, where m is much smaller than n. The density of points sampled in t1 is weighted more heavily in the region closer to t1¼0, where the S/ N is greater; however, FIDs at longer values of t1 are also sampled to provide information about the decay of the signal in t1. Thus, after MEM calculation to transform S(t1,f2) into S (f1,f2), better resolution is obtained in the f1 dimension of the resulting 2D spectrum when compared with the standard Fourier transformation of the spectrum obtained with t1 incremented from 0 to m/sw1. The above-mentioned methods of 2D NMR treat the t1 and t2 dimensions separately, so that if f1 spectral resolution is desired, the number of points in t1 must be increased. By removing the condition of treating t1 and t2 independently, as required by successive Fourier transformation with respect to t2 and then t1, a condition can be achieved where an increase in the number of points in t2 can have an impact on resolution in t1. This is graphically illustrated in Figure 10 (e). In the standard Fourier transformation method of processing 2D, separate traces along t1 contain information at one specific frequency in f2. To increase resolution in f1 more points are needed along each specific value of f2. If the two dimensions are linked in the transform from S(t1,t2) to S(f1, f2), then sampling additional data points in f2 can be used to provide more information with regard to the signal behavior in f1. On the left side of Figure 10(e), relatively few t1 increments (m typically 2–8) are sampled. Information along diagonal traces provides information about signals in both the f1 and f2 dimensions. Increasing the number of diagonal traces by increasing the number of t1, t2, or both increments provides additional information about the signal composition in both f1 and f2 dimensions. Of course, increasing the number of points in t2 has far less consequence on the total experiment time than increasing the number of points in t1. The Frydman group has devised a way to rapidly collect complete 2D NMR data sets in which they use PFGs to parse the sample into slices along the direction of the magnetic field (B0). Each slice is used to produce a signal corresponding to a

single t1 increment. The excitation scheme is organized so that in a single acquisition period a series of echoes is detected during the acquisition time; each echo corresponds to the signal from one of the sample slices, and as a consequence, contributes t1 increments to the final 2D data set. The resulting echo data is parsed into a 2D matrix and Fourier transformed to produce a 2D spectrum. With this method, a single excitation sequence can be used to produce the entire 2D data set in a matter of seconds. They have named this technique ultrafast 2D NMR. Recently, commercial dynamic nuclear polarization (DNP) equipment has become available to take advantage of nuclear Overhauser enhancement (NOE) of the signals from less sensitive nuclei such as 13C using the enormous polarization of electrons at low temperature (1–2 K). Under favorable circumstances the NMR signals are enhanced by a factor of 10 000-fold. The DNP process is inherently slow since the nuclear spins build up polarization at a rate comparable to their T1s, which are very long at low temperature. The polarization may take hours to develop for a single acquisition. Although the time-savings are enormous for collection of 1D spectra, the long polarization times and irreproducibility of the process makes DNP incompatible with standard 2D NMR data collection. However, the Frydman group has adapted their ultrafast 2D NMR methods to DNP. Ultrafast 2D techniques are ideally suited to DNP signal enhancement as a single DNP process can be used to produce the entire 2D spectrum.

Reduced dimensionality Finally, a number of methods have been devised to reduce the dimensionality of nD NMR to (nk)D NMR. In the simplest form, in a 3D H–X–Y correlation experiment selective pulses (described earlier) can be used to excite individual Y resonances that would normally define a third chemical shift dimension; a series of 2D planes might then be obtained in which the H–X correlations found would identify structures associated with a specific Y nucleus. For a structure with a relatively small number (y) of chemically distinct Y nuclei, y separate/selective 2D experiments could be performed much more quickly than the corresponding 3D NMR experiment that would sample all of the third dimension space. In an alternative approach, Szyperski’s group adapted the concepts originally associated with accordion NMR experiments, in which two (or more) evolution times are simultaneously incremented to convert the dimensionality of the experiment from nD NMR to (nk)D NMR. In its simplest form, the two evolution times t1 and t2 (containing information about the chemical shifts of A and B nuclei, respectively) in a 3D NMR experiment are simultaneously incremented to produce a 2D NMR spectrum in which the f1 dimension contains cross peaks at position corresponding to (nAþnB) and (nAnB). In a 4D experiment, a twofold reduction in dimensionality (k¼2) can be realized by simultaneously incrementing the t1, t2, and t3 evolution times to produce a (4–2)D spectrum in which the peaks would appear at (nAnBnC). The resonances are sorted based on the use of a G-matrix transform (GFT). For samples with relatively few

Two-Dimensional NMR

correlations, enormous reductions in dimensionality (k¼3 or 4) can be realized without jeopardizing the ability to resolve and identify signals.

See also: Chemical Shift and Relaxation Reagents in NMR; nDimensional NMR Methods; NMR Applications, 15N; NMR Data Processing; NMR Methods, 13C; NMR Parameter Survey, 13C; NMR Principles; NMR Pulse Sequences; NMR Spectrometers.

Further Reading Ardenkjaer-Larsen J, Fridlund B, Gram A, et al. (2003) Increase in signal-to-noise ratio of >10,000 times in liquid state NMR. Proceedings of the National Academy of Sciences of United States of America 100: 10158–10163. Aue WP, Bartholdi E, and Ernst RR (1976) Two-dimensional spectroscopy. Application to nuclear magnetic resonance. Journal of Chemical Physics 64: 2229. Berger S (1997) NMR techniques employing selective radiofrequency pulses in combination with pulsed field gradients. Progress in NMR Spectroscopy 30: 137–156. Bovey FA and Mirau PA (1996) NMR of Polymers. New York: Academic Press. Bradley SA and Krishnamurthy K (2005) A modified CRISIS-HSQC for band-selective IMPRESS. Magnetic Resonance in Chemistry 43: 117–123. Cavanagh J, Fairbrother WJ, Palmer III AG III, and Skelton NJ (1996) Protein NMR Spectroscopy Principles and Practice. San Diego: Academic Press. Clore GM and Gronenborn AM (1991) Applications of three and four-dimensional heteronuclear NMR spectroscopy to protein structure determination. Progress in NMR Spectroscopy 23: 43. Croasmun WR and Carlson RMK (1994) Two-Dimensional NMR Spectroscopy Applications for Chemists and Biochemists. Weinheimer and New York: VCH. Ernst RR and Anderson WA (1966) Application of Fourier transform spectroscopy to magnetic resonance. Review of Scientific Instruments 37: 93. Freeman RA (1997) A Handbook of Nuclear Magnetic Resonance, 2nd edn Harlow, UK: Longman. Freeman RA (1998) Shaped radiofrequency pulses in high resolution NMR. Journal of Progress in Nuclear Magnetic Resonance Spectroscopy 32: 59–106.

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Freeman R and Kupce E (2004) Distant echoes of the accordion: Reduced dimensionality, GFT-NMR, projection-reconstruction of multidimensional spectra. Concepts in Magnetic Resonance Part A 23A: 63. Frydman L and Blazina D (2007) Ultrafast two-dimensional nuclear magnetic resonance spectroscopy of hyperpolarized solutions. Nature Physics 3: 415–419. Griffiths PR (1978) Transform Techniques is Chemistry. New York: Plenum Press. Hu H and Krishnamurthy K (2008) Doubly compensated multiplicity-edited HSQC experiments utilizing broadband inversion pulses. Magnetic Resonance in Chemistry 46: 683–689. Jeener J (1971) Abstract. Basko Polje, Yugoslavia: AMPERE International Summer School. Kay LE, Keifer P, and Saarinen T (1992) Pure absorption gradient enhanced heteronuclear single quantum correlation spectroscopy with improved sensitivity. Journal of the American Chemical Society 114: 10663. Kupce E and Freeman R (1997) Compensation for spin-spin coupling effects during adiabatic pulses. Journal of Magnetic Resonance 127: 36–48. Kupce E, Nishida T, and Freeman R (2003) Hadamard NMR spectroscopy. Progress in Nuclear Magnetic Resonance Spectroscopy 42: 95–122. Mandelshtam VA (2001) FDM: The filter diagonalization method for data processing in NMR experiments. Progress in Nuclear Magnetic Resonance Spectroscopy 38: 159–196. Martin GE and Zektzer AS (1988) Two-dimensional NMR methods for establishing molecular connectivity. Weinheimer and New York: VCH. Muller L, Kumar A, and Ernst RR (1975) Two-dimensional carbon-13 NMR spectroscopy. Journal of Chemical Physics 63: 5490. Nyberg NT, Duus JO, and Sorensen OW (2005) Heteronuclear two-bond correlation: Suppressing heteronuclear three-bond or higher NMR correlations while enhancing two-bond correlations even for banishing 2JCH. Journal of the American Chemical Society 127: 6155–6156. Rovnyak D, Frueh DP, Sastry M, et al. (2004) Accelerated acquisition of high resolution triple resonance spectra using non-uniform sampling and maximum entropy reconstruction. Journal of Magnetic Resonance 170: 15–21. Schmidt-Rohr K and Spiess HW (1994) Multidimensional Solid State NMR and Polymers. New York: Academic Press. Silverstein RM, Webster FX, and Kiemle DJ (2005) Spectrometric Identification of Organic Compounds, 7th edn New York: Wiley. States DJ, Haberkorn RA, and Reuben DJ (1982) A Two-Dimensional Nuclear Overhauser Experiment with Pure Absorption Phase in Four Quadrants. Journal of Magnetic Resonance 48: 286–292.