Evolution dimension

Fig. 1 The Bracewell slice rojection theorem. The Fourier transform of a slice through the evolution dimension at an inclination a (left) is the projection of the corresponding frequency-domain spectrum at the same angle a (right)...

Because the number of time-domain slices (and hence the number of recorded projections) is relatively small, the density of sampling points is far lower than the density used in the conventional experiment, which must examine every point on the complete Cartesian grid while satisfying the Nyquist condition and the requirement for adequate resolution. This is where the critical time saving occurs. With this limited radial sampling [13], the speed advantage increases by an order of magnitude for each new evolution dimension beyond the first. This opens up the... 

When there is ambiguity in the three-dimensional spectrum, or where global isotopic enrichment in C and has been employed, a further evolution dimension may be introduced [18]. The problem can stUl be visuahzed as a cube in three-dimensional evolution space, neglecting any representation of the real-time direct acquisition dimension 14. The three evolution parameters are defined by... 

Most methods for determining residual dipolar couplings are based on the measurement of the displacement between cross-peak components in J-coupled spectra. However, for large macromolecules and macromolecular complexes, these methods are often unreliable since differential relaxation can significantly broaden one of the multiplet components and thereby make accurate determination of its position difficult. To overcome this problem, a J-evolved transverse relaxation optimized (JE-TROSY) method has been demonstrated for the determination of one-bond couplings that involves J-evolution of the sharpest crosspeak multiplet component selected in a TROSY experiment . Couplings are measured from the displacement of the TROSY component in the additional J-evolution dimension relative to a zero frequency origin. 

ROESY spectrum can be achieved with a BASHD-ROESY pulse sequence, incorporating band selection and homonuclear decoupling in the H" region of the spectra. Band selection in the evolution dimension is performed with the DPFGSE technique as described above. 

Hopfinger et al. [53, 54] have constructed 3D-QSAR models with the 4D-QSAR analysis formahsm. This formalism allows both conformational flexibility and freedom of alignment by ensemble averaging, i.e., the fourth dimension is the dimension of ensemble sampling. The 4D-QSAR analysis can be seen as the evolution of Molecular Shape Analysis [55, 56]. 

As mentioned above, CMLs are simple generalizations of generic CA systems. Confining ourselves for the time being to one-dimension for simplicity, we begin with a one-dimensional lattice of real-valued variables ai t) R whose temporal evolution is given by... 

Dimethyl sulphoxide has also been oxidized electrochemically, using either a platinum anode or a dimensionally stable anode containing iridium and selenium in 1 M sulphuric acid solution158. The former electrode requires a potential close to that required for oxygen evolution whilst the latter needed a potential 0.5 volts lower. Thus the dimension-... 

The evolution period tl is systematically incremented in a 2D-experiment and the signals are recorded in the form of a time domain data matrix S(tl,t2). Typically, this matrix in our experiments has the dimensions of 512 points in tl and 1024 in t2. The frequency domain spectrum F(o l, o 2) is derived from this data by successive Fourier transformation with respect to t2 and tl. 

In the one-dimensional NMR experiments discussed earlier, the FID was recorded immediately after the pulse, and the only time domain involved (ij) was the one in which the FID was obtained. If, however, the signal is not recorded immediately after the pulse but a certain time interval (time interval (the evolution period) the nuclei can be made to interact with each other in various ways, depending on the pulse sequences applied. Introduction of this second dimension in NMR spectroscopy, triggered byjeener s original experiment, has resulted in tremendous advances in NMR spectroscopy and in the development of a multitude of powerful NMR techniques for structure elucidation of complex organic molecules. 

Jeener s idea was to introduce an incremented time ti into the basic ID NMR pulse sequence and to record a series of experiments at different values of second dimension to NMR spectroscopy. Jeener described a novel experiment in which a coupled spin system is excited by a sequence of two pulses separated by a variable time interval <]. During these variable intervals, the spin system is allowed to evolve to different extents. This variable time is therefore termed the evolution time. The insertion of a variable time period between two pulses represents the prime feature distinguishing 2D NMR experiments from ID NMR experiments. 

Figure 5.5 shows the heteronuclear 2Dy-resolved spectrum of camphor. The broad-band decoupled C-NMR spectrum is plotted alongside it. This allows the multiplicity of each carbon to be read without difficulty, the F dimension containing only the coupling information and the dimension only the chemical shift information. If, however, proton broad-band decoupling is applied in the evolution period tx, then the 2D spectrum obtained again contains only the coupling information in the F domain, but the F domain now contains both the chemical shift and the coupling information (Fig. 5.6). Projection of the peaks onto the Fx axis therefore gives the Id-decoupled C spectrum projection onto the F axis produces the fully proton-coupled C spectrum.

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