Three-Spin Systems

If there are more than two nuclei exerting relaxation effects on one another, then it is convenient to consider them in pairs and to arrive at the overall effect by adding together the effects of various possible pairs. In the case of a three-spin system, we can consider two different situations (I) the nuclei //a, //b. and Hi. are arranged in a straight line, and (2) they are in a nonlinear arrangement. 

What is transient nOe, and why is it considered to provide a better estimate of the internuclear distance (r) than the normal nOe effect  

Let us first consider the nuclei //a, and He as lying in a straight line and equidistant from one another (Fig. 4.5). The central proton, can relax by interactions with two neighbors, //a and He, while //a and He can relax by interaction with only one neighbor, so H can relax twice as quickly as or He. If we assume that relaxation can occur only through dipole-dipole relaxation, and that an equilibrium steady state has been 

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Under these conditions, and neglecting external relaxation-interactions according to Eq. 4, the pairwise relaxation-contributions, py, are evaluated from Eqs. 10-12, provided that an equal number of experimentally determined values are available. For a weakly coupled, three-spin system,... 

It also is possible to determine py values from a single, nonselective experiment for more than three spin systems, by combining nonselective relaxation-rates with n.O.e. experiments. According to Noggle and Shir-mer, Eq. 5 can be solved under conditions in which, at time f = 0, spin i is observed with simultaneous saturation of spin j, resulting in the expression... 

As for nonselective-relaxation experiments, Eq. 15 can only be used for a three-spin system, because, for j>3, the problem is underdetermined. 

For a rigidly held, three-spin system, or when existing internal motion is very slow compared to the overall molecular tumbling, all relaxation methods appear to be adequate for structure determination, provided that the following assumptions are valid (a) relaxation occurs mainly through intramolecular, dipolar interactions between protons (b) the motion is isotropic and (c) differences in the relaxation rates between lines of a multiplet are negligibly small, that is, spins are weakly coupled. This simple case is demonstrated in Table V, which gives the calculated interproton distances for the bicycloheptanol derivative (52) of which H-1, -2, and -3 represent a typical example of a weakly coupled, isolated three-spin... 

Following is a linear three-spin system and the observed steady-state nOe s between three nuclei. The distance between nuclei A and B is... 

What happens in three-spin systems when the A-B-C angles 0 are 180° and 78°, respectively, = rgc = 1, and spin A is irradiated ... 

Three-spin effects arise when the nonequilibrium population of an enhanced spin itself acts to disturb the equilibrium of other spins nearby. For example, in a three-spin system, saturation of spin A alters the population of spin B from its equilibrium value by cross-relaxation with A. This change in turn disturbs the whole balance of relaxation at B, including its cross-relaxation with C, so that its population disturbance is ultimately transmitted also to C. This is the basic mechanism of indirect nOe, or the three-spin effect. 

The magnitude of the indirect nOe effect depends on the geometry of the three-spin system. It is maximum with a linear geometry of the spin system (0 = 180°). When 0 decreases, the distance also decreases, and the direct enhancement at C becomes more and more significant. At 0 = 78°, the direct and indirect nOe effects become equal. With smaller values of 0, the direct contribution rapidly starts to dominate the nOe at C, and a strong positive enhancement results. This means that indirect nOe effects are to be expected mainly when the spins are close to having a linear geometry. 

Figure 6.4 Three-dimensional spectrum of a three-spin system showing peak types appearing in a three-dimensional space. Three diagonal peaks, six (wi = Wj) and six (wj = w,) cross-signal peaks, six back-transfer peaks, and six cross-peaks are present in the cube, (a) The cubes (b-d) represent three planes in which crossdiagonal peaks and the back-transfer peaks appear on their respective (atj = 0)2), u>2 = cof), and ( >i = Wj) planes. (Reprinted from J. Mag. Reson. 84, C. Griesinger, et al., 14, copyright (1989), with permission from Academic Press, Inc.)...

Figure 7.17 Pulse sequence for soft H,H-COSY with two selective pulses exciting the / multiplet and the second mixing pulse exciting the li multiplet of a three-spin system (/, /,)). (Reprinted from Mag. Reson. Chem. 29, H. Kessler et al., 527,...

Three Spin Systems in [TbPc2]° Single-Ion Molecular Magnets (SIMMs)... 

Figure 8.9 TbPc2-based single ion magnets (SIMs) [90] (a) Schematic representation molecular localization of the three spin-systems characteristics of the [TbPc2]° complex J = 6, the uniaxial magnetic moment of the 4f configuration / = 3/2 nuclear spin of the only stable and naturally occurring 159Tb isotope S = 1/2 organic radical delocalized over the two Pc ligands. The radical...

Fig. 4. Numerical simulations showing the effect of 20% miscalibration of the rf amplitudes of two selective excitation pulses (Top) A 270° Gaussian of 30 ms duration (peak amplitudes from left to right 43 Hz, 54 Hz, and 65 Hz) and (Bottom) a Quaternion cascade of 30 ms duration (peak amplitudes from left to right 121 Hz, 151 Hz, and 181 Hz). The multiplets were obtained by simulating a three-spin system with couplings Jam = 7 Hz, Jax = 12 Hz, and Jmx = 0 Hz. The vertical axes show a constant arbitrary amplitude, whereas the horizontal axis gives the frequencies 1 a 50 Hz.

Fig. 7. Normal COSY spectrum (expansion of the ring-protons) of the trisaccharide 1 dissolved in CDCI3 (d). Individual COSY subspectra A, B and C ((a), (b) and (c)) of the three monomer units, obtained with the optimized 2D TOCSY-COSY experiment V. The three anomeric protons were used to selectively perturb the three spin systems. They are assigned and their connectivities to protons 2A, 2B and 2C, respectively, are indicated. (Continued...

Figure 6 Simulated 0-REDOR evolution curves for a fictive S-l two-spin system (A) and linear S-lj three-spin system (B) for the indicated pulse lengths (vr=10,000Hz, S= C ...

In recent years Kawamori and coworkers pursued their work on distance measurements in PS n383 509-515. An overview is given in383 detailing distances determined by pulse EPR. Approaches to analyze three-spin systems of PS II by PELDOR have recently been described.514,515 For the distances between Y5 and the Mn4 cluster in the S0 and in the S2 state a pronounced difference was observed and interpreted as a movement of the spin centre in the Mn4-cluster.513 For the S-2 and the S0 state almost the same distances from the Yp radical were found. 

Three-spin systems, (a) AB2 and AX2. The AX2 system results in a doublet and... 

Finally, the application of optimal control theory to DNP studies needs to be discussed. Optimal control theory is a means to systematically design and optimize pulse sequences to maximize the efficiency of transfer between spin states. While this method was initially introduced to benefit high-resolution NMR studies, it has recently been adapted to improve the electron-nuclear polarization transfer in DNP applications by considering simple two- or three-spin systems. " While no experimental implementation of DNP pulse sequences designed by optimal control methods has been reported, these methods have the great potential to enhance DNP performance at X-band, due to the powerful pulsed ESR hardware that is commercially available at these frequencies. 

It will help at this point to realize that the molecule consists of three spin systems with the insulation point at C-6 (see Section 3.5.2). The alkyl system consists of H-l, H-2, H-3, H-4, H-5, and H-9 an alkene system consists of H-7 and H-8 and another alkene system consists of H-10. The alkyl system accounts for the multiplets at the right of the spectrum, and the alkenes account for the multiplets at the left side. It will also help to reiterate that the protons of an alkyl CH2 group will be diastereotopic in the presence of a chiral center. As will be the protons of an alkene=CH2 group. 

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