Spin-dipolar interaction

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] ... 

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

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.

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. 

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 dipolar hyperfine interaction is a through-space interaction of the electron and nuclear spin magnetic moments. As such, it is similar to the nuclear spin-nuclear spin dipolar interaction discussed earlier in connection with the H2 molecule in its ground electronic state. We shall meet the dipolar hyperfine interaction in many examples described later, so at the risk of seeming somewhat pedantic and repetitive, we here... 

The second term in (8.244) describes the centrifugal correction to the spin-spin interaction, where the subscript + denotes the anticommutator we shall, however, omit this term from further consideration since its effects were very small. It is more convenient to take the spin spin dipolar interaction term to have the form... 

The magnetic hyperfine interaction is represented by the sum of two terms, representing the Fermi contact interaction, and 3Qiip representing the electron spin-nuclear spin dipolar interaction. They are written as follows ... 

The simplest molecular constant to understand is the nuclear spin dipolar interaction constant, to, which is found to be, within experimental error, that calculated from the classical interaction of two magnetic moments, i.e. gFgHfrwO /47t oc2)(7 3> =o. On the other hand, calculation of scalar electron-coupled spin-spin interaction constants is notoriously difficult, requiring a molecular electronic wave function of the highest quality. The best available calculation for HF quoted by Muenter and Klemperer is one due to O Reilly [96]. 

We show here the equivalence of the two forms of the nuclear spin dipolar interaction, equations (8.9) and (8.10). The most familiar representation of this interaction is equation (8.9),... 

Appendix 8.3. Electron spin-electron spin dipolar interaction... 

The third term in equation (8.510) describes only the electron-nuclear spin dipolar interaction, with the first-rank tensor T1 (.S. C2) being constructed so that... 

We shall however, represent the electron spin dipolar interaction by (9.101) and then relate our results to (9.99). Consequently the effective Hamiltonian can now be written... 

Finally the electron spin-nuclear spin dipolar interaction, which is more complicated, was given, initially, by equations (8.232) and (8.233) ... 

Note that Jefferts used d for y and / for c/. The first two terms contain contributions from both the Fermi contact interaction and the axial component of the electron spin-nuclear spin dipolar interaction, z being along the direction of the internuclear axis. 

This might appear to be a satisfactory conclusion, so far as the analysis of the observed spectrum is concerned. However, Carrington and Gammie [111] have reexamined the analysis and concluded that there does not appear to be any obvious reason why the nuclear spin-nuclear spin dipolar interaction should be neglected, since it is likely to be similar in magnitude to the nuclear spin-rotation interaction. This interaction was discussed for 112 in chapter 8, where it was represented, in spherical tensor form, by the term... 

Triplet signals were first noted by Kosower and Waits in concentrated solutions of stable pyridinyl radicals (4 ) and confirmed for (CHs) and 4 in MTHF glasses at 77 K K Three pairs of shoulders around the g = 2 signal at 3295G for the monomeric pyridinyl and a AM = 2 transition at 1645G suggested the presence of triplet pair. The zero-field parameters were D == 0.0098 cm and E = 0.0011 cm , for which a radical-radical separation of 6.5 A was estimated from the relationship for spin-spin dipolar interaction (Eq. 12). 

The terms in equation (4) are generally referred to as the orbital-dipolar interaction (o) between the orbital magnetic fields of the electrons and the nuclear spin dipole, the spin-dipolar interaction (D) between the spin magnetic moments of the electrons and nucleus and the Fermi contact interaction (c) between the electron and nuclear spins, respectively. Discussion of the mathematical forms of each of these three terms appears elsewhere. (3-9)... 

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