Scalar interactions

Fig. 6.6 Schematics of hydrogen bonds between purine and pyrimidine bases with indicated trans-hydrogen bond scalar interactions and related coupling constants, which can be measured using NMR. In addition to correlations between exchangeable protons and nitrogens, also a relayed transfer to nonexchangeable aromatic protons, shown by a dashed arrow, can be employed. The...

The Bloembergen-Morgan equations, Eqs. (14) and (15), predict that the electron spin relaxation rates should disperse at around msTy = 1. This will make the correlation times for the dipolar and scalar interaction, %ci and respectively, in Eq. (11) dependent on the magnetic field. A combination of the modified Solomon-Bloembergen equations (12) and (13), for nuclear relaxation rates with the Bloembergen-Morgan equations for the field dependence... 

The symbol Re(K ((o)) denotes the real part of the complex spectral density, corresponding to the autocorrelation of the dipolar interactions, while Re(i (co)) is its counterpart for the scalar interaction. The symbol Re(K (a>)) denotes the spectral density describing the cross-correlation of the two parts of the hyperfine interaction. The cross-correlation vanishes at the MSB level of the theory, but in the more complicated case of the lattice containing the electron spin, the cross term may be non-zero. A general expression for the dipolar spectral density is ... 

In most of the work using the slow-motion theory (except for some of the early work 77-79)), the interest was concentrated on the paramagnetic enhancement of the spin-lattice relaxation and the effects of the scalar interaction were neglected. The relevant special case of Eq. (33) then becomes ... 

An electron spin can relax by coupling with a neighboring electron spin. Therefore, when a paramagnetic metal ion interacts with a second paramagnetic metal ion, the electron relaxation rates of the two metal ions may be dramatically affected. If Si and S2 are the two spins coupled by a scalar interaction, new spin levels will be established due to the interaction, with total S varying in unitary steps from Si — S2I to Si + S2. The energies of these spin levels are given by )... 

The first line in this expression describes the rotational structure with color spin-doubling and the hyperflne interaction of the effective electron spin S with the nuclear spin I. B is the rotational constant, J is the electron-rotational angular momentum, A is the o -doubling constant. The second line describes the interaction of the molecule with the external fields B and E, (A is the unit vector directed from the heavy nucleus to the light one). The last line corresponds to the P-odd electromagnetic interaction of the electrons with the anapole moment of the nucleus described by the constant /ca [40], P,T-odd interaction of the electron EDM de with the interamolecular field, and P,T-odd scalar interactions of the electrons with the heavy nucleus [90]. 

The Tte of the 3Fe-4S centre in succinate ubiquinone reductase between 4 and 8 K is decreased by interaction with paramagnetic cytochrome b.98 To mitigate the impact of spectral diffusion the relaxation times were measured by a picket-fence sequence with 100 pulses. Analysis of the powder pattern distribution of relaxation times indicated that the anisotropic dipolar interaction dominated over isotropic scalar interaction and a lower limit of 10 A was estimated for the distance between the iron-sulfur cluster and the heme. 

Cross relaxation can occur in principle also between nuclei coupled by a time-dependent scalar interaction. In this case only wo can contribute to it (see Section 3.5). The correlation time for the reorientation is either chemical exchange or the relaxation rate of nucleus J if nucleus / is observed. The correlation time is generally too long to make the experiment successful. 

EXSY cross peaks are also obtained in TOCSY experiments (see later) because scalar interactions in the rotating frame are not separable from exchange interactions [7]. An EXSY experiment, performed using a TOCSY sequence (see Section 8.6) is reported in Fig. 8.7 relative to the complex 5Cl-Ni-SAL-MeDPT [5]. This complex, as shown in Fig. 8.8, displays a chemical equilibrium in which the two salicylaldiminate moieties exchange their non-equivalent positions [8]. It is interesting to learn that such complex interconversion occurs with times of the order of the spin-lock time (20 ms) or shorter. 

The COSY experiment is the most familiar to 2D NMR spectroscopists. The cross peaks connect protons which are coupled by scalar interactions. Under these... 

D spectra are in principle possible for heteronuclei coupled by either dipolar or scalar interactions. However, the magnetic moments of heteronuclei are sizably smaller than that of the proton, and since cross relaxation depends on the square of the magnetic moment it appears that this is a serious limitation for the observation of NOESY or ROESY cross peaks. However, as already discussed, in scalar-coupled systems the relevant coherences build up with sin(nJ/jt). Since Jjj in directly bound 13C- H and l5N- H moieties is of the order of 102 Hz, as opposed to about 10 Hz between proton pairs, it is conceivable that scalar correlation experiments are successful. Heterocorrelated spectra have the advantage of allowing one to detect signals of protons attached to carbons or nitrogens when they are within a crowded envelope. 

The scalar interaction is proportional to the square of the unpaired electron s wavefunction at the nucleus, i//(0) 2. In general, this quantity is not known or cannot be determined, making the scalar interaction difficult to predict except in certain and simple situations.4 The dipolar term is heavily dependent on the distance r between the two spins, leading to the distance (and time) dependence of the Overhauser effect. Further discussions of the dipolar interaction term are available in the literature.23... 

Figure 2 The four-level diagram for a system of two interacting spins, in this case an electron (S) and nucleus with a positive gyromagnetic ratio (/). The intrinsic electron and nuclear spin relaxation are given by p and w°, respectively, and the dipolar and/or scalar interactions between the electron and nuclear spin are represented by p, w0, w, and w2. The transition w0 is known as the zero-quantum transition, while w, is the singlequantum transition and w2 is the double-quantum transition. Nuclear and electronic relaxation through mechanisms other than scalar or dipolar coupling are denoted with w° — 1/Tio and p — 1/Tie, where Ti0 and T1e are the longitudinal relaxation times of the nucleus and electron in the absence of any coupling between them. Since much stronger relaxation mechanisms are available to the electron spin, the assumption p>p can be safely made. Adapted with permission from Ref. [24].

We will now extend the present model to include gravitational interactions by augmenting the model presented in the previous section with a general scalar interaction as follows in the basis m, rn) ... 

This term represents the scalar interaction between the spin of electron i and the magnetic field created by the spin and orbital motions of the other electrons. Substituting the first term of (3.133) for A [ yields... 

The major interaction between die nuclear spin magnetic moments in H2 and D2 is the dipolar interaction, equation (8.10). We should at least mention the existence of an electron-coupled scalar interaction this is very small compared with the dipolar interaction, and plays a very minor role in the gas phase measurements. In liquids, however, the dipolar interaction averages to zero, and the scalar coupling becomes the important observable interaction between nuclear spins. The power and range of applications of high-resolution n.m.r. in liquids depends ultimately upon the scalar shielding and spin-spin interactions. 

We come now to the second study, described ten years later [23]. The main development was the employment of a tunable dye laser to pump the A <— X + transition. Rotational levels in the ground state with J = 1 to 29, in the v = 0 vibrational level, were pumped by the laser and radioffequency hyperfine transitions studied. The range of J levels studied meant that the effective Hamiltonian required the addition of terms describing the dipolar and scalar interactions between the 23Na nuclear spins. These terms were given earlier in our discussion of the D2 molecule, and the complete effective Hamiltonian is ... 

The matrix elements of the scalar interaction between the nuclear spins in a homonuclear molecule were not given in our earlier discussion, but they are obtained very simply by noting that... 

The electron spin rotation interaction is similar to the nuclear spin rotation interaction we met earlier, and may be written as a simple scalar interaction,... 

As we shall see, each of these two terms, one for each nucleus, describes a second-rank scalar interaction between the electric field gradient at each nucleus and the nuclear quadrupole moment. De Santis, Lurio, Miller and Freund [44] included two other terms which involve the nuclear spins. One is the direct dipolar coupling of the 14N nuclear magnetic moments, an interaction which we discussed earlier in connection with the magnetic resonance spectrum of D2 its matrix elements were given in equation (8.33). The other is the nuclear spin-rotation interaction, also discussed in connection with H2 and its deuterium isotopes. It is represented by the term... 

The predominant isotope of cesium is 133Cs which has a nuclear spin I of 7/2 its quadrupole moment and g-factor will be denoted by Q and gi. The 19F nucleus has spin /2 of 1 /2 (and therefore no quadrupole moment) and a nuclear g-factor denoted g2. The nuclear hyperfine Hamiltonian used by English and Zorn [51] was the sum of five terms representing the 133Cs quadrupole interaction, the 133Cs nuclear spin-rotation interaction, the 19F nuclear spin-rotation interaction, the dipolar (tensorial) interaction between the 133Cs and 19F nuclear spins, and the scalar interaction between the two nuclear spins. Consistent with the conventions in use at the time, this Hamiltonian was written in the following form ... 

The final term in equation (8.282) represents the scalar interaction between the two nuclear spins its matrix elements are as follows ... 

In this section, we will only discuss a specific system the enhancement of proton spin relaxation in aqueous solutions of paramagnetic ions, which to our knowledge is the only case of paramagnetic relaxation, which has been studied with MD simulations. Just as in the description of dipole-dipole relaxation, we will only treat the through space dipole-dipole interaction, and not the scalar interaction. This is experimentally well motivated in the case of Ni ions [23]. 

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