Spin-dipolar

An anisotropic spin-dipolar contribution B arising from nonspherical distribution of the electronic spin density. 

It is well-known that the hyperfine interaction for a given nucleus A consists of three contributions (a) the isotropic Fermi contact term, (b) the spin-dipolar interaction, and (c) the spin-orbit correction. One finds for the three parts of the magnetic hyperfine coupling (HFC), the following expressions [3, 9] ... 

Each of the eight hyperfine resonances is an unresolved quadmpole doublet, due to the quadmpole interaction of Os in the hexagonal Os metal source. The authors have interpreted the hyperfine fields in terms of core polarization, orbital and spin-dipolar contributions. 

No electron Spin-spin dipolar External magnetic... 

Figure 5.9. Effect of spin-spin dipolar interaction and an external magnetic field on triplet levels.

Spin labels contain unpaired electrons that by highly efficient electron-nuclear spin dipolar coupling lead to accelerated transverse or longitudinal relaxation. The effect is rather far-reaching (at least up to 10 A), and its general use is described in Chapt. 15. 

Classical shielding arguments indicate an electron-rich phosphorus atom, or equally, an increase in coordination number. The silicon atom seems also to be electron-rich, while the carbon has a chemical shift in the range expected for a multiply bonded species. The coupling constant data are difficult to rationalize, as it is not possible to predict the influence of orbital, spin-dipolar, Fermi contact, or higher-order quantum mechanical contributions to the magnitude of the coupling constants. However, classical interpretation of the NMR data indicates that the (phosphino)(silyl)carbenes have a P-C multiple bond character. 

Hyperfine tensors are given in parts B and C of Table II. Although only the total hyperfine interaction is determined directly from the procedure outlined above, we have found it useful to decompose the total into parts in the following approximate fashion a Fermi term is defined as the contribution from -orbitals (which is equivalent to the usual Fermi operator as c -> < ) a spin-dipolar contribution is estimated as in non-relativistic theory from the computed expectation value of 3(S r)(I r)/r and the remainder is ascribed to the "spin-orbit" contribution, i.e. to that arising from unquenched orbital angular momentum. 

On the basis of their ENDOR results and crystal field theory. Brown and Hoffman(18) have made empirical estimates of these three factors for the copper hyperfine interaction these are shown in Table II. The spin dipolar contributions may be estimated from o, the population of the unpaired electron in the Cu dxy orbital ... 

Table II compares these empirical estimates with those obtained from the DSW calculation. Relativistic contributions have little effect on the spin-dipolar interactions, and both calculations are in reasonably good agreement with the empirical estimates. The spin-orbit contributions are also in moderately good agreement with the empirical estimates, showing that electron currents about the -axis are considerably more important than those about axes in the plane of the ligand. Indeed, in view of the approximations that enter into Equation 7, (, ), one might have as much confidence in the DSW result as in the empirical estimate given in the final column.

Figure 1. Three stages of resolution in a C-I3 spectrum of a cured epoxy. The top spectrum is obtained under conditions appropriate to a liquid-state spectrometer no dipolar decoupling and no magic angle spinning. Dipolar decoupling at 60 kHz is used for the middle spectrum and to that is added magic angle rotation at 2.2 kHz for the bottom figure.

According to NMR theory [30], contributions to spin-spin coupling come from paramagnetic spin-orbital (PSO), diamagnetic spin-orbital (DSO), Fermi contact (FC), and spin-dipolar (SD) interactions [30] ... 

The magnitude of the ZFS parameter D is largely determined by the spin-spin dipolar interaction of the two electrons at the divalent carbon atom. Accordingly, the fraction of the n spin density located at the carbenic center can be estimated from the D value of the carbene. In spite of the predominance of this one-center interaction, the spin density at atoms several bonds removed from the divalent carbon atom can also have a signihcant effect on the ZFS parameters. 

This interaction leads to "fine-structure" splittings in the spectra of atoms and molecules. For atoms and molecules in the S = 1 triplet state, the electron spin-electron spin dipolar interaction leads to the "D and E" fine-structure Hamiltonian. 

Nuclear spin nuclear spin dipolar interaction. 

We have seen that the dependence of Bn(R) on the vibrational coordinate causes a mixing of the vibrational level of interest with neighbouring levels. This mixing results in centrifugal distortion corrections to all the various parameters Xn(R) in the perturbation Hamiltonian 3C when combined in a cross term. The operator has the same form as in the original term, for example, (2/3) /6T 0(S, S) for the spin spin dipolar term, multiplied by N2. The coefficient which qualifies this term has the general form... 

These represent the nuclear spin Zeeman interaction, the rotational Zeeman interaction, the nuclear spin-rotation interaction, the nuclear spin-nuclear spin dipolar interaction, and the diamagnetic interactions. Using irreducible tensor methods we examine the matrix elements of each of these five terms in turn, working first in the decoupled basis set rj J, Mj /, Mi), where rj specifies all other electronic and vibrational quantum numbers this is the basis which is most appropriate for high magnetic field studies. In due course we will also calculate the matrix elements and energy levels in a ry, J, I, F, Mf) coupled basis which is appropriate for low field investigations. Most of the experimental studies involved ortho-H2 in its lowest rotational level, J = 1. If the proton nuclear spins are denoted I and /2, each with value 1 /2, ortho-H2 has total nuclear spin / equal to 1. Para-H2 has a total nuclear spin / equal to 0. 

We now show that the nuclear spin dipolar interaction has matrix elements of exactly the same form. We take the dipolar Hamiltonian to have the form given previously in equation (8.10) and find that its matrix elements are given by... 

The first three terms are present for both para- and ortho-H2 they represent the electron spin-rotation, electron spin-spin dipolar and spin orbit interactions respectively. The fourth term in (8.187) represents the magnetic hyperfine interactions, which we will come to a little later. We deal first, however, with the terms that do not involve nuclear spin interactions. 

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