Figure 1. (a) Schematic drawing of a three‐dimensional NMR experiment correlating the nuclei X, Y, and H. The basic pulse scheme (b) is repeated many times for sampling the multidimensional time‐domain grid (c). A recycling delay between successive repetitions (scans) is required for spin polarization recovery (longitudinal spin relaxation). The dependence of the experimental sensitivity on the recovery time, calculated from Equation 5, is shown in (d) for different flip angles and relaxation times: T ₁ = 1 s, β = 90° (straight line), T ₁ = 300 ms, β = 90° (dashed line), and T ₁ = 300 ms, β = 45° (dotted line). Alternative sparse data sampling schemes (random, radial, and linear undersampling) for two indirect dimensions (three‐dimensional experiment) are shown in insert (e).

Figure 2. Principles of single‐scan two‐dimensional NMR. (a) Spatiotemporal encoding associated to the indirect domain (left side) imparts helical windings depending on the particular shift of each site on the spins (Equation 8a). This spatial encoding is preserved during a homogeneous mixing process as either a phase or an amplitude modulation (center) and can be read out by an acquisition gradient G _a that provides, via the positions of the corresponding observable echoes in k =  ∫ G_a(t)dt, the associated distribution of signal intensity I(Ω) (in this figure, “red” and “green” sites are assumed to yield corresponding echoes). This procedure can be repeated multiple (N ₂) times, to yield a full S(F ₁, t ₂) interferogram within a single scan. (b) Combined gradient/RF manipulations leading to a discrete phase‐modulated (left) or a continuous amplitude‐modulated (right) version of the spatial encoding pattern. In both cases the amplitudes of the RF pulses need to be set so as to impart net π/2 or π rotations of the spins at all positions. (c) Processing the resulting single‐scan FID(t) interferogram into a two‐dimensional I(F ₁, F ₂) NMR spectrum involves rearranging the digitized points into their correct positions in the (F ₁, t ₂) space and one‐dimensional Fourier processing along t ₂. Notice that a square‐wave ±G _a modulation distributes the points along a zigzagging trajectory in the (F ₁, t ₂) space; if a FFT is employed, it is convenient to separate the data points arising from positive and negative echoes, Fourier transform these two sets individually, and then coadd their resulting two‐dimensional spectra for signal‐to‐noise improvement (right panel).

Figure 3. (a) Principle of real‐time two‐dimensional NMR: a series of two‐dimensional correlation spectra are recorded during a molecular kinetics event such as the folding of a protein from a highly disordered to a globular folded state. The series of two‐dimensional NMR spectra yield a movie of the protein's conformational changes with a time resolution given by the duration of a single two‐dimensional experiment. (b) The recorded two‐dimensional datasets can be coadded in pairs for artifact suppression using phase cycling between subsequent experiments and for increasing the apparent signal‐to‐noise ratio. (c) Sketch of a fast injection device that can be used to induce a kinetic reaction by mixing two different liquids (e.g., a protein dissolved in a denaturing environment with a refolding buffer).