Frequency dimension

Homonuclear shift-correlation spectroscopy (COSY) is a standard method for establishing proton coupling networks. Diagonal and off-diagonal peaks appear with respect to the two frequency dimensions. 

First, it is useful to understand what we mean by 1-D and 2-D experiments. If you consider a normal proton spectrum, it is plotted in two dimensions (chemical shift on the x axis and intensity on the y), so why is it called 1-D In fact, when NMR started, it wasn t because there was no need to distinguish it from what we now call 2-D. The dimensions that we are talking about are the number of frequency dimensions that the data set possesses. To try to understand we need to explain the basics of the pulse programme. If we take a simple example (e.g., 1-D proton) we can represent the pulse sequence in Figure 8.1. 

In the solid, dynamics occurring within the kHz frequency scale can be examined by line-shape analysis of 2H or 13C (or 15N) NMR spectra by respective quadrupolar and CSA interactions, isotropic peaks16,59-62 or dipolar couplings based on dipolar chemical shift correlation experiments.63-65 In the former, tyrosine or phenylalanine dynamics of Leu-enkephalin are examined at frequencies of 103-104 Hz by 2H NMR of deuterated samples and at 1.3 x 102 Hz by 13C CPMAS, respectively.60-62 In the latter, dipolar interactions between the 1H-1H and 1H-13C (or 3H-15N) pairs are determined by a 2D-MAS SLF technique such as wide-line separation (WISE)63 and dipolar chemical shift separation (DIP-SHIFT)64,65 or Lee-Goldburg CP (LGCP) NMR,66 respectively. In the WISE experiment, the XH wide-line spectrum of the blend polymers consists of a rather featureless superposition of components with different dipolar widths which can be separated in the second frequency dimension and related to structural units according to their 13C chemical shifts.63... 

D-NMR methods are highly useful for structure elucidation. Jeener described the principles of the first 2D-NMR experiment in 1971 [31]. In standard NMR nomenclature, a data set is referred to by one, i.e., less than the total number of actual dimensions, since the intensity dimension is implied. The 2D-data matrix therefore can be described as a plot containing two frequency dimensions. The inherent third dimension is the intensity of the correlations within the data matrix. This is the case in ID NMR data as well. The implied second dimension actually reflects the intensity of the peaks of a certain resonance... 

When performing 2D-NMR experiments one must keep in mind that the second frequency dimension (Fx) is digitized by the number of tx increments. Therefore, it is important to consider the amount of spectral resolution that is needed to resolve the correlations of interest. In the first dimension (F2), the resolution is independent of time relative to F. The only requirement for F2 is that the necessary number of scans is obtained to allow appropriate signal averaging to obtain the desired S/N. These two parameters, the number of scans acquired per tx increment and the total number of tx increments, are what dictate the amount of time required to acquire the full 2D-data matrix. 2D-homo-nuclear spectroscopy can be summarized by three different interactions, namely scalar coupling, dipolar coupling and exchange processes. 

Generally, a normal NMR-spectrum has amplitude plotted Vs just one frequency-dimension (the ppm scale). In 2D-NMR, the amplitude is plotted Vs two frequency-dimensions (two ppm scales), normally in the form of a counter plot, just like a topographic map. 

There was no special chapter devoted to NMR in the last edition in 1982, but such is the utility of the technique that many references were made to NMR studies in other chapters. Our main concern here will be work published after that date. Although commercial FT pulse spectrometers had been available for ten or more years, the techniques of spectroscopy in two frequency dimensions, 2D-NMR, were in their infancy11, whereas now they are almost commonplace. 

R. R. Ernst, G. Bodenhausen and A. Wokaum, Principles of Nuclear Magnetic Resonance in One and Two Frequency Dimensions, Clatendon Press, Oxford, 1987. 

For 20 years center stage has been occupied by two-dimensional (and now three-and four-dimensional) NMR techniques. 2D NMR and its offshoots offer two distinct advantages (1) relief from overcrowding of resonance lines, as the spectral information is spread out in a plane or a cube rather than along a single frequency dimension, and (2) opportunity to correlate pairs of resonances. In the latter respect 2D NMR has features in common with various double resonance methods, but as we shall see, 2D NMR is far more efficient and versatile. Hundreds of different 2D NMR techniques have been proposed in the literature, but most of these experiments can be considered as variations on a rather small number of basic approaches. Once we develop familiarity with the basic principles, it will be relatively easy to understand most variations of the standard 2D experiments. 

Up to a 90° phase shift in both frequency dimensions (which can be easily corrected for), the two resulting spectra are almost identical. Because the noise of the two spectra is uncorrelated (Cavanagh and Ranee, 1990b), it is increased only by a factor of yfl if the two spectra are added, whereas the signal is increased by a factor of 2. Overall, a sensitivity improvement of /2 can be achieved in the PEP version of the TOCSY experiment, relative to experiments where one of the two transverse magnetization components is simply eliminated. 

In multidimensional NMR experiments that contain several evolution and mixing periods, even more combinations are possible (Griesinger et al., 1987b). In these experiments, Hartmann-Hahn mixing periods with in-phase coherence transfer are of particular advantage, because the resolution is often limited in the indirectly detected frequency dimensions. 

A simple way of illustrating multidimensional NMR is through reference to hetero-nuclear correlation spectroscopy, in which two or more separate frequency dimensions are correlated with one another. For example, a particularly valuable 2D experiment is heteronuclear single quantum correlation (HSQC) spectroscopy, in which the resultant spectrum has two frequency axes, corresponding to and frequency dimensions, and one intensity axis. Analogous HSQC... 

In the single-pulse experiments described up to this point, a 90° pulse is followed by a period during which the free-induction decay is acquired (Figure 6-1 a). Fourier transformation of the time-dependent magnetic information into a frequency dimension provides the familiar spectrum of 8 values, henceforth called a one-dimensional (ID) spectrum. 

Figure 6-21 illustrates this procedure for an adamantane derivative. The H frequencies are on the vertical axis and the C frequencies are on the horizontal axis. The respective spectra are illustrated on the left and at the top. The 2D spectrum is composed only of cross peaks, each one relating a carbon to its directly bonded proton(s). There are no diagonal peaks (and no mirror symmetry associated with a diagonal), because two different nuclides are represented on the frequency dimensions. Quaternary carbons are invisible to the technique, as the fixed times A and A2 normally are set to values for one-bond couplings. This experiment often is a necessary component in the complete assignment of H and resonances. Its name, HETeronuclear chemical Shift CORrelation, usually is abbreviated as HETCOR, but other acronyms (e.g., HSC, for Heteronuclear Shift Correlation, and H, C-COSY, also are used. The method may be applied to protons coupled to many other nuclei, such as Si, and P, as well as C. 

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