Hamiltonian dipolar-coupling

In this section we consider the spin Hamiltonian appropriate to a biradical with weak dipolar coupling and see how ESR spectra of such species should appear. Obviously, it is possible to find triradicals, tetraradicals, etc. treatment of such species is similar, though of course somewhat more complicated. 

Within this framework, the dipolar coupling between two spins I and S may be described in terms of the Hamiltonian... 

The basic RFDR element consists of two 7i-pulses each centered in the middle of each rotor period in two consecutive rotor periods leading to a cyclic rf field. In the toggling frame of ideal (i.e., infinitely strong) 7i-pulses, the Hamiltonian of the chemical shift terms and the homonuclear dipolar coupling interaction is split up into three commuting parts ... 

As demonstrated by Griffin, Levitt, and coworkers in the late 1980s [21, 93], it is also possible to recouple homonuclear dipolar couplings through interference between isotropic chemical shifts and the rotor revolution. This phenomenon, called rotational resonance, occurs when the spinning frequency is adjusted to a submultiple of the isotropic chemical shift difference, i.e., ncor = ct> so o) °. To understand this experiment, the dipolar coupling Hamiltonian in (10) is transformed... 

The so-called HORROR experiment by Nielsen and coworkers [26] introduced continuous rf irradiation recoupling to homonuclear spin-pairs and initiated the later very widely used concept of /-encoded recoupling. Using a irreducible spherical approach as described above, the HORROR experiment (Fig. 2d) is readily described as starting out with the dipolar coupling Hamiltonian in (10) and x-phase rf irradiation in the form Hrf = ncor(Ix +SX), also here without initial constraint on n. The dipolar coupling Hamiltonian transforms into tilted frame (rotation n/2 around Iy + Sy)... 

The PAR experiment may be described by considering two heteronuclear dipolar coupling interactions with one proton (7) in common, i.e., a Hamiltonian of the type... 

In the simplest setup, the two strong field components may be set identical to Ci = Cs = C. The relatively large CIX or CSX term averages isotropic and anisotropic chemical shift effects as well as the heteronuclear dipolar coupling interaction between 15N or 13C and H. The difference of - or the sum of - the B coefficients selects the form of the recoupled heteronuclear dipole-dipole coupling interaction, as expressed in terms of the effective Hamiltonian in the interaction frame of the rf irradiation... 

The dipolar-coupling Hamiltonian (TTy) describes the through-space coupling between two nuclear spins l and Ij. The dipolar coupling has an rk- dependence, and is key to the determination of internuclear distances in both solid-state and solution-state NMR. The high-field truncated form of the dipolar Hamiltonian is given by... 

Rotational-echo double resonance (REDOR), originally introduced by Gullion and Schaefer [102], is a method to recouple heteronuclear spin pairs. The sequence relies on a train of rotor-synchronized n pulses applied to the I spin to interrupt the spatial averaging of the heteronuclear dipolar coupling under MAS to give a nonvanishing dipolar Hamiltonian over a full rotor cycle (Fig. 11.8). Typically, REDOR data are collected by col-... 

While keeping in mind the general picture of nuclear relaxation in paramagnetic systems as described in Section 3.1, it is appropriate to consider first the simple case of dipolar coupling between two point-dipoles as if the unpaired electrons were localized on the metal ion. The enhancement of the nuclear longitudinal relaxation rate Rim due to dipolar coupling with unpaired electrons can be calculated starting from the Hamiltonian for the system ... 

The Hamiltonian for the dipolar coupling between two magnetic moments associated with the spins I and J is given by an equation similar to Eq. (1.1)... 

Third, we deal with the dipolar coupling of the nuclear spins, which is evaluated below note the form of the dipolar Hamiltonian (equation (8.282)), which is the one appropriate for the particular angular momentum coupling scheme used. 

We first make use of the results derived for the dipolar coupling of two proton spins in appendix 8.1. If we replace I by I, I2 by S and note the overall change in sign, we see that the conventional dipolar Hamiltonian,... 

The simplest technique that enables phosphorus-phosphorus correlations for complex multi-component system and elucidation of structural constraints is the PDSD sequence.48 This experiment is relatively easy to set up however, the distances between spins cannot be established with satisfactory precision. One common way of determining distance type information using spin-7i NMR is based on the acquisition of the DQ filtered spectra as a function of the DQ mixing time.49 The recently introduced DQ homonuclear recoupling sequences based on C and/or R symmetry developed by Levitt and co-workers allows very precise analysis of intemuclear distances.50 The accuracy of distances obtained from these DQ experiments depends on the effective suppression of contributions to the spin Hamiltonian other than the required homonuclear dipolar coupling terms. 

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